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Another view on Gilbert damping in two-dimensional ferromagnets Anastasiia A. Pervishko1, Mikhail I. Baglai1,2, Olle Eriksson2,3, and Dmitry Yudin1 1ITMO University, Saint Petersburg 197101, Russia 2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75 121 Uppsala, Sweden 3School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden ABSTRACT A keen interest towards technological implications of spin-orbit driven magnetization dynamics requests a proper theoretical description, especially in the context of a microscopic framework, to be developed. Indeed, magnetization dynamics is so far approached within Landau-Lifshitz-Gilbert equation which characterizes torques on magnetization on purely phenomenological grounds. Particularly, spin-orbit coupling does not respect spin conservation, leading thus to angular momentum transfer to lattice and damping as a result. This mechanism is accounted by the Gilbert damping torque which describes relaxation of the magnetization to equilibrium. In this study we work out a microscopic Kubo-St ˇreda formula for the components of the Gilbert damping tensor and apply the elaborated formalism to a two-dimensional Rashba ferromagnet in the weak disorder limit. We show that an exact analytical expression corresponding to the Gilbert damping parameter manifests linear dependence on the scattering rate and retains the constant value up to room temperature when no vibrational degrees of freedom are present in the system. We argue that the methodology developed in this paper can be safely applied to bilayers made of non- and ferromagnetic metals, e.g., CoPt. Introduction In spite of being a mature field of research, studying magnetism and spin-dependent phenomena in solids still remains one of the most exciting area in modern condensed matter physics. In fact, enormous progress in technological development over the last few decades is mainly held by the achievements in spintronics and related fields1–11. However the theoretical description of magnetization dynamics is at best accomplished on the level of Landau-Lifshitz-Gilbert (LLG) equation that characterizes torques on the magnetization. In essence, this equation describes the precession of the magnetization, mmm(rrr;t), about the effective magnetic field, HHHeff(rrr;t), created by the localized moments in magnetic materials, and its relaxation to equilibrium. The latter, known as the Gilbert damping torque12, was originally captured in the form ammm¶tmmm, where the parameter adetermines the relaxation strength, and it was recently shown to originate from a systematic non-relativistic expansion of the Dirac equation13. Thus, a proper microscopic determination of the damping parameter a(or, the damping tensor in a broad sense) is pivotal to correctly simulate dynamics of magnetic structures for the use in magnetic storage devices14. From an experimental viewpoint, the Gilbert damping parameter can be extracted from ferromagnetic resonance linewidth measurements15–17or established via time-resolved magneto-optical Kerr effect18, 19. In addition, it was clearly demonstrated that in bilayer systems made of a nonmagnetic metal (NM) and a ferromagnet material (FM) the Gilbert damping is drastically enhanced as compared to bulk FMs20–24. A strong magnetocrystalline anisotropy, present in CoNi, CoPd, or CoPt, hints unambiguously for spin-orbit origin of the intrinsic damping. A first theoretical attempt to explain the Gilbert damping enhancement was made in terms of sdexchange model in Ref.25. Within this simple model, magnetic moments associated with FM layer transfer angular momentum via interface and finally dissipate. Linear response theory has been further developed within free electrons model26, 27, while the approach based on scattering matrix analysis has been presented in Refs.28, 29. In the latter scenario spin pumping from FM to NM results in either backscattering of magnetic moments to the FM layer or their further relaxation in the NM. Furthermore, the alternative method to the evaluation of the damping torque, especially in regard of first-principles calculations, employs torque-correlation technique within the breathing Fermi surface model30. While a direct estimation of spin-relaxation torque from microscopic theory31, or from spin-wave spectrum, obtained on the basis of transverse magnetic field susceptibility32, 33, are also possible. It is worth mentioning that the results of first-principles calculations within torque-correlation model34–38and linear response formalism39, 40reveal good agreement with experimental data for itinerant FMs such as Fe, Co, or Ni and binary alloys. Last but not least, an intensified interest towards microscopic foundations of the Gilbert parameter ais mainly attributed to the role the damping torque is known to play in magnetization reversal41. In particular, according to the breathing Fermi surface model the damping stems from variations of single-particle energies and consequently a change of the Fermi surfacearXiv:1807.07897v2 [cond-mat.mes-hall] 21 Nov 2018z yx FM NMFigure 1. Schematic representation of the model system: the electrons at the interface of a bilayer, composed of a ferromagnetic (FM) and a nonmagnetic metal (NM) material, are well described by the Hamiltonian (1). We assume the magnetization of FM layer depicted by the red arrow is aligned along the zaxis. shape depending on spin orientation. Without granting any deep insight into the microscopic picture, this model suggests that the damping rate depends linearly on the electron-hole pairs lifetime which are created near the Fermi surface by magnetization precession. In this paper we propose an alternative derivation of the Gilbert damping tensor within a mean-field approach according to which we consider itinerant subsystem in the presence of nonequilibrium classical field mmm(rrr;t). Subject to the function mmm(rrr;t)is sufficiently smooth and slow on the scales determined by conduction electrons mean free path and scattering rate, the induced nonlocal spin polarization can be approached within a linear response, thus providing the damping parameter due to the itinerant subsystem. In the following, we provide the derivation of a Kubo-St ˇreda formula for the components of the Gilbert damping tensor and illustrate our approach for a two-dimensional Rashba ferromagnet, that can be modeled by the interface between NM and FM layers. We argue that our theory can be further applied to identify properly the tensorial structure of the Gilbert damping for more complicated model systems and real materials. Microscopic framework Consider a heterostructure made of NM with strong spin-orbit interaction covered by FM layer as shown in Fig. 1, e.g., CoPt. In general FMs belong to the class of strongly correlated systems with partially filled dorforbitals which are responsible for the formation of localized magnetic moments. The latter can be described in terms of a vector field mmm(rrr;t)referred to as magnetization, that in comparison to electronic time and length scales slowly varies and interacts with an itinerant subsystem. At the interface (see Fig. 1) the conduction electrons of NM interact with the localized magnetic moments of FM via a certain type of exchange coupling, sdexchange interaction, so that the Hamiltonian can be written as h=p2 2m+a(sssppp)z+sss
MMM(rrr;t)+U(rrr); (1) where first two terms correspond to the Hamiltonian of conduction electrons, on condition that the two-dimensional momentum ppp= (px;py) =p(cosj;sinj)specifies electronic states, mis the free electron mass, astands for spin-orbit coupling strength, while sss= (sx;sy;sz)is the vector of Pauli matrices. The third term in (1) is responsible for sdexchange interaction with the exchange field MMM(rrr;t) =Dmmm(rrr;t)aligned in the direction of magnetization and Ddenoting sdexchange coupling strength. We have also included the Gaussian disorder, the last term in Eq. (1), which represents a series of point-like defects, or scatterers, hU(rrr)U(rrr0)i= (mt)1d(rrrrrr0)with the scattering rate t(we set ¯h=1throughout the calculations and recover it for the final results). Subject to the norm of the vector jmmm(rrr;t)j=1remains fixed, the magnetization, in broad terms, evolves according to (see, e.g., Ref.42), ¶tmmm=fffmmm=gHHHeffmmm+csssmmm; (2) where fffcorresponds to so-called spin torques. The first term in fffdescribes precession around the effective magnetic field HHHeffcreated by the localized moments of FM, whereas the second term in (2) is determined by nonequilibrium spin density of conduction electrons of NM at the interface, sss(rrr;t). It is worth mentioning that in Eq. (2) the parameter gis the gyromagnetic ratio, while c= (gmB=¯h)2m0=dis related to the electron gfactor ( g=2), the thickness of a nonmagnetic layer d, with mBand m0standing for Bohr magneton and vacuum permeability respectively. Knowing the lesser Green’s function, G<(rrrt;rrrt), one can easily evaluate nonequilibrium spin density of conduction electrons induced by slow variation of magnetization orientation, sm(rrr;t) =i 2Tr smG<(rrrt;rrrt) =Qmn¶tmn+:::; (3) 2/8where summation over repeated indexes is assumed ( m;n=x;y;z). The lesser Green’s function of conduction electrons can be represented as G<= GKGR+GA
=2, where GK,GR,GAare Keldysh, retarded, and advanced Green’s functions respectively. Kubo-St ˇreda formula We further proceed with evaluating Qmnin Eq. (3) that describes the contribution to the Gilbert damping due to conduction electrons. In the Hamiltonian (1) we assume slow dynamics of the magnetization, such that approximation MMM(rrr;t) MMM+(tt0)¶tMMMwith MMM=MMM(rrr;t0)is supposed to be hold with high accuracy, H=p2 2m+a(sssppp)z+sss
MMM+U(rrr)+(tt0)sss
¶tMMM; (4) where first four terms in the right hand side of Eq. (4) can be grouped into the Hamiltonian of a bare system, H0, which coincides with that of Eq. (1), provided by the static magnetization configuration MMM. In addition, the expression (4) includes the time-dependent term V(t)explicitly, as the last term. In the following analysis we deal with this in a perturbative manner. In particular, the first order correction to the Green’s function of a bare system induced by V(t)is, dG(t1;t2) =Z CKdtZd2p (2p)2gppp(t1;t)V(t)gppp(t;t2); (5) where the integral in time domain is taken along a Keldysh contour, while gppp(t1;t2) =gppp(t1t2)[the latter accounts for the fact that in equilibrium correlation functions are determined by the relative time t1t2] stands for the Green’s function of the bare system with the Hamiltonian H0in momentum representation. In particular, for the lesser Green’s function at coinciding time arguments t1=t2t0, which is needed to evaluate (3), one can write down, dG<(t0;t0) =i 2¥Z ¥de 2pZd2p (2p)2n gR pppsm¶g< ppp ¶e¶gR ppp ¶esmg< ppp+g< pppsm¶gA ppp ¶e¶g< ppp ¶esmgA pppo ¶tMm; (6) where m=x;y;z, while gR,gA, and g<are the bare retarded, advanced, and lesser Green’s functions respectively. To derive the expression (6) we made use of Fourier transformation gppp=Rd(t1t2)gppp(t1t2)eie(t1t2)and integration by parts. To finally close up the derivation we employ the fluctuation-dissipation theorem according to which g<(e) = [gA(e) gR(e)]f(e), where f(e) = [eb(em)+1]1stands for the Fermi-Dirac distribution with the Fermi energy m. Thus, nonequi- librium spin density of conduction electrons (3) within linear response theory is determined by Qmn=Q(1) mn+Q(2) mn, where Q(1) mn=1 4Trh sm¥Z ¥de 2pZd2p (2p)2n¶gR ppp ¶esngR pppgR pppsn¶gR ppp ¶e+gA pppsn¶gA ppp ¶e¶gA ppp ¶esngA pppo f(e)i ; (7) which involves the integration over the whole Fermi sea, and Q(2) mn=1 4Trh sm¥Z ¥de 2p ¶f(e) ¶eZd2p (2p)2n gR pppsngR ppp+gA pppsngA ppp2gR pppsngA pppoi ; (8) which selects the integration in the vicinity of the Fermi level. Generally, the form of Qmnbelongs to the class of Kubo-St ˇreda formula, and, in essence, represents the response to the external stimulus in the form of ¶tMn. We can immediately establish a quantitative agreement between the result given by Eq. (8) and the previous studies within a Kubo formalism40, 43–46which allow a direct estimation within the framework of disordered alloys. Formally, the expression (7) corresponds to the so-called St ˇreda contribution. Such a term was originally identified in Ref.47when studying quantum-mechanical conductivity. Notably, in Eq. (7) each term represents the product of either retarded or advanced Green’s functions. In this case the poles of the integrand function are positioned on the same side of imaginary plane, making disorder correction smaller in the weak disorder limit (see, e.g., Ref.48). Meanwhile, having no classical analog this contribution appears to be important enough when the spectrum of the system is gapped and the Fermi energy is placed exactly in the gap47. It is worth mentioning that the contribution due to Eq. (7) has never been discussed in this context before. In the meantime, Kubo-St ˇreda expression for the components of the Gilbert damping tensor has been addressed from the perspective of first-principles calculations49and current-induced torques50. 3/8Results and discussion Let us apply the formalism developed in the previous section to a prototypical model: we work out the Gilbert damping tensor for a Rashba ferromagnet with the magnetization mmm=zzzaligned along the zaxis. In the limit of weak disorder the Green’s function of a bare system can be expressed as gR ppp(e) = eH0SR
1=eeppp+id+a(sssppp)z+(D+ih)sz (eeppp+id)2a2p2(D+ih)2; (9) where eppp=p2=(2m)is the electron kinetic energy. We put the self-energy SRdue to scattering off scalar impurities into Eq. (9), which is determined from SR=i(dhsz)(see, e.g., Ref.51). In particular, for jej>jDjwe can establish that d=1=(2t) andh=0 in the weak disorder regime to the leading order. Without loss of generality, in the following we restrict the discussion to the regime m>jDj, which is typically satisfied with high accuracy in experiments. As previously discussed, the contribution owing to the Fermi sea, Eq. (7), can in some cases be ignored, while doing the momentum integral in Eq. (8) results in, 1 mtZd2p (2p)2gR ppp(e)sssgA ppp(e) =D2 D2+2ersss+Dd D2+2er(ssszzz)+D2er D2+2er(ssszzz)zzz; (10) where r=ma2. Thus, thanks to the factor of delta function d(em) =¶f(e)=¶e, to estimate Q(2) mnat zero temperature one should put e=min Eq. (10). As a result, we obtain, Q(2) mn=1 4pm D2+2mr0 @2tmr D 0 D 2tmr 0 0 0 2 tD21 A: (11) Meanwhile, to properly account the correlation functions which appear when averaging over disorder configuration one has to evaluate the so-called vertex corrections, which from a physical viewpoint makes a distinction between disorder averaged product of two Green’s function, hgRsngAidis, and the product of two disorder averaged Green’s functions, hgRidissnhgAidis, in Eq. (8). Thus, we further proceed with identifying the vertex part by collecting the terms linear in dexclusively, GGGs=Asss+B(ssszzz)+C(ssszzz)zzz; (12) provided A=1+D2=(2er),B= (D2+2er)Dd=(D2+er)2, and C=D2=(2er)er=(D2+er). To complete our derivation we should replace snin Eq. (8) by Gs nand with the aid of Eq. (10) we finally derive at e=m, Q(2) mn=0 @Qxx Qxy 0 QxyQxx 0 0mtD2=(4pmr)1 A: (13) We defined Qxx=mtmr=[2p(D2+mr)]andQxy=mD(D2+2mr)=[4p(D2+mr)2], which unambiguously reveals that account of vertex correction substantially modifies the results of the calculations. With the help of Eqs. (3), (11), and (13) we can write down LLG equation. Slight deviation from collinear configurations are determined by xandycomponents ( mxand myrespectively, so that jmxj;jmyj1). The expressions (11) and (13) immediately suggest that the Gilbert damping at the interface is a scalar, aG, ¶tmmm=˜gHHHeffmmm+aGmmm¶tmmm; (14) where the renormalized gyromagnetic ratio and the damping parameter are, ˜g=g 1+cDQxy;aG=cDQxx 1+cDQxycDQxx: (15) In the latter case we make use of the fact that mc1for the NM thickness d100mm — 100 nm. In Eq. (14) we have redefined the gyromagnetic ratio g, but we might have renormalized the magnetization instead. From physical perspective, this implies the fraction of conduction electrons which become associated with the localized moment owing to sdexchange interaction. With no vertex correction included one obtains aG=mc 2p¯htmrD D2+2mr; (16) 4/8t=1ns t=10ns D=0.2meV D=0.3meV D=1meV 501001502002503000.0000.0010.0020.0030.004 T,KaGFigure 2. Gilbert damping, obtained from numerical integration of Eq. (8), shows almost no temperature dependence associated with thermal redistribution of conduction electrons. Dashed lines are plotted for D=1meV for t=1andt=10ns, whereas solid lines stand for D=0:2, 0:3, and 1 meV for t=100 ns. while taking account of vertex correction gives rise to a different result, aG=mc 2p¯htmrD D2+mr: (17) To provide a quantitative estimate of how large the St ˇreda contribution in the weak disorder limit is, on condition that m>jDj, we work out Q(1) mn. Using ¶gR=A(e)=¶e=[gR=A(e)]2and the fact that trace is invariant under cyclic permuattaions we conclude that only off-diagonal components m6=ncontribute. While the direct evaluation results in Q(1) xy=3mD=[2(D2+2mr)]in the clean limit. It has been demonstrated that including scattering rates dandhdoes not qualitatively change the results, leading to some smearing only52. Interestingly, within the range of applicability of theory developed in this paper, the results of both Eqs. (16) and (17) depend linearly on scattering rate, being thus in qualitative agreement with the breathing Fermi surface model. Meanwhile, the latter does not yield any connection to the microscopic parameters (see, e.g., Ref.53for more details). To provide with some quantitative estimations in our simulations we utilize the following set of parameters. Typically, experimental studies based on hyperfine field measurements equipped with DFT calculations54reveal the sdStoner interaction to be of the order of 0.2 eV , while the induced magnetization of s-derived states equals 0.002–0.05 (measured in the units of Bohr magneton, mB). Thus, the parameter of sdexchange splitting, appropriate for our model, is D0.2–1 meV . In addition, according to first-principles simulations we choose the Fermi energy m3 eV . The results of numerical integration of (8) are presented in Fig. 2 for several choices of sdexchange and scattering rates, t. The calculations reveal almost no temperature dependence in the region up to room temperature for any choice of parameters, which is associated with the fact that the dominant contribution comes from the integration in a tiny region of the Fermi energy. Fig. 2 also reveal a non-negligible dependence on the damping parameter with respect to both Dandt, which illustrates that a tailored search for materials with specific damping parameter needs to address both the sdexchange interaction as well as the scattering rate. From the theoretical perspective, the results shown in Fig. 2 correspond to the case of non-interacting electrons with no electron-phonon coupling included. Thus, the thermal effects are accounted only via temperature-induced broadening which does not show up for m>jDj. Conclusions In this paper we proposed an alternative derivation of the Gilbert damping tensor within a generalized Kubo-St ˇreda formula. We established the contribution stemming from Eq. (7) which was missing in the previous analysis within the linear response theory. In spite of being of the order of (mt)1and, thus, negligible in the weak disorder limit developed in the paper, it should be properly worked out when dealing with more complicated systems, e.g., gapped materials such as iron garnets (certain half metallic Heusler compounds). For a model system, represented by a Rashba ferromagnet, we directly evaluated the Gilbert damping parameter and explored its behaviour associated with the temperature-dependent Fermi-Dirac distribution. In essence, the obtained results extend the previous studies within linear response theory and can be further utilized in first-principles calculations. We believe our results will be of interest in the rapidly growing fields of spintronics and magnonics. 5/8References 1.Žuti´c, I., Fabian, J. & Das Sarma, S. Spintronics: Fundamentals and applications. Rev. Mod. Phys. 76, 323 (2004). 2.Bader, S. D. & Parkin, S. S. P. Spintronics. Annu. Rev. Condens. Matter Phys. 1, 71 (2010). 3.MacDonald, A. H. & Tsoi, M. 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Atomistic spin dynamics: Foundations and applications (Oxford University Press, Oxford, 2017). 54.Brooks, M. S. S. & Johansson, B. Exchange integral matrices and cohesive energies of transition metal atoms. J. Phys. F: Met. Phys. 13, L197 (1983). 7/8Acknowledgements A.A.P. acknowledges the support from the Russian Science Foundation Project No. 18-72-00058. O.E. acknowledges support from eSSENCE, the Swedish Research Council (VR), the foundation for strategic research (SSF) and the Knut and Alice Wallenberg foundation (KAW). D.Y . acknowledges the support from the Russian Science Foundation Project No. 17-12-01359. Author contributions statement D.Y . conceived the idea of the paper and contributed to the theory. A.A.P. wrote the main manuscript text, performed numerical analysis and prepared figures 1-2. M.I.B. and O.E. contributed to the theory. All authors reviewed the manuscript. Additional information Competing interests The authors declare no competing interests. 8/8

2018-07-20

A keen interest towards technological implications of spin-orbit driven magnetization dynamics requests a proper theoretical description, especially in the context of a microscopic framework, to be developed. Indeed, magnetization dynamics is so far approached within Landau-Lifshitz-Gilbert equation which characterizes torques on magnetization on purely phenomenological grounds. Particularly, spin-orbit coupling does not respect spin conservation, leading thus to angular momentum transfer to lattice and damping as a result. This mechanism is accounted by the Gilbert damping torque which describes relaxation of the magnetization to equilibrium. In this study we work out a microscopic Kubo-St\v{r}eda formula for the components of the Gilbert damping tensor and apply the elaborated formalism to a two-dimensional Rashba ferromagnet in the weak disorder limit. We show that an exact analytical expression corresponding to the Gilbert damping parameter manifests linear dependence on the scattering rate and retains the constant value up to room temperature when no vibrational degrees of freedom are present in the system. We argue that the methodology developed in this paper can be safely applied to bilayers made of non- and ferromagnetic metals, e.g., CoPt.

Another view on Gilbert damping in two-dimensional ferromagnets

1807.07897v2

Large deviations for the Langevin equation with strong damping Sandra Cerraiy, Mark Freidlinz Department of Mathematics University of Maryland College Park, MD 20742 USA Abstract We study large deviations in the Langevin dynamics, with damping of order 1and noise of order 1, as #0. The damping coecient is assumed to be state dependent. We proceed rst with a change of time and then, we use a weak convergence approach to large deviations and their equivalent formulation in terms of the Laplace principle, to determine the good action functional. Some applications of these results to the exit problem from a domain and to the wave front propagation for a suitable class of reaction diusion equations are considered. 1 Introduction For every>0, let us consider the Langevin equation 8 >>< >>:q(t) =b(q(t))(q(t)) _q(t) +(q(t))_B(t); q(0) =q2Rd;_q(0) =p2Rd:(1.1) HereB(t) is ar-dimensional standard Wiener process, dened on some complete stochastic basis ( ;F;fFtg;P). In what follows, we shall assume that bis Lipschitz continuous and andare bounded and continuously dierentiable, with bounded derivative. Moreover, is invertible and there exist two constants 0 < 0< 1such thata0(q)1, for all q2Rd. Equation (1.1) can be rewritten as the following system in R2d 8 >>< >>:_q(t) =p(t); q(0) =q2Rd; _p(t) =b(q(t))(q(t)) _q(t) +(q(t))_B(t); p(0) =p2Rd; Key words : Large deviations, Laplace principle, over damped stochastic dierential equations yPartially supported by the NSF grant DMS 1407615. zPartially supported by the NSF grant DMS 1411866. 1arXiv:1503.01027v1 [math.PR] 3 Mar 2015and, due to our assumptions on the coecients, for any >0,T >0 andk1, the system above admits a unique solution z= (q;p)2Lk( ;C([0;T];R2d)), which is a Markov process. Now, if we do a change of time and dene q(t) :=q(t=),t0, we have 8 >< >:2q(t) =b(q(t))(q(t)) _q(t) +p(q(t)) _w(t); q(0) =q2Rd;_q(0) =p 2Rd;(1.2) wherew(t) =pB(t=),t0, is another Rr-valued Wiener process, dened on the same stochastic basis ( ;F;fFtg;P). In the present paper, we are interested in studying the large deviation principle for equation (1.2), as #0. Namely, we want to prove that the family fqg>0satises a large deviation principle in C([0;T];H), with the same action functional Iand the same normalizing factor that describe the large deviation principle for the rst order equation _g(t) =b(g(t)) (g(t))+p(g(t)) (g(t))_w(t); g(0) =q2Rd: (1.3) In particular, as shown in Section 4, this implies that the asymptotic behavior of the exit time from a basin of attraction for the over damped Langevin dynamics(1.1) can be described by the quasi potential Vassociated with I, as well as the asymptotic behavior of the solutions of the degenerate parabolic and elliptic problems associated with the Langevin dynamics. Moreover, in Section 4, we will show how these results allow to prove that in reaction- diusion equations with non-linearities of KPP type, where the transport is described by the Langevin dynamics itself, the interface separating the areas where uis close to 1 and to 0, as#0, is given in terms of the action functional I, as in the classical case, when the vanishing mass approximation is considered. In [8] and [3], the system 8 >< >:q;(t) =b(q;(t))(q;(t)) _q;(t) +p(q;(t)) _w(t); q;(0) =q2Rd; _q;(0) =p 2Rd;(1.4) for 0<;<< 1, has been studied, under the crucial assumption that the friction coecient is independent of q. It has been proven that, in this case, the so-called Kramers-Smoluchowski approxi- mation holds, that is for any xed  > 0 the solution q;of system (1.4) converges in L2( ;C([0;T];Rd)), as#0, tog, the solution of the rst order equation (1.3). More- over, it has been proven that, if V(q;p) is the quasi-potential associated with the family fq;g>0, for>0 xed, then lim !0inf p2RdV(q;p) =V(q); 2whereVis the quasi-potential associated with the action-functional I. In [9], equation (1.4) with non constant friction has been considered and it has been shown that in this case the situation is considerably more delicate. Actually, the limit of q;toghas only been proven via a previous regularization of the noise, which has led to the convergence of q;to the solution ~ gof the rst order equation with Stratonovich integral. Finally, we would like to mention that in the recent paper [12], by Lyv and Roberts, an analogous problem has been studied for the stochastic damped wave equation in a bounded regular domain DRd, withd= 1;2;3, 8 >>>< >>>:@2u(t;x) @t2=
u(t;x) +f(u(t;x))@u(t;x) @t+@w(t;x) @t u(t;x) = 0; x2@D; u (0;x) =u0(x);@u(0;x) @t=v0;(x) where >0 is a small parameter, the friction coecient is constant ( = 1),w(t;x) is a smooth cylindrical Wiener process and fis a cubic non-linearity. By using the weak convergence approach, the authors show that the family fug>0satises a large deviation principle in C([0;T];L2(D)), with normalizing factor 2and the same action functional that describes the large deviation principle for the stochastic parabolic equation. As mentioned above, in the present paper we are dealing with the case of non-constant frictionand=2. Dealing with a non-constant friction coecient turns out to be important in applications, as it allows to describes new eects in reaction-diusion equations and exit problems (see section 4). Here, we will study the large deviation principle for equation(1.2) by using the approach of weak convergence (see [1] and [2]) and we will show the validity of the Laplace principle, which, together with the compactness of level sets, is equivalent to the large deviation principle. At this point, it is worth mentioning that one major diculty here is handling the integralZt 0exp Zt s(q(r))dr (q(s))dw(s); and proving that it converges to zero, as #0, inL1( ;C([0;T];Rd)). Actually, as is non- constant, the integral above cannot be interpreted as an It^ o's integral and in our estimates we cannot use It^ o's isometry. Nevertheless, due to the regularity of q(t), we can consider the integral above as a pathwise integral, and with appropriate integrations by parts, we can get the estimates required to prove the Laplace principle. 2 The problem and the method We are dealing here with the equation 8 >< >:2q(t) =b(q(t))(q(t)) _q(t) +p(q(t)) _w(t); q(0) =q2Rd;_q(0) =p 2Rd;(2.1) 3Herew(t),t0, is ar-dimensional Brownian motion and the coecients b,andsatisfy the following conditions. Hypothesis 1. 1. The mapping b:Rd!Rdis Lipschitz-continuous and the map- ping:Rd!L(Rr;Rd)is continuously dierentiable and bounded, together with its derivative. Moreover, the matrix (q)is invertible, for any q2Rd, and1:Rd! L(Rr;Rd)is bounded. 2. The mapping :Rd!Rbelongs toC1 b(Rd)and inf x2Rd(x) =:0>0: (2.2) In view of the conditions on the coecients ,bandassumed in Hypothesis 1, for every xed >0, equation (2.6) admits a unique solution z= (q;p)2Lk(0;T;Rd), with T >0 andk1. Now, for any predictable process utaking values in L2([0;T];Rr), we introduce the problem _gu(t) =b(gu(t)) (gu(t))+(gu(t)) (gu(t))u(t); gu(0) =q2Rd: (2.3) The existence and uniqueness of a pathwise solution guto problem (2.3) in C([0;T];Rd) is an immediate consequence of the conditions on the coecients b,andthat we have assumed in Hypothesis 1. In what follows, we shall denote by Gthe mapping G:L2([0;T];Rr)!C([0;T];Rd); u7!G(u) =gu: Moreover, for any f2C([0;T];Rd) we shall dene I(f) =1 2infZT 0ju(t)j2dt:f=G(u); u2L2([0;T];Rr) ; with the usual convention inf ;= +1. This means that I(f) =1 2ZT 0(f(s))1(f(s)) _f(s)b(f(s)) (f(s))2 ds; (2.4) for allf2W1;2(0;T;Rd). If we denote by gthe solution of the stochastic equation _g(t) =b(g(t)) (g(t))+p(g(t)) (g(t))_w(t); g(0) =q2Rd; (2.5) we have that Iis the large deviation action functional for the family fgg>0in the space of continuous trajectories C([0;T];Rd) (for a proof see e.g. [11]). This means that the level setsfI(f)cgare compact in C([0;T];Rd), for anyc>0, and for any closed subset FC([0;T];Rd) and any open set GC([0;T];Rd) it holds lim sup !0+logP(g2F)I(F); lim inf !0+logP(g2G)I(G); 4where, for any subset AC([0;T];Rd), we have denoted I(A) = inf f2AI(f): The main result of the present paper is to prove that in fact the family of solutions q of equation (1.2) satises a large deviation principle with the same action functional Ithat describes the large deviation principle for the family of solutions gof equation (2.5). And, due to the fact that q(t) =q(t),t0, this allows to describe the behavior of the over damped Langevin dynamics (1.1) (see Section 4 for all details). Theorem 2.1. Under Hypothesis 1, the family of probability measures fL(q)g>0, in the space of continuous paths C([0;T];Rd), satises a large deviation principle with action func- tionalI. In order to prove Theorem 2.1, we follow the weak convergence approach, as developed in [1], (see also [2]). To this purpose, we need to introduce some notations. We denote by PTthe set of predictable processes in L2( [0;T];Rr), and for any T > 0 and >0, we dene the sets S T= f2L2(0;T;Rd) :ZT 0jf(s)j2ds  A T= u2PT:u2S T;Pa.s. : Next, for any predictable process utaking values in L2([0;T];Rr), we denote by qu (t) the solution of the problem 8 >< >:2qu (t) =b(qu (t))(qu (t)) _qu (t) +p(qu (t)) _w(t) +(qu (t))u(t); qu (0) =q2Rd;_qu (0) =p 2Rd:(2.6) As well known, for any xed  >0 and for any T > 0 andk1, this equation admits a unique solution qu inLk( ;C([0;T];Rd)). By proceeding as in the proof of [2, Theorem 4.3], the following result can be proven. Theorem 2.2. Letfug>0be a family of processes in S Tthat converge in distribution, as #0, to someu2S T, as random variables taking values in the space L2(0;T;Rd), endowed with the weak topology. If the sequencefqug>0converges in distribution to gu, as#0, in the space of contin- uous paths C([0;T];Rd), then the family fL(q)g>0satises a large deviation principle in C([0;T];Rd), with action functional I. Actually, as shown in [2], the convergence of qutoguimplies the validity of the Laplace principle with rate functional I. This means that, for any continuous mapping  :C([0;T];Rd)!Rit holds lim !0logEexp 1 (q) = inf f2C([0;T];Rd)( (f) +I(f) ): And, as the level sets of Iare compact, this is equivalent to say that fL(q)g>0satises a large deviation principle in C([0;T];Rd), with action functional I. 53 Proof of Theorem 2.1 As we have seen in the previous section, in order to prove Theorem 2.1, we have to show that iffug>0is a family of processes in S Tthat converge in distribution, as #0, to someu2 S T, as random variables taking values in the space L2(0;T;Rd), endowed with the weak topology, then the sequence fqug>0converges in distribution to gu, as#0, in the spaceC([0;T];Rd). In view of the Skorohod representation theorem, we can rephrase such a condition in the following way. On some probability space (  ;F;P), consider a Brownian motion  wt, t0, along with the corresponding natural ltration fFtgt0. Moreover, consider a family offFtg-predictable processes fu;ug>0inL2( [0;T];Rd), taking values in S T,P-a.s., such that the joint law of ( u;u;w), under P, coincides with the joint law of ( u;u;w ), under P, and such that lim !0u= u; Pa.s. (3.1) asL2(0;T;R)-valued random variables, endowed with the weak topology. Let  qube the solution of a problem analogous to (2.6), with uandwreplaced respectively by  uand w. Then, we have to prove that lim !0qu=gu; Pa.s. inC([0;T];Rd). In fact, we will prove more. Actually, we will show that lim !0Esup t2[0;T]jqu(t)gu(t)j= 0: (3.2) In order to prove (3.2), we will need some preliminary estimates. For any  >0, we dene the process H(t) =peA(t)Zt 0eA(s)(qu (s))dw(s); t0: (3.3) Lemma 3.1. Under Hypothesis 1, for any T > 0,k1and > 0, there exists 0>0 such that for any u2S Tand2(0;0] sup stEjH(t)jkck; (T)(jqjk+jpjk+ 1)3k 2+ckk 2tk 2ek0t 2: (3.4) Moreover, we have Esup t2[0;T]jH(t)jpc (T)(1 +jqj+jpj): (3.5) Proof. Equation (2.6) can be rewritten as the system 8 >< >:_qu (t) =pu (t); qu (0) =q 2_pu (t) =b(qu (t))(qu (t))pu (t) +p(qu (t)) _w(t) +(qu (t))u(t); pu (0) =p : 6Thus, if for any 0 stand>0 we dene A(t;s) :=1 2Zt s(qu (r))dr; A(t) :=A(t;0); we have pu (t) =1 eA(t)p+1 2Zt 0eA(t;s)b(qu (s))ds +1 2Zt 0eA(t;s)(qu (s))u(s)ds+1 2H(t):(3.6) Integrating with respect to t, this yields qu (t) =q+1 Zt 0eA(s)pds+1 2Zt 0Zs 0eA(s;r)b(qu (r))drds +1 2Zt 0Zs 0eA(s;r)(qu (r))u(r)drds +1 2Zt 0H(s)ds:(3.7) Thanks to the Young inequality, this implies that for any t2[0;T] jqu (t)jjqj+jpj+cZt 0(1 +jqu (s)j)ds+Zt 0ju(s)jds+1 2Zt 0jH(s)jds c (T)(jqj+jpj+ 1) +1 2Zt 0jH(s)jds+Zt 0jqu (s)jds; and from the Gronwall lemma we can conclude that jqu (t)jc (T) (1 +jqj+jpj) +c(T)1 2Zt 0jH(s)jds: This implies that for any k1 jqu (t)jkck; (T)(jqjk+jpjk+ 1) +ck; (T)2kZt 0jH(s)jkds; 2(0;1]: (3.8) Now, due to (3.6), we have jpu(t)j1 e0t 2jpj+1 2Zt 0e0(ts) 2(1 +jqu (s)j)ds +1 2Zt 0e0(ts) 2ju(s)jds+1 2jH(t)j; so that, thanks to (3.8), for any 2(0;1] we get jpu (t)j1 e0t 2jpj+c (T)(jqj+jpj+ 1) +1 2Zt 0e0(ts) 2ju(s)jds+c(T)1 2jH(t)j: (3.9) 7As well known, if f2C1([0;t]) andg2C([0;t]), then the Stiltjies integral Zt 0f(s)dg(s); t0; is well dened and, if g(0) = 0, the following integration by parts formula holds Zt 0f(s)dg(s) =Zt 0(g(t)g(s))h0(s)ds+g(t)h(0); t0: (3.10) Now, the mapping [0;+1)!L(Rr;Rd); s7!eA(s)(qu (s)); is dierentiable, P-a.s., so that the stochastic integral in (3.3) is in fact a pathwise integral. In particular, we can apply formula (3.10), with h(s) =eA(s)(qu (s)); g (s) =w(s); and we get H(t) =pZt 0(w(t)w(s))eA(t;s)(qu (s)) 2+0(qu (s))pu (s) ds +pw(t)eA(t)(q):(3.11) Thanks to (3.9), this yields for any 2(0;1] jH(t)jcpZt 0jw(t)w(s)je0(ts) 2 2 1 +2jpu (s)j
ds+cpjw(t)je0t 2 c (T)(jqj+jpj+ 1)pZt 2 0jw(t)w(t2s)je0sds +pc (T)Zt 2 0jw(t)w(t2s)je0sjH(t2s)jds+cpjw(t)je0t 2; and hence, for any k1, we have jH(t)jkck; (T)(jqjk+jpjk+ 1)k 2Zt 2 0jw(t)w(t2s)jke0sds +k 2ck; (T)Zt 2 0jw(t)w(t2s)jke0sjH(t2s)jkds+ckk 2jw(t)jkek0t 2: 8By taking the expectation, due to the independence of jw(t)w(t2s)jwithjH(t2s)j andRt2s 0jH(r)jkdr, this implies that for any 2(0;1] EjH(t)jkck; (T)(jqjk+jpjk+ 1)3k 2Zt 2 0sk 2e0sds +3k 2ck; (T)Zt 2 0sk 2e0sEjH(t2s)jkds+ckk 2tk 2ek0t 2 ck; (T)(jqjk+jpjk+ 1)3k 2+ckk 2tk 2ek0t 2+3k 2ck; (T) sup stEjH(s)jk: Therefore, if we pick 02(0;1] such that 3k 2ck; (T)<1 2; we get (3.4). Now, let us prove (3.5). From (3.11), we have jH(t)jpcsup t2[0;T]jw(t)j 1 +Zt 0e0(t2) 2jpu(s)jds pcsup t2[0;T]jw(t)j 1 +Zt 0jpu(s)j2ds1 2! ; and hence Esup t2[0;T]jH(t)jpc(T) 1 + EZt 0jpu(s)j2ds1 2! : Thanks to (3.9), as a consequence of the Young inequality, we get Zt 0jpu(s)j2dsc (T)(1 +jqj2+jpj2) +1 4c(T)Zt 0jH(s)j2ds; (3.12) so that Esup t2[0;T]jH(t)jpc (T)(1 +jqj+jpj) +1pc(T)Zt 0EjH(s)j2ds1 2 : Therefore, (3.5) follows from (3.4). Lemma 3.2. Under Hypothesis 1, for any T >0,k1and >0there exists 0>0such that for any u2S Tand2(0;0)we have Esup t2[0;T]jqu (t)jkck; (T)(jqjk+jpjk+ 1)k 2+ck; (T)23k 2: (3.13) 9Proof. Estimate (3.13) follows by combining together (3.4) and (3.8). Now, we are ready to prove (3.2), that, in view of Theorem 2.2, implies Theorem 2.1. Theorem 3.3. Letfug>0be a family of predictable processes in S Tthat converge P-a.s., as#0, to someu2S T, with respect to the weak topology of L2(0;T;Rd). Then, we have lim !0Esup t2[0;T]jqu(t)gu(t)j= 0: (3.14) Proof. Integrating by parts in (3.7), we obtain qu(t) =q+Zt 0b(qu(s)) (qu(s))ds+Zt 0(qu(s)) (qu(s))u(s)ds+R(t); where R(t) =p Zt 0eA(s)ds1 (qu(t))Zt 0eA(t;s)b(qu(s))ds+pZt 0(qu(s)) (qu(s))dw(s) +Zt 0Zs 0eA(s;r)b(qu(r))dr1 2(qu(s))hr(qu(s));pu(s)ids 1 (qu(t))H(t) +Zt 01 2(qu(s))H(s)hr(qu(s));pu(s)ids=:6X k=1Ik (t): This implies that qu(t)gu(t) =Zt 0b(qu(s)) (qu(s))b(gu(s)) (gu(s)) ds+Zt 0(qu(s)) (qu(s))(gu(s)) (gu(s)) u(s)ds +Zt 0(gu(s)) (gu(s))[u(s)u(s)]ds+R(t): (3.15) Due to the Lipschitz-continuity and the boundedness of the functions and 1=, we have that= is bounded and Lipschitz continuous. Then, as u2S T, we obtain jqu(t)gu(t)j2 cZt 0(gu(s)) (gu(s))[u(s)u(s)]ds2 +cjR(t)j2+c(T)Zt 0jqu(s)gu(s)j2ds +c(T)Zt 0jqu(s)gu(s)j2ds Zt 0ju(s)j2ds+ sup s2[0;t]jgu(s)j2! cZt 0(gu(s)) (gu(s))[u(s)u(s)]ds2 +cjR(t)j2+c (T)Zt 0jqu(s)gu(s)j2ds: 10By the Gronwall lemma, this allows to conclude that sup t2[0;T]jqu(t)gu(t)j c (T) sup t2[0;T]Zt 0(gu(s)) (gu(s))[u(s)u(s)]ds+c (T) sup t2[0;T]jR(t)j:(3.16) Now, for any >0, we dene (t) =Zt 0(gu(s)) (gu(s))[u(s)u(s)]ds: For any 0<s<t we have (t)(s) =Zt s(gu(r)) (gu(r))[u(r)u(r)]dr; so that, as uanduare both in S T, j(t)(s)jc p ts; > 0: As (0) = 0, this implies that the family of continuous functions is fg>0is equibounded and equicontinuous, so that, by the Ascoli-Arzel a theorem, there exists n#0 andv2 C([0;T];Rd) such that lim n!0sup t2[0;T]jn(t)v(t)j= 0;Pa.s. On the other hand, as (3.1) holds, for any h2Rdwe have lim !0h(t);hi= lim !0 uu;(gu(
)) (gu(
))h L2(0;T;Rd)= 0; so that we can conclude that v= 0 and lim !0Esup t2[0;T]j(t)j= 0: Thanks to (3.16), this implies that lim sup !0Esup t2[0;T]jqu(t)gu(t)jclim sup !0Esup t2[0;T]jR(t)j; so that (3.14) follows if we show that lim !0Esup t2[0;T]jR(t)j= 0: (3.17) We have jI1 (t)j=jpj Zt 0eA(s)dscjpj1Zt 0e0s 2dscjpj: (3.18) 11Moreover jI2 (t)j=1 j(qu(t))jZt 0eA(t;s)b(qu(s))ds cZt 0e0(ts) 2(1 +jqu(s)j)dsc2 1 + sup t2[0;T]jqu(t)j! : Thanks to (3.13), this implies Esup t2[0;T]jI2 (t)jc (T)(jpj+jqj+ 1)3 2; 2(0;1]: (3.19) Next Esup t2[0;T]jI3 (t)j=pEsup t2[0;T]Zt 0(qu(s)) (qu(s))dw(s)c(T)p: (3.20) Concerning I4(t), we have jI4 (t)j=Zt 0Zs 0eA(s;r)b(qu(r))dr1 2(qu(s))hr(qu(s));pu(s)ids 2c 1 + sup t2[0;T]jqu(t)j!Zt 0jpu(s)jds; so that, due to (3.13) we obtain Esup t2[0;T]jI4 (t)j2c (T)(jqj+jpj+ 1)1 2 EZt 0jpu(s)j2ds1 2 : As a consequence of (3.4) and (3.12), this yields Esup t2[0;T]jI4 (t)jc (T)(jqj2+jpj2+ 1); 2(0;0]: (3.21) Concerning I5 (t), according to (3.5) we have Esup t2[0;T]jI5 (t)jcEsup t2[0;T]jH(t)jpc (T)(1 +jqj+jpj): (3.22) Finally, it remains to estimate I6 (t). We have jI6 (t)j=Zt 01 2(qu(s))H(s)hr(qu(s));pu(s)idscZt 0jH(s)jjpu(s)jds; so that Esup t2[0;T]jI6 (t)jcZT 0EjH(s)j2dsZT 0Ejpu(s)j2ds1 2 : 12By using (3.12), this gives Esup t2[0;T]jI6 (t)jc (T)(1 +jqj+jpj)ZT 0EjH(s)j2ds1 2 +1 2ZT 0EjH(s)j2ds; so that, from (3.4) we get Esup t2[0;T]jI6 (t)jc (T)(1 +jqj+jpj); 2(0;0]: This, together with (3.18), (3.19), (3.20), (3.21) and (3.22), implies (3.17) and (3.14) follows. 4 Some applications and remarks LetGbe a bounded domain in Rd, with a smooth boundary @G. We consider here the exit problem for the process q(t) dened as the solution of equation (1.1). For every >0 we dene := minft0 :q(t)=2Gg; := minft0 :q(t)=2Gg; whereq(t) =q(t=) is the solution of equation (2.6). It is clear that =1 ; q() =q(): In what follows, we shall assume that the dynamical system _q(t) =b(q(t)); t0; (4.1) satises the following conditions. Hypothesis 2. The pointO2Gis asymptotically stable for the dynamical system (4.1) and for any initial condition q2Rd lim t!1q(t) =O: Moreover, we have hb(q);(q)i>0; q2@G; where(q)is the inward normal vector at q2@G. Now, we introduce the quasi-potential associated with the action functional Idened in (2.4) V(q) = infn I(f); f2C([0;T];Rd); f(0) =O; f (T) =q; T > 0o =1 2inf(ZT 0(f(s))1(f(s)) _f(s)b(f(s)) (f(s))2 ds; f (0) =O; f (T) =q; T > 0) : It is easy to check that, under our assumptions on (q), the quasi-potential Vcoincides with 1 2infZT 01(f(s)) _f(s)(f(s))b(f(s))2 ds; f (0) =O; f (T) =q; T > 0 :(4.2) 13Theorem 4.1. Under Hypotheses 1 and 2, for each q2fq2G:V(q)V0gandp2Rd, we have lim !0logE(q;p)= lim !0logE(q;p)=V0; (4.3) and lim !0log= lim !0log=V0;in probability ; (4.4) where V0:= min q2@GV(q): Moreover, if the minimum of Von@Gis achieved at a unique point q?2@G, then lim !0q() = lim !0q() =q?: (4.5) Proof. First, note that q(t) is the rst component of the 2 d-dimensional Markov process z(t) = (q(t);p(t)). Because of the structure of the p-component of the drift of this process and our assumptions on the vector eld b, starting from ( q;p)2R2d, the trajectory of z(t) spends most of the time in a small neighborhood of the point q=Oandp= 0, with probability close to 1, as 0 <  << 1. From time to time, the process z(t) deviates from this point and, as proven in Theorem 2.1, the deviations of q(t) are governed by the large deviation principle with action functional I, dened in (2.4). This allows to prove the validity of (4.3), (4.4) and (4.5) in the same way as Theorems 4.41, 4.42 and 4.2.1 from [11] are proven. We omit the details. As an immediate consequence of (4.2) and [11, Theorem 4.3.1], we have the following result. Theorem 4.2. Assumea(q) :=(q)?(q) =Iand(q)b(q) =rU(q) +l(q), for any q2Rd, for some smooth function U:Rd!Rhaving a unique critical point (a minimum) atO2Rdand such that hrU(q);l(q)i= 0; q2Rd: Then V(q) = 2U(q); q2Rd: From Theorems 4.1 and 4.2, it is possible to get a number of results concerning the asymptotic behavior, as #0, of the solutions of the degenerate parabolic and the elliptic problems associated with the dierential operator Ldened by Lu(q;p) =1 2dX i;j=1ai;j(q)@2u @pi@pj(q;p) + b(q)1 (q)p
rpu(q;p) +p
rqu(q;p): Assume now that the dynamical system (4.1) has several asymptotically stable attrac- tors. Assume, for the sake of brevity, that all attractors are just stable equilibriums O1, O2,. . . ,Ol. Denote byEthe set of separatrices separating the basins of these attractors, and assume the setEto have dimension strictly less than d. Moreover, let each trajectory q(t), 14starting at q02RdnE, be attracted to one of the stable equilibriums Oi,i= 1;:::;l , as t!1 . Finally, assume that the projection of b(q) on the radius connecting the origin in Rdand the point q2Rdis directed to the origin and its length is bounded from below by some uniform constant >0 (this condition provides the positive recurrence of the process z(t) = (q(t);p(t)),t0). In what follows, we shall denote V(q1;q2) =1 2infZT 0(f(s))1(f(s)) _f(s)b(f(s)) (f(s))2 ds; f (0) =q1; f(T) =q2; T > 0 and Vij=V(Oi;Oj); i;j2f1;:::;lg: In a generic case, the behavior of the process ( q(t);p(t)), on time intervals of order exp(1), > 0 and 0<  << 1, can be described by a hierarchy of cycles as in [11] and [6]. The cycles are dened by the numbers Vij. For (almost) each initial point qand a time scale, these numbers dene also the metastable state Oi?,i?=i?(q;), whereq(t) spends most of the time during the time interval [0 ;exp(1)]. Slow changes of the eld b(q) and/or of the damping coecient (q) can lead to stochastic resonance (compare with [7]). Consider next the reaction diusion equation in Rd 8 >< >:@u @t(t;q) =Lu(t;q) +c(q;u(t;q))u(t;q); u(0;q) =g(q); q2Rd; t> 0:(4.6) HereLis a linear second order uniformly elliptic operator, with regular enough coecients. Letq(t) be the diusion process in Rdassociated with the operator L. The Feynman-Kac formula says that ucan be seen as the solution of the problem u(t;q) =Eqg(q(t)) expZt 0c(q(s);u(ts;q(s))ds: (4.7) Reaction-diusion equations describe the interaction between particle transport dened byq(t) and reaction which consists of multiplication (if c(q;u)>0) and annihilation (if c(q;u)<0) of particles. In classical reaction-diusion equations, the Langevin dynamics which describes a diusion with inertia is replaced by its vanishing mass approximation. If the transport is described by the Langevin dynamics itself, equation (4.6) should be replaced by an equation in R2d. Assuming that the drift is equal to zero ( b(q) = 0), and the damping is of order1, as#0, this equation has the form8 >>>>>>>>< >>>>>>>>:@u @t(t;q;p ) =1 2dX i;j=1ai;j(q)@2u @pi@pj(q;p)1 (q)p
rpu(q;p) +p
rqu(q;p) +c(q;u(t;q;p ))u(t;q;p ); t> 0;(q;p)2R2d; u(0;q;p) =g(q)0;(q;p)2R2d:(4.8) 15Now, we dene R(t;q) = supZt 0c(f(s);0)dsIt(f) :f(0) =q; f(t)2G0 ; where It(f) =1 2Zt 02(f(s))a1(f(s))_f(s)
_f(s)ds; andG0= suppfg(q); q2Rdg. Denition 4.3. 1. We say that Condition (N) is satised if R(t;x)can be characterized, for anyt>0andx2t=fq2Rd; R(t;q) = 0g, as supZt 0c(f(s);0)dsIt(f); f(0) =q; f(t)2G0; R(ts;f(s))0;0st : 2. We say that the non-linear term f(q;u) =c(q;u)uin equation (4.8) is of KPP (Kolmogorov-Petrovskii-Piskunov) type if c(q;u)is Lipschitz-continuous, c(q;0) c(q;u)>0, for any 0<u< 1,c(q;1) = 0 andc(q;u)<0, for anyu>1. Theorem 4.4. Let the non-linear term in (4.8) be of KPP type. Assume that Condition (N) is satised and assume that the closure of G0= suppfg(q); q2Rdgcoincides with the closure of the interior of G0. Then, lim !0u(t=;q;p ) = 0;ifR(t;q)<0; (4.9) and lim !0u(t=;q;p ) = 1;ifR(t;q)>0; (4.10) so that equation R(t;q) = 0 inR2ddenes the interface separating the area where u, the solution of (4.8) , is close to 1and to 0, as#0. Proof. If we dene u(t;q;p ) =u(t=;q;p ), the analog of (4.7) yields u(t;q;p ) =E(q;p)g(q(t)) exp1 Zt 0c(q(s);u(ts;q(s);p(s))ds ; (4.11) wherez(t) = (q(s);p(s)) is the solution to equation (2.6). By taking into account our assumptions on c(q;u), we derive from (4.11) u(t;q;p )E(q;p)g(q(t)) exp1 Zt 0c(q(s);0)ds : Theorem 2.1 and the Laplace formula imply that the right hand side of the above inequality is logarithmically equivalent , as #0, to exp1 R(t;q)
and this implies (4.9). In order to prove (4.10), rst of all one should check that if R(t;q) = 0, then for each >0 u(t;q;p )exp 1  ; (4.12) 16when>0 is small enough. This follows from (4.11) and Condition (N), if one takes into account the continuity of c(q;u). The strong Markov property of the process ( q(t);p(t)) and bound (4.12) imply (4.10) (compare with [5]). Consider, as an example, the case c(q;0) =c= const. Then R(t;q) =ctinffIt(f); f(0) =q; f(t)2G0g: The inmum in the equality above coincides with 1 2t2(q;G 0); (see, for instance, [5] for a proof), where (q1;q2),q1;q22Rd, is the distance in the Riemaniann metric ds=(q)vuutdX i;j=1ai;j(q)dqidqj: This implies that the interface moves according to the Huygens principle with the constant speedp 2c, if calculated in the Riemannian metric ds. If(q) = 0 in a domain G1Rd, the points of G1should be identied. The Riemaniann metric in Rdinduces now, in a natural way, a new metric ~ in this space with identied points. The motion of the interface, in this case, can be described by the Huygens principle with constant velocityp 2cin the metric ~ . Ifc(q;0) is not constant, the motion of the interface, in general, cannot be described by a Huygens principle. Actually, the motion can have jumps and other specic features (compare with [5]). Finally, if the Condition (N) is not satised, the function R(t;q) should be replaced by another one. Dene ~R(t;q) = sup min 0atZa 0c(f(s);0)dsIa(f) :f(0) =q; f(t)2G0 : The function ~R(t;q) is Lipschitz continuous and non-positive and if Condition (N) is satis- ed, then ~R(t;q) = minfR(t;q);0g: By proceeding as in [6], it is possible to prove that lim !0u(t=;q;p ) = 0;ifR(t;q)<0; and lim !0u(t=;q;p ) = 1; if (t;q) is in the interior of the set f(t;q) :t>0; q2Rd;~R(t;q) = 0g. Finally, we would like to mention a few generalizations. 171. The arguments that we we have used in the proof of Theorem 2.1, can be used to prove the same result for the equation 8 >>< >>:q(t) =b(q(t))(q(t)) _q(t) +1 (q(t))_B(t); q(0) =q2Rd;_q(0) =p2Rd; for any <1=2. As a matter of fact, with the very same method we can show that also in this case the family fqg>0satises a large deviation principle in C([0;T];Rd) with action functional Iand with normalizing factor 12. 2. The damping can be assumed to be anisotropic. This means that the coecient (q) can be replaced by a matrix (q), with all eigenvalues having negative real part. 3. Systems with strong non-linear damping can be considered. Namely, let ( q;p) be the time-inhomogeneous Markov process corresponding to the following initial-boundary value problem for a degenerate quasi-linear equation on a bounded regular domain GRd 8 >>>>>>>>>< >>>>>>>>>:@u(t;p;q ) @t=1 2dX i;j=1ai;j(q)@2u(t;q;p ) @pi@pj+b(q)
rpu(t;q;p ) (q;u(t;q;p )) p
rpu(t;q;p ) +p
rqu(t;q;p ): u(0;q;p) =g(q); u(t;q;p )jq2@G= (q); Existence and uniqueness of such degenerate problem, under some mild conditions, follows from [4, Chapter 5]. The non-linearity of the damping leads to some pecu- larities in the exit problem and in metastability. In particular, in the generic case, metastable distributions can be distributions among several asymptotic attractors and the limiting exit distributions may have a density (see [10]). References [1] P. Dupuis, R. Ellis, A weak convergence approach to the theory of large deviations , Wiley Series in Probability and Statistics, John Wiley and Sons, Inc., New York, 1997. [2] M. Bou e, P. Dupuis, A variational representation for certain functionals of Brownian motion , Annals of Probability 26 (1998), no. 4, 1641{1659. [3] Z. Chen, M.I. Freidlin, Smoluchowski-Kramers approximation and exit problems Stochastics and Dynamycs 5 (2005), pp. 569{585. [4] M.I. Freidlin, Functional integration and partial differential equations , Annals of Mathematics Studies, 109, Princeton University Press, 1985. 18[5] M.I. Freidlin, Limit theorems for large deviations and reaction-diusion equations , An- nals of Probability 13 (1985), pp. 639{675. [6] M.I. Freidlin, Coupled reaction-diusion equations , Annals of Probability 19 (1991), pp. 29{57. [7] M.I. Freidlin, Quasi-deterministic approximation, metastability and stochastic reso- nance , Physica D 137 (2000), pp. 333{352. [8] M.I. Freidlin, Some remarks on the Smoluchowski-Kramers approximation , Journal of Statistical Physics 117, pp. 617{634, 2004. [9] M.I. Freidlin, W. Hu, Smoluchowski-Kramers approximation in the case of variable friction , Journal of Mathematical Sciences, 179 (2011), pp. 184{207. [10] M.I. Freidlin, L. Koralov, Nonlinear stochastic perturbations of dynamical systems and quasi-linear parabolic PDE's with a small parameter , Probability Theory andRelated Fields 147 (2010), pp. 273{301. [11] M.I. Freidlin, A.D. Wentzell, Random perturbations of dynamical systems , Third Edition, Springer, Heidelberg, 2012. [12] Y. Lyv, A.J. Roberts, Large deviation principle for singularly perturbed stochastic damped wave equations , Stochastic Analysis and Applications, 32 (2014), pp. 50-60. 19

2015-03-03

We study large deviations in the Langevin dynamics, with damping of order $\e^{-1}$ and noise of order $1$, as $\e\downarrow 0$. The damping coefficient is assumed to be state dependent. We proceed first with a change of time and then, we use a weak convergence approach to large deviations and their equivalent formulation in terms of the Laplace principle, to determine the good action functional. Some applications of these results to the exit problem from a domain and to the wave front propagation for a suitable class of reaction diffusion equations are considered.

Large Deviations for the Langevin equation with strong damping

1503.01027v1

Inertia, diffusion and dynamics of a driven skyrmion Christoph Sch ¨utte,1Junichi Iwasaki,2Achim Rosch,1and Naoto Nagaosa2, 3, 1Institut f ¨ur Theoretische Physik, Universit ¨at zu K ¨oln, D-50937 Cologne, Germany 2Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 3RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: January 5, 2015) Skyrmions recently discovered in chiral magnets are a promising candidate for magnetic storage devices because of their topological stability, small size ( 3100nm), and ultra-low threshold current density (  106A/m2) to drive their motion. However, the time-dependent dynamics has hitherto been largely unexplored. Here we show, by combining the numerical solution of the Landau-Lifshitz-Gilbert equation and the analysis of a generalized Thiele’s equation, that inertial effects are almost completely absent in skyrmion dynamics driven by a time-dependent current. In contrast, the response to time-dependent magnetic forces and thermal fluctuations depends strongly on frequency and is described by a large effective mass and a (anti-) damping depending on the acceleration of the skyrmion. Thermal diffusion is strongly suppressed by the cyclotron motion and is proportional to the Gilbert damping coefficient . This indicates that the skyrmion position is stable, and its motion responds to the time-dependent current without delay or retardation even if it is fast. These findings demonstrate the advantages of skyrmions as information carriers. PACS numbers: 73.43.Cd,72.25.-b,72.80.-r I. INTRODUCTION Mass is a fundamental quantity of a particle determining its mechanical inertia and therefore the speed of response to ex- ternal forces. Furthermore, it controls the strength of quantum and thermal fluctuations. For a fast response one usually needs small masses and small friction coefficients which in turn lead to large fluctuations and a rapid diffusion. Therefore, usually small fluctuations and a quick reaction to external forces are not concomitant. However a “particle” is not a trivial object in modern physics, it can be a complex of energy and mo- mentum, embedded in a fluctuating environment. Therefore, its dynamics can be different from that of a Newtonian parti- cle. This is the case in magnets, where such a “particle” can be formed by a magnetic texture1,2. A skyrmion3,4is a rep- resentative example: a swirling spin texture characterized by a topological index counting the number of times a sphere is wrapped in spin space. This topological index remains un- changed provided spin configurations vary slowly, i.e., dis- continuous spin configurations are forbidden on an atomic scale due to high energy costs. Therefore, the skyrmion is topologically protected and has a long lifetime, in sharp con- trast to e.g. spin wave excitations which can rapidly decay. Skyrmions have attracted recent intensive interest because of their nano-metric size and high mobility5–14. Especially, the current densities needed to drive their motion ( 106A/m2) are ultra small compared to those used to manipulate domain walls in ferromagnets ( 101112A/m2)15–19. The motion of the skyrmion in a two dimensional film can be described by a modified version of Newton’s equation. For sufficiently slowly varying and not too strong forces, a sym- metry analysis suggests the following form of the equations of motion, G_R+D_R+mR+R=Fc+Fg+Fth:(1) Here we assumed translational and rotational invariance of the linearized equations of motion. The ‘gyrocoupling’ G=G^e?is an effective magnetic field oriented perpendicular to the plane,is the (dimensionless) Gilbert damping of a sin- gle spin,Ddescribes the friction of the skyrmion, mits mass andRits centre coordinate. parametrizes a peculiar type of damping proportional to the acceleration of the particle. We name this term ‘gyrodamping’, since it describes the damping of a particle on a cyclotron orbit (an orbit with R/G_R), which can be stronger ( parallel to G) or weaker (antipar- allel to G) than that for linear motion. Our main goal will be to describe the influence of forces on the skyrmion arising from electric currents ( Fc), magnetic field gradients (Fg) and thermal fluctuations (Fth). By analyzing the motion of a rigid magnetic structure M(r;t) =M0(rR(t))forstatic forces, one can obtain analytic formulas for G;D;FcandFgusing the approach of Thiele19–22,24. In Ref. [25], an approximate value for the mass of a skyrmion was obtained by simulating the motion of a skyrmion in a nanodisc and by estimating contributions to the mass from internal excitations of the skyrmion. For rapidly changing forces, needed for the manipulation of skyrmions in spintronic devices, Eq. (1) is however not suffi- cient. A generalized version of Eq. (1) valid for weak but also arbitrarily time-dependent forces can be written as G1(!)V(!) =Fc(!) +Fg(!) +Fth(!) (2) =Sc(!)vs(!) +Sg(!)rBz(!) +Fth(!) HereV(!) =R ei!t_R(t)dtis the Fourier transform of the velocity of the skyrmion, vs(!)is the (spin-) drift velocity of the conduction electrons, directly proportional to the cur- rent,rBz(!)describes a magnetic field gradient in frequency space. The role of the random thermal forces, Fth(!), is spe- cial as their dynamics is directly linked via the fluctuation- dissipation theorem to the left-hand side of the equation, see below. The 22matrix G1(!)describes the dynam- ics of the skyrmion; its small- !expansion defines the terms written on the left-hand side of Eq. (1). One can expectarXiv:1501.00444v1 [cond-mat.str-el] 2 Jan 20152 FIG. 1: When a skyrmion is driven by a time dependent external force, it becomes distorted and the spins precess resulting in a de- layed response and a large effective mass. In contrast, when the skyrmion motion is driven by an electric current, the skyrmion ap- proximately flows with the current with little distortion and preces- sion. Therefore skyrmions respond quickly to rapid changes of the electric current. strongly frequency-dependent dynamics for the skyrmion be- cause the external forces in combination with the motion of the skyrmion can induce a precession of the spin and also ex- cite spinwaves in the surrounding ferromagnet, see Fig. 1. We will, however, show that the frequency dependence of the right-hand side of the Eq. (2) is at least as important: not only the motion of the skyrmion but also the external forces excite internal modes. Depending on the frequency range, there is an effective screening or antiscreening of the forces described by the matrices Sc(!)andSg(!). Especially for the current- driven motion, there will be for all frequencies an almost exact cancellation of terms from G1(!)andSc(!). As a result the skyrmion will follow almost instantaneously any change of the current despite its large mass. In this paper, we study the dynamics of a driven skyrmion by solving numerically the stochastic Landau-Lifshitz-Gilbert (LLG) equation. Our strategy will be to determine the param- eters of Eq. (2) such that this equation reproduces the results of the LLG equation. Section II introduces the model and outlines the numerical implementation. Three driving mecha- nisms are considered: section III studies the diffusive motion of the skyrmion due to thermal noise, section IV the skyrmion motion due to time-dependent magnetic field gradient and sec- tion V the current-driven dynamics. We conclude with a sum- mary and discussion of the results in Sec. VI. II. MODEL Our study is based on a numerical analysis of the stochastic Landau-Lifshitz-Gilbert (sLLG) equations27defined by dMr dt= Mr[Be+b (t)]  MMr(Mr[Be+b (t)]):(3) Here is the gyromagnetic moment and the Gilbert damp- ing;Be=H[M] Mris an effective magnetic field created by the surrounding magnetic moments and b (t)a fluctuating,stochastic field creating random torques on the magnetic mo- ments to model the effects of thermal fluctuations, see below. The Hamiltonian H[M]is given by H[M] =JX rMr
Mr+aex+Mr+aey
X r MrMr+aex
ex+MrMr+aey
ey
B
X rMr (4) We useJ= 1 , = 1 ,jMrj= 1 ,= 0:18Jfor the strength of the Dzyaloshinskii-Moriya interaction and B= (0;0;0:0278J)for all plots giving rise to a skyrmion with a radius of about 15lattice sites, see Appendix A. For this pa- rameter set, the ground state is ferromagnetic, thus the single skyrmion is a topologically protected, metastable excitation. Typically we simulate 100100spins for the analysis of dif- fusive and current driven motion and 200200spins for the force-driven motion. For these parameters lattice effects are negligible, see appendix B. Typical microscopic parameters used, areJ= 1meV (this yields Tc10K) which we use to estimate typical time scales for the skyrmion motion. Following Ref. 27, we assume that the field bfl r(t)is gen- erated from a Gaussian stochastic process with the following statistical properties bfl r;i(t) = 0 bfl r;i(t)bfl r0;j(s) = 2kBT Mijrr0(ts) (5) whereiandjare cartesian components and h:::idenotes an average taken over different realizations of the fluctuating field. The Gaussian property of the process stems from the in- teraction of Mrwith a large number of microscopic degrees of freedom (central limit theorem) which are also responsi- ble for the damping described by , reflecting the fluctuation- dissipation theorem. The delta-correlation in time and space in Eq. (5) expresses that the autocorrelation time and length of bfl r(t)is much shorter than the response time and length scale of the magnetic system. For a numerical implementation of Eq. (3) we follow Ref. 27 and use Heun’s scheme for the numerical integration which converges quadratically to the solution of the general system of stochastic differential equations (when interpreted in terms of the Stratonovich calculus). For static driving forces, one can calculate the drift veloc- ity_Rfollowing Thiele20. Starting from the Landau-Lifshitz Gilbert equations, Eq. (3), we project onto the translational mode by multiplying Eq. (3) with @iMrand integrating over space21–23. G=~M0Z dr n
(@xn@yn) D=~M0Z dr(@xn
@xn+@yn
@yn)=2 Fc=Gvs+Dvs; Fg=MsrB; M s=M0Z dr(1nz) (6)3 where nis the direction of the magnetization, M0the lo- cal spin density, vsthe (spin-) drift velocity of the conduc- tion electrons proportional to the electric current, and Ms is the change of the magnetization induced by a skyrmion in a ferromagnetic background. The ’gyrocoupling vector’ G= (0;0;G)TwithG=~M04is given by the winding number of the skyrmion, independent of microscopic details. III. THERMAL DIFFUSION Random forces arising from thermal fluctuations play a de- cisive role in controlling the diffusion of particles and there- fore also the trajectories R(t)of a skyrmion. To obtain R(t) and corresponding correlation functions we used numerical simulations based on the stochastic Landau-Lifshitz-Gilbert equation27. These micromagnetic equations describe the dy- namics of coupled spins including the effects of damping and thermal fluctuations. Initially, a skyrmion spin-texture is embedded in a ferromagnetic background. By monitoring the change of the magnetization, we track the center of the skyrmion R(t), see appendix A for details. Our goal is to use this data to determine the matrix G1(!) and the randomly fluctuating thermal forces, Fth(!), which together fix the equation of motion, Eq. (2), in the presence of thermal fluctuations ( rBz=vs= 0). One might worry that this problem does not have a unique solution as both the left-hand and the right-hand side of Eq. (2) are not known a priori. Here one can, however, make use of the fact that Kubo’s fluctuation-dissipation theorem26constraints the ther- mal forces on the skyrmion described by Fthin Eq. (2) by linking them directly to the dissipative contributions of G1. On averagehFth= 0i, but its autocorrelation is proportional to the temperature and friction coefficients. In general it is given by hFi th(!)Fj th(!0)i=kBT[G1 ij(!) +G1 ji(!)]2(!+!0): (7) For small ! one obtainshFx th(!)Fx th(!0)i = 4kBTD(!+!0)while off-diagonal correla- tions arise from the gyrodamping hFx th(!)Fy th(!0)i= 4i!kBT(!+!0). Using Eq. (7) and demand- ing furthermore that the solution of Eq. (2) reproduces the correlation function h_Ri(t)_Rj(t0)i(or, equivalently, h(Ri(t)Rj(t0))2i) obtained from the micromagnetic simulations, leads to the condition26 Gij(!) =1 kBTZ1 0(tt0)h_Ri(t)_Rj(t0)i (8) ei!(tt0)d(tt0): We therefore determine first in the presence of thermal fluc- tuations (rBz=vs= 0) from simulations of the stochastic LLG equation (3) the correlation functions of the velocities and use those to determine Gij(!)using Eq. (8). After a sim- ple matrix inversion, this fixes the left-hand side of the equa- tion of motion, Eq. (2), and therefore contains all information 0 5 10 15 20 25t ωp00.511.522.5 <ΔR2>α=0.01 α=0.05 α=0.1 α=0.15 α=0.2FIG. 2: Time dependence of the correlation function (Ri(t0+t)Ri(t0))2 forT= 0:1Jand different values of the Gilbert damping (!p=B= 0:0278Jis the frequency for cyclotron motion). on the (frequency-dependent) effective mass, gyrocoupling, damping and gyrodamping of the skyrmion. Furthermore, the corresponding spectrum of thermal fluctuations is given by Eq. (7). Fig. 2 showsh(
R)2it=h(Rx(t0+t)Rx(t0))2i. As expected, the motion of the skyrmion is diffusive: the mean squared displacement grows for long times linearly in time h(
R)2it= 2Dt, whereDis the diffusion constant. Usu- ally the diffusion constant of a particle grows when the fric- tion is lowered26. For the skyrmion the situation is opposite: the diffusion constant becomes small for the small friction, i.e., small Gilbert damping . This surprising observation has its origin in the gyrocoupling G: in the absence of friction the skyrmion would be localized on a cyclotron orbit. From Eq. (1), we obtain D=kBTD G2+ (D)2(9) The diffusion is strongly suppressed by G. As in most materi- alsis much smaller than unity while DG , the skyrmion motion is characterized both by a small diffusion constant and a small friction. Such a suppressed dynamics has also been shown to be important for the dynamics of magnetic vortices28. For typical parameters relevant for materials like MnSi we estimate that it takes 106sto105sfor a skyrmion to diffusive over an average length of one skyrmion diameter. To analyze the dynamics on shorter time scales we show in Fig. 3 four real functions parametrizing G1(!): a frequency- dependent mass m(!), gyrocouplingG(!), gyrodamping (!)and dissipation strength D(!)with G1(!) = D(!)i!m(!)G(!) +i!(!) G(!)i!(!)D(!)i!m(!) For!!0one obtains the parameters of Eq. (1). All pa- rameters depend only weakly on temperature, Gandmare ap- proximately independent of , while the friction coefficients4 0 1 2 3 4 5 ω / ωp04812 α thermal diffusion 0 1 2 3 4 5 ω / ωp00.51-G / 4 π 0 1 2 3 4 5 ω / ωp0100200 α Γ0 1 2 3 4 5 ω / ωp050100 m current driven motion force driven motion FIG. 3: Dissipative tensor D, massm, gyrocouplingGand gyro- dampingas functions of the frequency !for the diffusive motion atT= 0:1(solid lines). They differ strongly from the “apparent” dynamical coefficients (see text) obtained for the force driven (red dashed line) and current driven motion (green dot-dashed line). We use= 0:2,= 0:1. The error bars reflect estimates of systematic errors arising mainly from discretization effects, see appendix B. 00.05 0.1 0.15 0.2α01234 αT=0.15 T=0.2 00.05 0.1 0.15 0.2α00.51-G / 4 π 00.05 0.1 0.15 0.2α010203040 α Γ00.05 0.1 0.15 0.2α0255075100 mT=0.05 T=0.1 FIG. 4: Dissipative strength D, massm, gyrocouplingGand gy- rodampingas functions of the Gilbert damping for different temperatures T. Dandare linear in , see Fig. 4. In the limit T!0, G(!!0)takes the value4, fixed by the topology of the skyrmion15,20. Both the gyrodamping and and the effective mass m have huge numerical values. A simple scaling analysis of the Landau-Lifshitz-Gilbert equation reveals that both the gyro- couplingGandDare independent of the size of the skyrmion, while andmare proportional to the area of the skyrmion, and frequencies scale with the inverse area, see appendix B. For the chosen parameters (the field dependence is dis-cussed in the appendix B), we find m0:3N ipm0and 0:7N ipm0, wherem0=~2 Ja2is the mass of a sin- gle flipped spin in a ferromagnet ( 1in our units) and we have estimated the number of flipped spins, N ip, from the total magnetization of the skyrmion relative to the ferromagnetic background. As expected the mass of skyrmions grows with the area (consistent with an estimate29formobtained from the magnon spectrum of skyrmion crystals), the observation that the damping rate Dis independent of the size of skyrmions is counter-intuitive. The reason is that larger skyrmions have smoother magnetic configurations, which give rise to less damping. For realistic system parameters J= 1meV (which yields a paramagnetic transition temperature TC10K, but there are also materials, i.e. FeGe, where the skyrmion lattice phase is stabilised near room-temperature16) anda= 5 ˚A and a skyrmion radius of 200 ˚A one finds a typical mass scale of 1025kg. The sign of the gyrodamping is opposite to that of the gyrocouplingG. This implies that describes not damp- ing but rather antidamping: there is less friction for cyclotron motion of the skyrmion than for the linear motion. The nu- merical value for the antidamping turns out to be so large thatDm+ G<0. This has the profound consequence that the simplified equation of motion shown in Eq. (1) cannot be used: it would wrongly predict that some oscillations of the skyrmion are not damped, but grow exponentially in time due to the strong antidamping. This is, however, a pure artifact of ignoring the frequency dependence of G1(!), and such oscillations do not grow. Fig. 3 shows that the dynamics of the skyrmion has a strong frequency dependence. We identify the origin of this fre- quency dependence with a coupling of the skyrmion coordi- nate to pairs of magnon excitations as discussed in Ref. 31. Magnon emission sets in for ! > 2!pwhere!p=Bis the precession frequency of spins in the ferromagnet (in the pres- ence of a bound state with frequency !b, the onset frequency is!p+!b, Ref. 31). This new damping channel is most ef- ficient when the emitted spin waves have a wavelength of the order of the skyrmion radius. As a test for this mechanism, we have checked that only this high-frequency damping survives for !0. In Fig. 5 we show the frequency dependent damping D(!)for various bare damping coefficients . For small!it is proportional to as predicted by the Thiele equation. For !>2!p, however, an extra dampling mechanism sets in: the skyrmion motion can be damped by the emission of pairs of spin waves. This mechanism is approximately independent of and survives in the!0limit. This leads necessarily to a pronounced frequency dependence of the damping and therefore to the ef- fective mass m(!)which is related by the Kramers-Kronig relationm(!) =1 !R1 1D(!0) !0!d!0 toD(!). Note also that the largeindependent mass m(!!0)is directly related to theindependent damping mechanism for large !. Also the frequency dependence of m(!)andG(!)can be traced back to the same mechanism as these quantities can be computed fromD(!)and(!)using Kramers-Kronig relations. For large frequencies, the effective mass practically vanishes and5 0 1 2 3 4 5 ω/ωp05101520αD(ω)α=0.05 α=0.1 α=0.2 FIG. 5: Effective damping, D(!)for= 0:2,0:1and0:05. 0 1 2 3 4 5 ω / ωp-400-2000200400Mz tot600Re/Im SgRe Sg11 Re Sg21 Im Sg11 Im Sg21 FIG. 6: Dynamical coupling coefficients for the force driven motion (= 0:2). In the static limit everything but the real part of the diago- nal vanishes. R eS11 g(!)however approaches the total magnetization Mz totas expected. The error bars reflect estimates of systematic er- rors, see appendix B. the ‘gyrocoupling’ Gdrops by a factor of a half. IV . FORCE-DRIVEN MOTION Next, we study the effects of an oscillating magnetic field gradient rBz(t)in the absence of thermal fluctuations. As the skyrmion has a large magnetic moment Mz totrelative to the ferromagnetic background, the field gradient leads to a force acting on the skyrmion. In the static limit, the force is exactly given by Fg(!!0) =Mz totrBz: (10) Using G1(!)determined above, we can calculate how the effective force Sg(!)rBz(!)(see Eq. 2) depends on fre- quency. Fig. 6 shows that for !!0one obtains the expectedresultSg(!!0) =ijMz tot, while a strong frequency de- pendence sets in above the magnon gap, for !&!p. This is the precession frequency of spins in the external magnetic field. In general, both the screening of forces (parametrized bySg(!)) and the internal dynamics (described by G1(!)) determines the response of skyrmions, V(!) = G(!)Sg(!)rBz(!). Therefore it is in general not possi- ble to extract, e.g., the mass of the skyrmion as described by G1(!)from just a measurement of the response to field gra- dients. It is, however, instructive to ask what “apparent” mass one obtains, when the frequency dependence of Sg(!)is ig- nored. We therefore define the “apparent” dynamics G1 a(!) byGa(!)Sg(!= 0) = G(!)Sg(!). The matrix elements ofG1 a(!)are shown in Fig. 3 as dashed lines. The appar- ent mass for gradient-driven motion, for example, turns out to be more than a factor of three smaller then the value ob- tained from the diffusive motion clearly showing the impor- tance of screening effects on external forces. The situation is even more dramatic when skyrmions are driven by electric currents. V . CURRENT-DRIVEN MOTION Currents affect the motion of spins both via adiabatic and non-adiabatic spin torques30. Therefore one obtains two types of forces on the spin texture even in the static limit19–22,24. The effect of a time-dependent, spin-polarized current on the magnetic texture can be modelled by supplementing the right hand side of eq. (3) with a spin torque term TST, TST=(vs
r)Mr+ M[Mr(vs
r)Mr]:(11) The first term is called the spin-transfer-torque term and is derived under the assumption of adiabaticity: the conduction- electrons adjust their spin orientation as they traverse the mag- netic sample such that it points parallel to the local magnetic moment Mrowing toJHandJsd. This assumptions is justi- fied as the skyrmions are rather large smooth objects (due to the weakness of spin-orbit coupling). The second so called - term describes the dissipative coupling between conduction- electrons and magnetic moments due to non-adiabatic effects. Bothandare small dimensionless constants in typical ma- terials. From the Thiele approach one obtains the force Fc(!!0) =Gvs+Dvs: (12) For a Galilei-invariant system one obtains =. In this special limit, one can easily show that an exact solution of the LLG equations in the presence of a time-dependent current, described by vs(t)is given by M(rRt 1vs(t0)dt0)pro- vided, M(~ r)is a static solution of the LLG equation for vs= 0. This implies that for =, the skyrmion motion exactly follows the external current, _R(t) =vs(t). Using Eq. (2), this implies that for =one has G1(!) =Sc(!). Defin- ing the apparent dynamics, as above, Ga(!)Sc(!= 0) = G(!)Sc(!)one obtains a frequency independent G1 a(!) =6 0 1 2 3 4 5 ω / ωp-4 π-10-50β D(0)510Re/Im ScIm Sc11 Im Sc21Re Sc11 Re Sc21 FIG. 7: Dynamical coupling coefficients (symbols) for the current- driven motion ( = 0:2,= 0:1,J= 1,= 0:18J,B= 0:0278 ). These curves follow almost the corresponding matrix elements of G1(!)shown as dashed lines. A deviation of symbols and dashed line is only sizable for Re S11 c. 0 1 2 3 4 5 ω/ωp-5051015 mα=0.2,β=0 α=0.2,β=0.1 0 1 2 3 4 5 ω/ωp0510 α Γα=0.2,β=0.15 α=0.2,β=0.19 α=0.2,β=0.3 FIG. 8: Mass m(!)and gyrodamping (!)as functions of the driving frequency !for the current-driven motion. Note that both M andvanish for=. Sc(!= 0) =D 1iyG: the apparent effective mass and gyrodamping are exactly zero in this limit and the skyrmion follows the current without any retardation. For 6=, the LLG equations predict a finite apparent mass. Numerically, we find only very small apparent masses, ma c/, see dot-dashed line in upper-right panel of Fig. 3, where the case = 0:2,= 0:1is shown. This is anticipated from the anal- ysis of the=case: As the mass vanishes for == 0, it will be small as long as both andare small. Indeed even for6=this relation holds approximately as shown in Fig. 7. The only sizable deviation is observed for Re S11 c for which the Thiele equation predicts Re S11 c(!!0) =D while Re G111(!!0) =Das observed numerically. A better way to quantify that the skyrmion follows the cur-rent even for 6=almost instantaneously is to calculate the apparent mass and gyrodamping for current driven mo- tion, where only results for = 0:2and= 0:1have been shown. As these quantities vanish for =, one can ex- pect that they are proportional to at least for small ;. This is indeed approximately valid at least for small frequen- cies as can be seen from Fig. 8. Interestingly, one can even obtain negative values for  >  (without violating causal- ity). Most importantly, despite the rather large values for  andused in our analysis, the apparent effective mass and gyrodamping remain small compared to the large values ob- tained for force-driven motion or the intrinsic dynamics. This shows that retardation effects remain tiny when skyrmions are controlled by currents. VI. CONCLUSIONS In conclusion, we have shown that skyrmions in chiral mag- nets are characterised by a number of unique dynamical prop- erties which are not easily found in other systems. First, their damping is small despite the fact that skyrmions are large composite objects. Second, despite the small damping, the diffusion constant remains small. Third, despite a huge iner- tial mass, skyrmions react almost instantaneously to external currents. The combination of these three features can become the basis for a very precise control of skyrmions by time- dependent currents. Our analysis of the skyrmion motion is based on a two- dimensional model where only a single magnetic layer was considered. All qualitative results can, however, easily be generalized to a film with NLlayers. In this case, all terms in Eq. (1) get approximately multiplied by a factor NLwith the exception of the last term, the random force, which is en- hanced only by a factorpNL. As a consequence, the diffu- sive motion is further suppressed by a factor 1=pNLwhile the current- and force-driven motion are approximately unaf- fected. An unexpected feature of the skyrmion motion is the an- tidamping arising from the gyrodamping. The presence of antidamping is closely related to another important property of the system: both the dynamics of the skyrmion and the ef- fective forces acting on the skyrmion are strongly frequency dependent. In general, in any device based on skyrmions a combination of effects will play a role. Thermal fluctuations are always present in room-temperature devices, the shape of the device will exert forces13,14and, finally, we have identified the cur- rent as the ideal driving mechanism. In the linear regime, the corresponding forces are additive. The study of non-linear effects and the interaction of several skyrmions will be impor- tant for the design of logical elements based on skyrmions and this is left for future works. As in our study, we expect that dynamical screening will be important in this regime.7 30 40 50 60 70 30 40 50 60 70 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 FIG. 9: Skyrmion density based on the normalized z-component of the magnetization. Acknowledgments The authors are greatful for insightful discussions with K. Everschor and Markus Garst. Part of this work was funded through the Institutional Strategy of the University of Cologne within the German Excellence Initiative” and the BCGS. C.S. thanks the University of Tokyo for hospitality during his re- search internship where part of this work has been performed. N.N. was supported by Grant-in-Aids for Scientific Research (No. 24224009) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and by the Strategic International Cooperative Program (Joint Research Type) from Japan Science and Technology Agency. J.I. is sup- ported by Grant-in-Aids for JSPS Fellows (No. 2610547). Appendix A: Definition of the Skyrmion’s centre coordinate In order to calculate the Green’s function, Eq. (3), one needs to calculate the velocity-velocity correlation function. Therefore it is necessary to track the skyrmion position throughout the simulation. Mostly two methods have been used so far for this25: (i) tracking the centre of the topological charge and (ii) tracking the core of the Skyrmion (reversal of magnetization). The topological charge density top(r) =1 4^ n(r)
(@x^ n(r)@y^ n(r)) (A1) integrates to the number of Skyrmions in the system. There- fore for our case of a single Skyrmion in the ferromagnetic background this quantity is normalized to 1. The center of topological charge can therefore be defined as R=Z d2rtop(r)r (A2) For the case of finite temperature this method can, however, not be used directly. Thermal fluctuations in the ferromagnetic background far away from the skyrmion lead to a large noise to this quantity which diverges in the thermodynamic limit. A similar problem arises when tracking the center using the magnetization of the skyrmion.One therefore needs a method which focuses only on the region close to the skyrmion center. To locate the skyrmion, we use thez-component of the magnetization but take into ac- count only points where Mz(r)<0:7(the magnetization of the ferromagnetic background at T= 0is+1). We therefore use (r) = (1Mz(r)) [Mz(r)0:7] (A3) where [x]is the theta function. A first estimate, Rest=RV, for the radius is obtained from RA=R Ar(r)d2rR A(r)d2r(A4) by integrating over the full sample volume V.Restis noisy due to the problems mentioned above but for the system sizes simulated one nevertheless obtains a good first esti- mate for the skyrmion position. This estimate is refined by using in a second step for the integration area only D= r2R2jjrRestj<r whereris choosen to be larger than the radius of the skyrmion core (we use r= 1:3p N<=, whereN<is the number of spins with Mz<0:7). Thus we obtain a reliable estimate, R=RD, not affected by spin fluctuations far away from the skyrmion. From the resulting R(t), one can obtain the velocity-velocity correlation function hVi(t0+t)VJ(t0)i. Appendix B: Scaling invariance To obtain analytic insight into the question of how param- eters depend on system size and to check the numerics for lattice artifacts it is useful to perform a scaling analysis of the sLLG equations and the effective equations of motion for the skyrmion. We investigate a scaling transformation, where the radius of the skyrmion is enlarged by a factor ,M(r)!~M(r) = M(r=). For the scaling analysis, we use the continuum limit of Eq. (4) (setting a= 1) H[M] =Z d2rJ 2(rM)2+M
r MB
M The three terms scale with 0,and2, respectively. To ob- tain a larger skyrmion, we therefore have to rescale != andB!B=2. This implies that the Beterm in the sLLG equation scales with 1=2and therefore also the time axis has to be rescaled, t!2t, implying that all time scales are a factor of2longer and all frequencies a factor 1=2smaller. Similiarly, the driving current, i.e. vsis reduced by a factor 1while the temperature remains unscaled. This implies that when M(r;t)is a solution for a given value of ,Bandvs andG(!)the corresponding velocity-correlation function of the skyrmion, then M(r=;t=2)is a solution for =,B=2, vs=with correlation function G(!2). Accordingly, the !!0limit and therefore the gyrocou- plingG, the friction constant Dand the diffusion constant of8 0.03 0.035 0.04 Bz00.511.5 α D 0.03 0.035 0.04 Bz00.51-G / 4 π 0.03 0.035 0.04 Bz05101520 α Γ0.03 0.035 0.04 Bz020406080 m FIG. 10: Field dependence of the dissipative tensor D, the massm, the gyrocouplingGand the gyrodamping forJ= 1,= 0:18J. The main effect is that both mandshrink when the size of the skyrmion shrinks with increasing Bz. the skyrmion are independent of, consistent with the analyt- ical formulas eq. (6). In contrast, the mass of the skyrmion, m, and the gyrodamping scale with2. They are therefore proportional to the number of spins constituting the skyrmion consistent with our numerical findings. When one does, how- ever, change the external magnetic field for fixed strength of the DM interaction, the internal structure of the skyrmion and therefore also G(!)change quantitatively in a way not predictable by scaling. In Fig. 10 we therefore show the B-dependence of D,m,Gand. For increasing Bthe size of the skyrmion shrinks. From our previous analysis, it is there- fore not surprising that both the effective mass mand the gy- rodampingshrink substantially while DandGare less affected . Quantitatively, this change of mandis how- ever not proportional to the number of spins constituting the skyrmion. We have tested numerically the scaling properties for = 1:2and find that all features are quantitatively reproduced. Small variations on the level of a few percent do, however, occur reflecting the typical size of features arising from the discretization of the continuum theory. A conservative esti- mate of such systematic discretization effects for the diffusive motion is given by the error bars in Figs. 3 (all statistical er- rors are smaller than the thickness of the line). For the field- driven motion (Fig. 3 and Fig. 6) spatial discretization effects lead to a different source of errors. For very small field gradi- ents and high frequencies the displacement of the skyrmion is much smaller than the lattice spacing and the response is af- fected by a tiny pinning of the skyrmion to the discreet lattice. For larger gradients, however, nonlinear effects set in and for small frequencies the skyrmion starts to approach the edge of the simulated area. In Fig. 3, we therefore used for the force- driven motionrB= 0:0005 for!<2!pandrB= 0:0015 for! >2!p. Error bars have been estimated from variations of the numerical values when rBwas varied from 0:0001 to 0:0015 . For the current-driven motion errors are so tiny that they are not shown. Electronic address: nagaosa@riken.jp 1Hubert, A. & Sch ¨afer, R. Magnetic Domains: The Analysis of Magnetic Microstructures (Springer, Berlin, 1998). 2Malozemoff, A. P. & Slonczewski, J.C. Magnetic Domain Walls in Bubble Materials (Academic Press, New York, 1979). 3Skyrme, T. H. R. A Non-Linear Field Theory. Proc. Roy. Soc. London A 260, 127–138 (1961). 4Skyrme, T. H. R. A unified field theory of mesons and baryons. Nuc. Phys. 31,556–569 (1962). 5Bogdanov, A. N. & Yablonskii, D. A. Thermodynamically stable ”vortices” in magnetically ordered crystals. The mixed state of magnets. Sov. Phys. JETP 68,101–103 (1989). 6M¨uhlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919 (2009). 7Yu, X. Z. et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901–904 (2010). 8Heinze, S. et al. Spontaneous atomic-scale magnetic skyrmion lat- tice in two dimensions. Nature Phys. 7,713–718 (2011). 9Seki, S., Yu, X. Z., Ishiwata, S. & Tokura, Y . Observation of skyrmions in a multiferroic material. Science 336, 198–201 (2012). 10Fert, A., Cros, V . & Sampaio, J. Skyrmions on the track. Nature Nanotech. 8,152–156 (2013). 11Lin, S. -Z., Reichhardt, C., Batista, C. D. & Saxena, A. Driven skyrmions and dynamical transitions in chiral magnets. Phys. Rev. Lett. 110, 207202 (2013). 12Lin, S. -Z., Reichhardt, C., Batista, C. D. & Saxena, A. Particle model for skyrmions in metallic chiral magnets: Dynamics, pin-ning, and creep. Phys. Rev. B 87,214419 (2013). 13Iwasaki, J., Mochizuki, M. & Nagaosa, N. Current-induced skyrmion dynamics in constricted geometries. Nature Nanotech. 8,742–747 (2013). 14Iwasaki, J., Koshibae, W. & Nagaosa, N. Colossal spin transfer torque effect on skyrmion along the edge. Nano letters 14 (8), 4432–4437 (2014). 15Jonietz, F. et al. Spin transfer torques in MnSi at ultralow current densities. Science 330, 1648–1651 (2010). 16Yu, X. Z. et al. Skyrmion flow near room temperature in an ul- tralow current density. Nat. Commun. 3,988 (2012). 17Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain-wall racetrack memory. Science 320, 190–194 (2008). 18Yamanouchi, M., Chiba, D., Matsukura, F. & Ohno, H. Current- induced domain-wall switching in a ferromagnetic semiconductor structure. Nature 428, 539–542 (2004). 19Schulz, T. et al. Emergent electrodynamics of skyrmions in a chi- ral magnet. Nature Phys. 8,301–304 (2012). 20Thiele, A. A. Steady-state motion of magnetic domains. Phys. Rev. Lett. 30,230–233 (1973). 21Everschor, K. et al. Current-induced rotational torques in the skyrmion lattice phase of chiral magnets. Phys. Rev. B 86,054432 (2012). 22Everschor, K. et al. Rotating skyrmion lattices by spin torques and field or temperature gradients. Phys. Rev. B 86,054432 (2012). 23He, J. et al. Current-driven vortex domain wall dynamics by mi- cromagnetic simulations. Phys. Rev. B 73,184408 (2006). 24Iwasaki, J., Mochizuki, M. & Nagaosa, N. Universal current-9 velocity relation of skyrmion motion in chiral magnets. Nat. Com- mun. 4,1463 (2013). 25Makhfudz, I., Kr ¨uger, B. & Tchernyshyov, O. Inertia and chiral edge modes of a skyrmion magnetic bubble. Phys. Rev. Lett. 109, 217201 (2012). 26Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 29,255–284 (1966). 27Garc ´ıa-Palacios, J. L. and L ´azaro, F. J. Langevin-dynamics study of the dynamical properties of small magnetic particles. Phys. Rev. B58,14937–14958 (1998).28Clarke, D. J. and Tretiakov, O. A. and Chern, G.-W. and Bazaliy, Y . B. and Tchernyshyov, O., Phys. Rev. B 78, 134412 (2008). 29Petrova, O. and Tchernyshyov O., Phys. Rev. B 84, 214433 (2011). 30Tatara, G., Kohno, H., & Shibata, J. Microscopic approach to current-driven domain wall dynamics. Physics Reports 468, 213– 301 (2008). 31Sch¨utte, C. and Garst, M. Magnon-skyrmion scattering in chiral magnets. arxiv 1405.1568 (2014).

2015-01-02

Skyrmions recently discovered in chiral magnets are a promising candidate for magnetic storage devices because of their topological stability, small size ($\sim 3-100$nm), and ultra-low threshold current density ($\sim 10^{6}$A/m$^2$) to drive their motion. However, the time-dependent dynamics has hitherto been largely unexplored. Here we show, by combining the numerical solution of the Landau-Lifshitz-Gilbert equation and the analysis of a generalized Thiele's equation, that inertial effects are almost completely absent in skyrmion dynamics driven by a time-dependent current. In contrast, the response to time-dependent magnetic forces and thermal fluctuations depends strongly on frequency and is described by a large effective mass and a (anti-) damping depending on the acceleration of the skyrmion. Thermal diffusion is strongly suppressed by the cyclotron motion and is proportional to the Gilbert damping coefficient $\alpha$. This indicates that the skyrmion position is stable, and its motion responds to the time-dependent current without delay or retardation even if it is fast. These findings demonstrate the advantages of skyrmions as information carriers.

Inertia, diffusion and dynamics of a driven skyrmion

1501.00444v1

arXiv:1805.06535v3 [math.AP] 28 Nov 2018Stabilization rates for the damped wave equation with H¨ older-regular damping Perry Kleinhenz Abstract We study the decay rate of the energy of solutions to the damped w ave equation in a setup where the geometric control condition is violated. We cons ider damping coefficients which are 0 on a strip and vanish like polynomials, xβ. We prove that the semigroup cannot be stable at rate faster than 1 /t(β+2)/(β+3)by producing quasimodes of the associated stationary damped wave equation. We also prove that the semigroup is stable at rate at least as fast as 1 /t(β+2)/(β+4). These tworesults establish an explicit relation between the rate of vanishing of the damping and rate of de cay of solutions. Our result partially generalizes a decay result of Nonnemacher in whic h the damping is an indicator function on a strip. 1 Introduction LetM= (M,g) be a Riemannian manifold. Fix some W∈L∞(M),W≥0. We study the asymptotic behavior as t→ ∞of solutions to the damped wave equation ∂2 tu−∆u+W(x)∂tu= 0 inM×R+ (u,∂tu)|t=0= (u0,u1) inM u|∂M= 0 if ∂M/ne}ationslash=∅.(1) The quantity of particular interest is the energy E(u,t) =1 2/parenleftBig ||∇u(·,t)||2 L2+||∂tu(·,t)||2 L2/parenrightBig . (2) In this paper we will work on the square M= [−b,b]×[−b,b] or torusM=T2, which we parametrize by ( x,y). We will give detailed proofs in the case of the square then s how how those results can be applied to the torus. For some fixed β≥0 anda,σ>0, such that a+σ<b, we study damping W∈L∞(M) of the form W(x,y) = 0 |x|<a (|x|−a)βa<|x|<a+σ c(|x|)>0a+σ<|x|<b,(3) 1In particular note that the damping is invariant in the ydirection. Remark. The particular form of c(x) does not affect our result so long as c(x)∈ L∞(a+σ,b) and it is uniformly bounded away from 0. However choosing a csuch thatW is smooth for |x|>aandW=C >0 for|x|>a+2σis perhaps the most interesting case in the context of existing results. Definition 1. Letf(t) be a function such that f(t)→0 ast→ ∞. We say that (1) is stable at rate f(t) if there exists a constant C >0 such that for all ( u0,u1)∈(H2(M)∩ H1 0(M))×H1 0(M) (orH2(M)×H1(M) if∂M=∅) ifusolves (1) with ( u0,u1) as Cauchy data then E(u,t)≤Cf(t)2/parenleftBig ||u0||2 H2(M)+||u1||2 H1(M)/parenrightBig for allt>0. Our main result is Theorem 1.1. For allε>0, withWas in(3)the equation (1)cannot be stable at rate t−β+2 β+3−ε. More precisely if we use m1(t) to denote the best possible ffor which definition 1 holds then this result along with Lemma 4.6 of [AL+14] and Proposition 3 of [BD08] show that m1(t)≥C/(1+t)β+2 β+3for someC >0, where we use the notation of [BD08]. We also show that Theorem 1.2. ForWas in(3)withβ >0the equation (1)is stable at rate t−β+2 β+4. Again using Lemma 4.6 of [AL+14] and Proposition 3 of [BD08] this shows m1(t)≤ C/(1+t)β+2 β+4for someC >0. The decay of energy of the damped wave equation is a well studi ed question. The strongestpossibledecay isuniformstabilization, whichi sdefinedastheexistence of F(t)→ 0 ast→ ∞such that, for all usolving (1) with initial data in H1(M)×L2(M) E(u,t)≤F(t)E(u,0), t≥0. It was established by [RT75] that uniform stabilization occ urs withF=Ce−κtfor someκ,C >0, when∂M=∅,W∈C0(M) and supp Wsatisfies the geometric control condition (GCC). We recall that a set Usatisfies the GCC if there exists T >0 such that for every geodesic γonMof lengthT,γ∩U/ne}ationslash=∅. The reverse implication, that uniform stabilization with any Fimplies supp Wsatisfies the GCC, was shown in [Ral69]. These results were extended to the case M/ne}ationslash=∅by [CBR92] and [BG97]. This in turn guarantees that when uniform stabilization occurs one can always let F=Ce−κtfor someκ,C >0. For a finer discussion of when uniform stabilization occurs f orL∞damping see [BG18]. 2A natural next question to ask is what occurs when the GCC does not hold for supp W. Because of the necessity of the GCC for uniform stabilizatio n, as soon as it does not hold we must adjustthe kindof decay wehopefor. Thenextbest thin gis thestability definedin Definition 1, which comes from [Leb96] and requires initial d ata with an additional spatial derivative. In [Bur98], the author showed that the energy of a solution to (1) decays at least logarithmically ( f(t) = 1/log(2 +t)) as soon as the damping W(x)≥c >0 on some open, nonempty set. In [Leb96] the author gave explicit exam ples of domains on which f(t) = 1/log(2+t) is the exact decay rate, in particular when M=S2and{W >0}does not intersect a neighborhood of the equator. For related wor k see also [LR97]. In the case of the square when the damped region contains a ver tical strip, [LR05] established a decay rate of f(t) = (log(t)/t)1/2. This was expanded to the case of partially rectangular domains when {W >0}contains a neighborhood of the nonrectangular part in [BH07]. Additionally in [BH07] a relation between vanish ing rate of the damping and decay rate for the damped wave equation was established. These results were refined by [AL+14]. The authors established a decay rate of f(t) = 1/t1/2for the damped wave equation in a more general setting, so lon g as the associated Schr¨ odinger equation is controllable. This includes the c ase of not identically vanishing damping on the 2 dimensional square (or torus) as a consequen ce of [Jaf90] (resp. [Mac10], [BZ12]). Continuing in the case of the 2 dimensional square [AL+14] also show that the system can not be stable at rate f(t) = 1/t1+εfor anyε>0, when{W >0}does not satisfy GCC, (this condition is referred to as the GCC being strongly viol ated). They further show the existence of a smooth damping coefficient which strongly viol ates the GCC for which the energy decays at rate f(t) = 1/t1−εfor anyε>0. In an appendix to [AL+14], Nonnenmacher shows that when the damping is the in- dicator function on a strip on the square or torus the system c annot be stable at rate f(t) = 1/t2/3+ε, for anyε>0. The complementary result was shown in [Sta17] to estab- lishf(t) = 1/t2/3as the exact rate of decay when damping is a strip on the square or torus. The difference in behavior between the smooth and discontinuo us damping led the authors of [AL+14] to pose the problem of establishing an explicit relation between the vanishing rate of the damping and the decay rate. Anexplicitrelation wasestablishedby[LL17]inaslightly differentsetting. Theauthors study the case of a general manifold in which the damping is su pported everywhere but a flat subtorus. In the 2 dimensional case this is an example of t he GCC not holding but also not being strongly violated. The damping is required to be in variant along this subtorus and must satisfy W(x)≤C|x|βnear where it vanishes. When this is the case the authors show that the system cannot be stable at rate f(t) = 1/tβ+2 β+ε, for anyε >0. They also show that if the vanishing behavior of the damping is further limited toC−1 1|x|β≤W(x)≤ C1|x|βthe system is stable at exactly the rate f(t) = 1/tβ+2 β(See also [BZ15]). 3Note that in [LL17] decreasing βcorrespondsto faster vanishing(i.e. less regular damp- ing) which produces faster decay, which is counter to the beh avior exhibited in [AL+14], [BH07] and our own result, namely that faster vanishing (i.e . less regular damping) pro- duces slower decay. However the dynamics in 2 dimensions in t he two situations are different, with only one undamped orbit in the former as oppose d to a whole family in the latter. Our paper provides a partial answer to the problem posed by [A L+14]. We establish an explicit relation between the rate of vanishing of the dampi ng and the stability rate of the system, in a case where the GCC is strongly violated on the squ are orT2. Our work also partially extends that of Nonnenmacher in the appendix to [A L+14], which agrees with our first theorem when β= 0, although that result follows from the existence of modes of the stationary equation where we produce quasimodes. Our two results provide further evidence for the fact, discussed in [AL+14], [LL17], [BH07], that once the support of the damping is fixed the rate of vanishing of the damping is the mos t significant feature when determining the decay rate. We note that our second result improves that of Theorem 1.2 of [BH07], which gives a decay rate of f(t) = 1/tβ/(β+4)(see also [AL+14] Theorem 2.6). We also note that there is a gap between our two results. Closing this gap would be an i nteresting area for further work. In the next section we outline the proof of Theorem 1.1. Secti ons 3, 4 and 5 contain the details of the proof. Section 6 contains the proof of Theo rem 1.2. Acknowledgements The author would like to thank Jared Wunsch for his advice and guidance. The author would also like to thank Matthieu L´ eautaud for comments on an early manuscript. The author would also like to thank the r eferees for their prompt, detailed and constructive comments. This research was supp orted in part by the National Science Foundation grant ”RTG: Analysis on manifolds” at No rthwestern University. 2 Outline of Proof of Theorem 1.1 To prove Theorem 1.1 we rely on the following result from [AL+14] (Proposition 2.4) and [BT10] which relates energy decay of the damped wave equatio n to resolvent estimates of thestationary dampedwave equation. Withthisresultitiss ufficienttoproducesufficiently good quasimodes, to do so we reduce the problem to one dimensi on and then the interval [0,b]. We will then show that we can use solutions to a related prob lem, the complex absorbing potential, on the half line to produce the desired quasimodes. We finally show that suchsolutionsof thecomplex absorbingpotential prob lemexistbyproducingsolutions on (0,a) and (a,∞) separately, the latter following from a rescaling argumen t, we are able to glue these solutions together via a compatibility condit ion which we satisfy via the implicit function theorem. 4Proposition 2.1. Fixα, if there exist sequences {qj} ∈C,{uj} ∈H2(M)∩H1 0(M),(or H2(M)if∂M=∅) and some j0∈N, such that for all j >|j0| /vextendsingle/vextendsingle/vextendsingle/vextendsingle−∆uj+iqjW(x)uj−q2 juj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(M)≤C |Re(qj)|2/α/parenleftBig ||uj||2 H1(M)+|qj|2||uj||2 L2(M)/parenrightBig ,(4) and |qj| → ∞,|Im(qj)| ≤C |Re(qj)|1/α, (5) then for all ε>0the system (1)is not stable at rate 1/tα+ε. Remark. Although Proposition 2.1 has ||uj||2 H1on the right hand side of (4) the quasimodes ujwe will apply it to satisfy a stronger inequality, namely /vextendsingle/vextendsingle/vextendsingle/vextendsingle−∆uj+iqjW(x)uj−q2 juj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(M)≤C |Re(qj)|2||uj||2 L2(M). (6) ThestrengthoftheconclusionweobtainfromapplyingPropo sition2.1tothesequasimodes is instead limited by the qjfor which we have such an estimate due to (5). Notethatproducingsequences qjandujwhichsatisfythehypothesesofthisproposition withα=β+2 β+3proves Theorem 1.1. We will make two simplifications before proceeding. First we will reducethe problem to obtaining quasimodes of an ordinary differential equation on [−b,b]. We will then further restrict our attention to the same equation on [0 ,b]. After making these simplifications we will introduce three key parameters and the complex absor bing potential problem on (0,∞), solutions of which we will use to produce our desired quasi modes. For the first simplification note that for any sequence of inte gersmjif/tildewideujis a sequence of functions on [ −b,b] which satisfy /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2 x/tildewideuj+iqjW/tildewideuj+/parenleftbigg 4π2m2 j b2−q2 j/parenrightbigg /tildewideuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(−b,b)≤C |Re(qj)|2||/tildewideuj||2 L2(−b,b)asj→ ∞ /tildewideuj(x) = 0|x|=b, (7) thenuj(x,y) =/tildewideuj(x)sin/parenleftBig2πmjy b/parenrightBig satisfy (6). Therefore it is enough for us to find functions which satisfy (7) with qjwhich satisfy (5) with α=β+2 β+3. The second simplification we make is to limit our attention to [0,b] from [−b,b]. Since our damping is even, if we find integers mjand functions /tildewideujon [0,b] which satisfy /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2 x/tildewideuj+iqjW/tildewideuj+/parenleftbigg 4π2m2 j b2−q2 j/parenrightbigg /tildewideuj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)≤C |Re(qj)|2||/tildewideuj||2 L2(0,b)asj→ ∞ /tildewideuj(x) = 0x=b /tildewideuj(0) = 0 or/tildewideu′ j(0) = 0(8) 5we can extend the /tildewideujto−b≤x<0 by setting/tildewideuj(−x) =−/tildewideuj(x) (or/tildewideuj(−x) =/tildewideuj(x) resp.) and the resulting functions satisfy (7). Therefore it is eno ugh for us to find functions which satisfy (8) with qjwhich satisfy (5) with α=β+2 β+3. Before we introduce the complex absorbing potential we intr oduce three new parame- ters. Leth∈[0,1) be a small parameter which will be sent to 0 and haveb 2πh2∈N. Let lbe a bounded parameter, when working in the case u(0) = 0 we impose l∈Zand when u′(0) = 0 we impose l+1 2∈Zbut otherwise leave lfree in relation to h. We define λh=πlh a+Chh(β+4)/(β+2)Ch=Oh(1)∈C. (9) We will eventually specify Chmore completely in Section 4. When appropriate we will refer to sequences of these parameters as hj,λhj,ljandChj, where we emphasize that λhj andChjdepend onhj. Now we introduce the complex absorbing potential problem on (0,∞) /braceleftBigg 0 =−h2∂2 xv+i(x−a)β +v−λ2 hv v(0) = 0 orv′(0) = 0.(10) In order to relate this to (8) we make an ansatz for relations b etween the parameters. Ifvjare a sequence of solutions of (10) for some hj,lj,λhj,Chj, we define qjandmjas follows mj=b 2πh2 j∈N (11) qj=1 h2 j+λ2 hj 2=1 h2 j+π2l2 jh2 j a2+2Chjπlj ah(2β+6)/(β+2) j +C2 hjh(2β+8)/(β+2) j . Note that in this regime Re(qj) =1 h2 j+O(h2 j) Im(qj) =2Im(Chj)πlj ah(2β+6)/(β+2) j +O(h(2β+8)/(β+2) j ), so qj→ ∞and|Im(qj)| ≤C |Re(qj)|(β+3)/(β+2)asj→ ∞. As we will see shortly, solutions of (10) in this regime satis fy the inequality in (8) but not necessarily the boundary condition at x=b. In order to ensure they do we multiply these solutions by a cutoff function which is 0 in a neighborho od ofb. We will see the resulting functions still satisfy the inequality in (8) as t he solutions of (10) in this regime have rapid decay on the support of the potential (see Lemma 3. 1), which is exactly where errors introduced by the cutoff function appear. 6Remark We note also that it is because we are working with solutions t o (10) on the half-line that we will only produce quasimodes rather than r eal modes. It is necessary for us to work on the half-line in order to perform a rescaling tha t allows us to set h= 0, if we were working on a finite interval the rescaling would make the interval depend on hwhich our approach is not well adapted to address. Fixδ>0 such that a+σ<b−2δ; we define φ∈C∞(0,∞) to satisfy φ(x) =/braceleftBigg 1x<b−2δ 0x>b−δ(12) Proposition 2.2. FixM >0, let{vj} ∈H2(0,∞)be a sequence of solutions of (10)with eigenvalues λhj=πljhj a+Chjh(β+4)/(β+2) j, Chj=O(1)∈C, where|lj| ≤Mandhj→0asj→ ∞andb 2πh2 j∈N. Set uj(x) =φ(x)vj(x). Then forjlarge enough so that hj<σβ/2the functions ujwithqj,mjas defined in (11) satisfy(8)and(5). It remains to be seen that we can indeed find solutions to the co mplex absorbing potential problem with eigenvalues of this form. Theorem 2.3. For alll∈Z,(orl+1 2∈Z), there exists h0>0andK >0such that for allh∈(0,h0), there exists a Chwith|Ch|<Kandv∈H2(0,∞)∩H1 0(0,∞),v/ne}ationslash= 0(resp. H2(0,∞)) satisfying (10)withv(0) = 0(resp.v′(0) = 0) withλgiven by (9). Using Theorem 2.3 we obtain a sequence {vj}of solutions of (10) which satisfy the hy- potheses of Proposition 2.2 which in turn produces sequence s which satisfy the hypotheses of Propositions 2.1 which in turn proves Theorem 1.1. Remark. It is straightforward to extend these results to the case M=T2. We parametrize T2by [−b,b]×[−b,b] with parallel edges identified. Thus it is enough to show that the quasimodes we produced on the square satisfy period ic boundary conditions and are thus functions on the torus. Our quasimodes are of the for m u(x,y) =vj(x)φ(x)sin/parenleftbigg2πmjy b/parenrightbigg , so it is straightforward to see they satisfy periodic bounda ry conditions in yandx(as u(x,y)≡0 for|x−b|<δand|b+x|<δ). We will prove Proposition 2.2 in Section 3, we will then prove Theorem 2.3 in Section 4. We prove a necessary estimate in Section 5. We finally prove Theorem 1.2 in Section 6. 73 Proof of Proposition 2.2 We begin by stating an estimate necessary for the proof. Lemma 3.1. Letv∈H1(0,∞)be a solution of (10)with eigenvalue λ=O(h)and letφ be as in (12). Fixs∈Rthen forh<σβ/2for allNthere exists CN,s>0such that ||φv||2 Hs h(a+σ,b)≤CN,shN||φv||2 L2(0,b). (13) This will be proved in Section 5 using the semiclassical elli pticity of −h2∂2 x+i(x−a)β +− λ2 hon (a+σ/4,b). Proof of Proposition 2.2. We have a sequence vjof solutions of 0 =−h2 j∂2 xvj+i(x−a)β +vj−λ2 hjvj, x∈(0,∞), with λhj=πlhj a+O/parenleftBig h(β+4)/(β+2) j/parenrightBig , hj→0 asj→ ∞. It is clear that uj=φvjhasuj(b) =φ(b)vj(b) = 0. Recalling (11) and the subsequent discussionqj,mjsatisfy (5). It remains to be seen that ujsatisfies the inequality in (8). By (11) and (10) ujsatisfies −∂2 xuj+iqjW(x)uj+/parenleftBigg 4π2m2 j b2−q2 j/parenrightBigg uj =φ/parenleftBigg iλ2 hj 2(x−a)β +vj−λ4 hj 4vj/parenrightBigg −φ′′vj−2φ′v′ j+iqj/parenleftBig W(x)−(x−a)β +/parenrightBig φvj. Thus /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2 xuj+iqjW(x)uj+/parenleftBigg 4π2m2 j a2−q2 j/parenrightBigg uj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)≤λ4 hj 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(x−a)β +uj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)+λ8 hj 16||uj||2 L2(0,b)+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′vj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b) +4/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′v′ j/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)+|qj|2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(W−(x−a)β +)φvj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b). Sinceb−δ>a+σ, by Lemma 3.1 for any N >0 /vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′′vj/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)+/vextendsingle/vextendsingle/vextendsingle/vextendsingleφ′v′ j/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)≤C h2||φvj||H2 h(b−δ,b)≤CNhN||φvj||2 L2(0,b). Furthermore by Lemma 3.1 for any N >0 /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(W−(x−a)β +)φvj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)≤C||φvj||2 L2(a+σ,b)dx≤CNhN||φvj||2 L2(0,b). 8Therefore /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−∂2 xuj+iqj(x−a)β +uj+/parenleftBigg 4π2m2 j b2−q2 j/parenrightBigg uj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,b)≤Cλ4 hj 4||uj||2 L2(0,b)+λ8 hj 16||uj||2 L2(0,b)+CNhN||uj||2 L2(0,b) ≤C |Re(qj)|2||uj||2 L2(0,b). Where we used that |Re(qj)|= 1/h2 j+O(h2 j) andλhj=O(hj). We now show that there are solutions of (10) with the desired e igenvalues. 4 Proof of Theorem 2.3 From this point on we focus on solutions of (10) on (0 ,∞). We begin by considering the cases when v(0) = 0 orv′(0) = 0, but focus on the case v(0) = 0 for the bulk of the section. We then explain how the proof changes for v′(0) = 0. In order to produce solutions to (10) with the desired eigenv alues we will solve it on (0,a) and (a,∞) separately. That is given solutions vl,vr∈H2of (10) on (0 ,a) and (a,∞) respectively, with the same values of λhandh, if there exists B∈Csuch that /braceleftBigg vl(a) =Bvr(a) v′ l(a) =Bv′ r(a),(14) then v=/braceleftBigg vl(x)x<a Bvr(x)a<x, solves (10) on (0 ,∞) with the same λhandhandv∈H2(0,∞)∩H1 0(0,∞) (orH2(0,∞) ifv′ l(0) = 0) . We will refer to equations (14) as the compatibility condition. We will explicitly solve (10) on (0 ,a). We will then use a rescaling of the equation on (a,∞) and the implicit function theorem to show that the compatib ility condition can be satisfied when λhis of the form (9) and his small enough. On (0,a) (10) is solved by vl(x) =eiλh(x−a)/h+Ref(λh,h)e−iλh(x−a)/h, where we choose Ref( λh,h) to ensure the boundary condition at 0 is satisfied. That is Ref(λh,h) =/braceleftBigg −e−2iλha/hv(0) = 0, e−2iλha/hv′(0) = 0. We will work now specify to the case where v(0) = 0 and work through it in detail and then summarize how the proof changes for v′(0) = 0. 9We now rescale the equation on ( a,∞). IfFsolves /braceleftBigg 0 =−F′′(x)+/parenleftBig ixβ−λ2 h h2β/(β+2)/parenrightBig F(x)x∈(0,∞) F′(0) = 1,(15) thenvr(x) =F(h−2/(β+2)(x−a)) solves (10) on ( a,∞) withv′ r(a) =h−2/(β+2). This follows immediately from the definition of Fand (10). Remark. This rescaling is necessary; in order to show the compatibil ity condition can be satisfied for all hin a neighborhood of 0 we will apply the implicit function the orem and so must be able to set h= 0. This can not be done in a satisfactory way with solutions o f (10); however for λhof the form (9), (15) is well defined at h= 0 as λ2 h h2β/(β+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=0=1 h2β/(β+2)Ch2+O(h(2β+6)/(β+2))/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=0= 0. Before we apply the implicit function theorem we first establ ish an inequality for u∈ H1(0,∞) using some facts about the Neumann spectrum of −∂2 x+xβon (0,∞). We then use this inequality to establish the existence and uniquene ss ofH2(0,∞) solutions to (15) and to show the boundary value F(0) is bounded away from 0 uniformly in the spectral parameter. We then explain how we can use the implicit functi on theorem to satisfy the compatibility condition and make use of these properties of Fto do so. LetσN(−∂2 x+xβ) be the spectrum of −∂2 x+xβon (0,∞) with Neumann boundary conditions, and let /tildewiderλ1= infσN(−∂2 x+xβ). Lemma 4.1. /tildewiderλ1>0 and /tildewiderλ1||u||2 L2(0,∞)≤/vextendsingle/vextendsingle/vextendsingle/vextendsingleu′/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,∞), (16) for allu∈H1(0,∞). Proof.The Neumann spectrum is discrete (this follows for instance from [HS12] Theorems 5.10 and 10.7) so we know that /tildewiderλ1is the lowest eigenvalue of the operator. We also know that the spectrum doesn’t contain 0, since ||∂xu||2 L2(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,∞)>0, for nontrivial u∈H1. Thus/tildewiderλ1>0. The inequality follows immediately from the variational pr inciple for the spectrum of self adjoint operators (see [HS12] Corollary 12.2). 10Lemma 4.2. For any |η|</tildewiderλ1, there exists a unique H2(0,∞)solution of /braceleftBigg 0 =−F′′(x)+ixβF(x)−ηF(x) F′(0) = 1.(17) Furthermore if we let F(0,η)be the value of this function at x= 0there exists C >0such that for all |η| ≤/tildewiderλ1 2 1/C≤ |F(0,η)| ≤C. andF(0,η)is holomorphic in ηon that same neighborhood. Proof.We first show the existence of a solution. Let ψ∈C∞ 0(0,∞) withψ′(0) = 1,ψ(0) = 0. DefineQ(η,ψ) as Q(η,ψ) :=ψ′′−ixβψ+ηψ. Now letJsolve /braceleftBigg −J′′+ixβJ−ηJ=Q J′(0) = 0,(18) and note that F=ψ+Jsolves (17). We will apply the Lax-Milgram theorem to show th e existence of solutions to (18). Let H=H1(0,∞)∩x−β/2L2(0,∞), and define the norm ||u||2 H=||u||2 H1(0,∞)+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexβ/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2(0,∞), noting that His a Hilbert space with this norm. We define the sesquilinear form B:H×H→C B[u,v] =ˆ∞ 0u′¯v′+ixβu¯v−ηu¯v. For anyu,v∈H |B[u,v]| ≤ˆ |u′||v′|+xβ|u||v|+|η||u||v| ≤C||u||H||v||H. Furthermore for u∈H⊂H1using (16) |B[u,u]| ≥ˆ |u′|2+xβ|u|2−|η||u|2dx≥/parenleftbigg 1−|η| /tildewiderλ1/parenrightbiggˆ |u′|2+xβ|u|2dx≥C||u||2 H. Therefore by Lax-Milgram for any Q∈Hthere exists a unique J∈Hsuch that B[J,v] =ˆ Q¯vdx, 11for allv∈H. Therefore there exists an F∈Hsolving (17) given by F=J+ψ. Now to show that F(0,η) is holomorphic in ηwe restate the result of our application of Lax-Milgram. We have shown that when |η|</tildewiderλ1, for allQ∈Hthere exists a unique J∈Hsuch that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(−∂2 x+ixβ−η)−1Q/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle H=||J||H≤C||Q||H. Therefore( −∂2 x+ixβ−η)isbijective withaboundedinversefor |η|</tildewiderλ1. Thustheresolvent (−∂2 x+ixβ−η)−1exists for |η|</tildewiderλ1and by [HS12] Theorem 1.2 it is holomorphic in η there as well. Now recall that the trace operator T:H1(0,∞)→Ris linear and continuous on H. ThusTJ=J(0,η) is also holomorphic in η. Recall that F(0,η) =J(0,η)+ψ(0) =J(0,η), soF(0,η) is holomorphic in η. To see that Fis unique assume otherwise, so there exists F1∈H1(0,∞) which also solves (17) with F1(0)/ne}ationslash=F(0). SetF2=F1−F, thenF′ 2(0) = 0,F2∈H1(0,∞) and 0 =−∂2 xF2+ixβF2−ηF2. Multiply both sides by ¯F2and integrate then integrate by parts 0 =ˆ∞ 0|∂xF2|2+ixβ|F2|2−η|F2|2dx. Then using (16) 0≥ˆ |∂xF2|2+xβ|F2|2−|η||F2|2dx≥/parenleftbigg 1−|η| /tildewiderλ1/parenrightbiggˆ |∂xF2|2+xβ|F2|2dx>0, sinceF2∈H1(0,∞). The inequality is strict since |η|</tildewiderλ1, which is a contradiction. To establish the stated regularity of Fnote that the equation (17) is elliptic and the left hand side is 0 so in fact F∈H∞(0,∞) and thusF∈H2(0,∞) in particular. Now we show that F(0,η) is bounded away from 0 and ∞. The upper bound follows immediately by the holomorphy of F(0,η) and the boundedness of η. To see|F(0,η)|>1/Cit is enough to show that F(0,η)/ne}ationslash= 0 sinceFis continuous and we are considering ηin a compact set. Assume otherwise, so F(0,η0) = 0 for some η0∈C,|η0| ≤/tildewiderλ1/2. Multiply both sides of (17) by ¯Fthen integrate and integrate by parts 0 =ˆ∞ 0|∂xF|2+ixβ|F|2−η0|F|2dx. Then using (16) (noting that here F∈H1 0⊂H1) 0≥ˆ |∂xF|2+xβ|F|2−|η0||F|2dx≥/parenleftbigg 1−|η0| /tildewiderλ1/parenrightbiggˆ |∂xF|2+xβ|F|2dx>0, We note that the inequality is strict since |η0|</tildewiderλ1, which is a contradiction. 12Now we introduce µh∈C. We use it to clarify the dependence of λhonhand will implicitly solve for it in terms of hin order to show that the compatibility condition can be satisfied. If F0is the Dirichlet data of the unique H2solution of (15) for h=λh= 0, we setC1=A1+µhso our definition of λhin (9) becomes λh=πlh a+A1h(β+4)/(β+2)+µhh(β+4)/(β+2), wherel∈Zand A1=πlF0 a2. Now takevr(x) =F(h−2/(β+2)(x−a)),whereFis the unique H2solution of (15) with the aboveλh. Recalling the explicit form of vl(x), the compatibility condition becomes iλh h(1−Ref(µh,h)) =Bh−2/(β+2) 1+Ref(µh,h) =BF(0,µh,h), whereF(0,µh,h) denotes the Dirichlet data of Fand Ref(µh,h) = Ref(λh,h) and both are written this way to emphasize their dependence on µhandh. Divide the top equation by the bottom /parenleftbiggπli a+A1ih2/(β+2)+iµh2/(β+2)/parenrightbigg/parenleftBig 1+exp(−2πil−2A1iah2/(β+2)−2iaµhh2/(β+2))/parenrightBig F(0,µh,h) =h−2/(β+2)/parenleftBig 1−exp(−2πil−2A1iah2/(β+2)−2iaµhh2/(β+2))/parenrightBig . Now Taylor expand the exponentials for small h /parenleftbiggπli a+A1ih2/(β+2)+iµhh2/(β+2)/parenrightbigg/parenleftBig 2−2A1iah2/(β+2)−2iaµhh2/(β+2)+g(h)/parenrightBig F(0,µh,h) =h−2/(β+2)/parenleftBig 2A1iah2/(β+2)+2iaµhh2/(β+2)−g(h)/parenrightBig , whereg(h) is the remainder term from the Taylor expansion with g(h) =O(h4/(β+2)). So in order to prove Theorem 2.3 it is enough to establish that for allhnear 0∈[0,∞) there exists µh∈Csuch that the following function has a zero at ( µh,h); G(µ,h) =/parenleftbiggπli a+A1ih2/(β+2)+iµh2/(β+2)/parenrightbigg/parenleftBig 2−2A1iah2/(β+2)−2iaµh2/(β+2))+g(h)/parenrightBig F(0,µ,h) −2A1ia−2iaµ+g(h)h−2/(β+2). (19) To do so we apply the implicit function theorem with weak regu larity hypotheses to solve forµh. We recall the implicit function theorem as stated in Theore m 11.1 of [LS90] (pp. 166) can be applied if there exists some h0such that 131.G(0,0) = 0 2.Gis continuous on [0 ,1]×[0,h0) 3.DµGexists and is continuous on [0 ,1]×[0,h0) 4.DµG(0,0) is invertible. To begin we see immediately that G(0,0) =2πli aF0−2A1ia=2πli aF0−2πli aF0= 0. In the following two lemmas we show that points 2 and 3, and 4 ar e satisfied. Lemma 4.3. There exists h0>0such thatGas defined in (19)is continuous and∂ ∂µG exists and is continuous on {µ∈C;|µ|<1}×[0,h0). Proof.To see that Gis continuous it will be enough to show that F(0,µ,h) is continuous as the other terms in G(µ,h) are clearly continuous in µandh. Similarly to see that∂ ∂µG exists and is continuous it is enough to see that∂ ∂µF(0,µ,h) exists and is continuous in µ andh. We recall that F(0,µh,h) is the Dirichlet data for the L2solution of (17) with spectral parameter η=λ2 h h2β/(β+2)=π2l2h4/(β+2)+(2A1πl+2πlµ)h6/(β+2) a+(A2 1+µ2+A1µ)h8/(β+2). By Lemma 4.2 the Dirichlet data for the L2solution is holomorphic in |η|</tildewiderλ1. Thisηis a sum of functions which are jointly continuous in µandhand continuously differentiable inµ, thereforeF(0,µ,h) is as well. Furthermore since |µ|<1 and there is a positive power of hin each term of ηthere exists some h0such that |η|</tildewiderλ1/2 for|µ|<1 andh∈[0,h0). Lemma 4.4. WithGdefined as in (19) ∂ ∂µG(µ,h)/vextendsingle/vextendsingle/vextendsingle/vextendsingle h=0,µ=0/ne}ationslash= 0, and thus is invertible. Proof.Note G(µ,0) =2πli aF(0,µ,0)−2A1ai−2aiµ so ∂ ∂µG(µ,0) =2πli a/parenleftbigg∂ ∂µF(0,µ,0)/parenrightbigg −2ai. 14Whenh= 0 the equation Fsolves is 0 = −F′′(x)+ixβF(x).Therefore when h= 0 there is no dependence on µso∂ ∂µF(0,µ,h= 0) = 0. Thus ∂ ∂µG(µ= 0,h= 0) =−2ai/ne}ationslash= 0. Now that we have shown we can apply the implicit function theo rem we conclude the proof of Theorem 2.3. That is given some lwe leth0be as in the proof of Lemma 4.3 and letK=π|F0| a2+1. Then we choose an h∈(0,h0) and use the implicit function theorem to produce aµh∈[0,1] such that G(µh,h) = 0 and |µh|<1. We setCh=πF0 a2+µhwe are then able to solve (10) on (0 ,a) and (a,∞) forλh=πlh a+A1h(β+4)/(β+2)+µhh(β+4)/(β+2) such that the compatibility conditions are satisfied giving us a solution v∈H2(0,∞)∩ H1 0(0,∞),v/ne}ationslash= 0 with |Ch|<Kandλof the appropriate form. 4.1 Case u′(0) = 0 We now discuss how these proofs change when u′(0) = 0. When this is the case Ref(λh,h) =e−2iλha/h. Because of this we take l+ 1/2∈Zrather than l∈Z. This changes the specific steps taken when going from the compatibility condition to the defi nition ofGin (19) but the eventual definition of Gis the same. The specific form of lis otherwise not used so the remaining proofs in this section hold unchanged. 5 Proof of Lemma 3.1 Proof.To show this result we consider our operator Ph=−h2∂2 x+i(x−a)β +−λ2 hon (a+σ/4,b). We note that the coefficients are smooth away from aand so it makes sense to look at the pricipal symbol |ξ|2+i(x−a)β +. It is straightforward to see from this that Phis semiclassically elliptic on ( a+σ/4,b). Recall in our definition of φthat we required a+σ<b−2δand in (12) that φsatisfies φ(x) =/braceleftBigg 1x<b−2δ 0b−δ<x. We now define ψ∈C∞(0,b) satisfying ψ=/braceleftBigg 0x<a+σ/2 1a+σ<x<b, 15so thatφψ∈Ψ0 h(0,b) withWFh(φψ)⊂ellh(Ph). Ifvis asolution of Phv= 0, by standardsemiclassical elliptic estimates (see fori nstance [Zwo12] Theorem 7.1) there exists χ∈C∞ 0(a+σ/4,b) such that for all s∈R ||φψv||Hs h(a+σ/2,b)≤O(h∞)||χv||L2(a+σ/4,b) In particular ||φv||Hs h(a+σ,b)≤O(h∞)||v||L2(0,b). It remains to show that ||v||L2(0,b)/lessorsimilar||φv||L2(0,b). To proceed we multiply both sides of (10) by ¯ vthen integrate and integrate by parts 0 =h2ˆ∞ 0|∂xv|2dx+iˆ∞ 0(x−a)β +|v|2dx−λ2 hˆ∞ 0|v|2dx. Take the imaginary part of both sides and rearrange ˆ∞ 0(x−a)β +|v|2dx= Im(λ2 h)ˆ∞ 0|v|2dx≤O(h2)ˆ∞ 0|v|2dx. Furthermore ˆ∞ a+σσβ|v|2dx≤ˆ∞ a+σ(x−a)β +|v|2dx≤ˆ∞ 0(x−a)β +|v|2dx. So ˆ∞ a+σ|v|2dx≤h2 σβˆ∞ 0|v|2dx. Add´∞ 0|v|2dxto both sides and rearrange /parenleftbigg 1−h2 σβ/parenrightbiggˆ∞ 0|v|2dx≤ˆa+σ 0|v|2dx. Notice that ( b−2δ,b)⊂(0,∞) and (0,a+σ)⊂(0,b−2δ) so /parenleftbigg 1−h2 σβ/parenrightbiggˆb b−2δ|v|2dx≤/parenleftbigg 1−h2 σβ/parenrightbiggˆ∞ 0|v|2dx≤ˆa+σ 0|v|2dx≤ˆb−2δ 0|v|2dx. Therefore for h<σβ/2 ˆb 0|v|2dx=ˆb−2δ 0|v|dx+ˆb b−2δ|v|2dx≤/parenleftbigg 1+σβ σβ−h2/parenrightbiggˆb−2δ 0|v|2dx≤Cˆb 0|φv|2dx. 166 Proof of Theorem 1.2 In this section we establish a rate of decay of the energy by ad apting and improving Section 3 of [BH07] for our particular setup. That result and our result rely heavily on an observability result by Burq and Zworski (Proposition 6.1) in [BZ04]. To prove Theorem 1.2 we again rely on Proposition 2.4 of [AL+14] (see also [BT10]) which we state a variant of. Proposition 6.1. If there exists C >0andq0≥0such that /vextendsingle/vextendsingle/vextendsingle/vextendsingle(−∆+iqW(x)−q2)−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle L2→L2≤C|q|1 α−1(20) for allq∈R,|q| ≥q0then(1)is stable at rate 1/tα. Proof of Theorem 1.2. Consideru∈H2(M) solving (−∆+iqW−q2)u=f∈L2(M), u|∂M= 0, q≫1. (21) Let 0≤χ∈C∞ 0(R) be a cutoff function with χ= 0 for|x| ≥2 andχ= 1 for|x| ≤1. Now letχq=χ(qγW(x)) withγ=β β+2. Note that χqis identically 0 for |x|>a+σ/4 for qlarge enough since W >0 on [a+σ,b]. Because of this χqand it’s derivatives are only supported where Whas the form ( |x|−a)β +. Furthermore on the support of χ′ q(x) W(x)∼1 qγ, q≫1. (22) Remark. The proof in [BH07] uses an analogous setup with γ= 1. The key change we make is to set γ=β β+2. The rest of our argument is similar to the proof in [BH07] but we detail it for the convenience of the reader and to explain why this value of γis ideal. The function χqustill vanishes on ∂M(or if∂M=∅it still satisfies the periodicity condition) and satisfies on M (−∆−q2)χqu=χqf+χ′′ qu−2∂x(χ′ qu)−iqW(x)χqu. (23) We apply Proposition 6.1 of [BZ04] to this equation choosing the control region ωx= [a+σ/4,a+σ/2] and setting ω:=wx×[−b,b] to obtain ||χqu||2 L2≤C/parenleftBig/vextendsingle/vextendsingle/vextendsingle/vextendsingleχqf+χ′′ qu−2∂x(χ′ qu)−iqW(x)χqu/vextendsingle/vextendsingle/vextendsingle/vextendsingle H−1 xL2y(M)/parenrightBig +||χqu|ω||2 L2(ω) ≤C/parenleftBig ||χqf||2 L2+/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′′ qu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2+/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′ qu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2+||qWχqu||2 L2/parenrightBig . (24) We emphasize that χqvanishes on ω, which allowed us to drop that term. We now estimate the remaining terms on the right hand side. Using that Wis exactly ( |x| −a)β +on the support ofχqand its derivatives we obtain the following bound on the deri vative ofχq, |χ′ q|=|qγχ′ q(qγW(x))W′(x)| ≤Cqγ/β, (25) 17and similarly |χ′′ q| ≤Cq2γ/β. Note that on the support of χ′ qandχ′′ qthe damping Wis smooth, so this computation is valid for all β >0. Now write χ′ qu=χ′ qW1/2u W1/2, then using (25) and (22)/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′ q W1/2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤qγ(1/2+1/β), and consequently/vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′ qu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2≤Cqγ(1+2/β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2. (26) We estimate the L2norm ofχ′′ quin a similar way /vextendsingle/vextendsingle/vextendsingle/vextendsingleχ′′ qu/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2≤O(1)qγ(1+4/β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2. (27) Finally a similar argument shows ||qWχqu||2 L2≤O(1)q2−γ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2. (28) The smaller the terms on the right of (26), (27) and (28) are th e stronger the resolvent estimate is. Because of this we would like to minimize max{2−γ,γ(1+4/β),γ(1+2/β)}. This is attained when 2−γ=γ(1+4/β), i..e.γ=β/(β+2). Therefore (26), (27), (28) along with (24) give ||χqu||2 L2≤O(1)/parenleftbigg ||f||2 L2+q(β+4)/(β+2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleW1/2u/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 L2/parenrightbigg . Now note that pairing (21) with ¯ u |q|ˆ W|u|2dx≤ ||f||L2||u||L2. (29) Therefore ||χqu||2 L2≤O(1)/parenleftBig ||f||2 L2+q2/(β+2)||f||L2||u||L2/parenrightBig . 18It remains to control the L2norm of (1 −χq)u. To do so we remark that 1 −χqis supported in the set where W≥1/qγ. Using (29) again ||(1−χq)u||2 L2≤qγˆ (1−χq)W|u|2dx≤qγ−1||f||L2||u||L2. Therefore ||u||2 L2≤O(1)/parenleftBig ||f||2 L2+q1/(β+2)||f||L2||u||L2/parenrightBig and thus we obtain ||u||L2≤O(1)q2/(β+2)||f||L2, q∈R,|q| ≫1, which along with Proposition 6.1 gives stability at the stat ed rate. References [AL+14] N. Anantharaman, M. L´ eautaud, et al. Sharp polynomial d ecay rates for the damped wave equation on the torus. Anal. PDE , 7(1):159–214, 2014. [BD08] C. J. K. Batty and T. Duyckaerts. Non-uniform stabili ty for boundedsemi-groups on banach spaces. Journal of Evolution Equations , 8(4):765–780, Nov 2008. [BG97] N. Burq and P. G´ erard. Condition n´ ecessaire et suffis ante pour la contrˆ olabilit´ e exacte des ondes. Comptes Rendus de l’Acadˆ emie des Sciences - Series I - Math- ematics, 325(7):749 – 752, 1997. [BG18] N. Burq and P. G´ erard. Stabilisation of wave equatio ns on the torus with rough dampings. arXiv preprint arXiv:1801.00983 , 2018. [BH07] N. Burq and M. Hitrik. Energy decay for damped wave equ ations on partially rectangular domains. Mathematical Research Letters , 14(1):35–47, 2007. [BT10] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Mathematische Annalen , 347(2):455–478, 2010. [Bur98] N. Burq. D´ ecroissance de l’´ energie locale de l’´ e quation des ondes pour le probl` eme ext´ erieur et absence de r´ esonance au voisinage du r´ eel. Acta Math. , 180(1):1–29, 1998. [BZ04] N.BurqandM.Zworski. Geometriccontrol intheprese nceofablack box. Journal of the American Mathematical Society , 17(2):443–471, 2004. [BZ12] N. Burq and M. Zworski. Control for Schr¨ odinger oper ators on tori. Mathematical Research Letters , 19, 06 2012. 19[BZ15] N. Burq and C. Zuily. Laplace eigenfunctions and damp ed wave equation on prod- uct manifolds. Applied Mathematics Research eXpress , 2015(2):296–310, 2015. [CBR92] G. Lebeau C. Bardos and J. Rauch. Sharp sufficient cond itions for the obser- vation, control, and stabilization of waves from the bounda ry.SIAM Journal on Control and Optimization , 30(5):1024–1065, 1992. [HS12] P. Hislop and I. Sigal. Introduction to spectral theory: With applications to Schr¨ odinger operators , volume 113. Springer Science & Business Media, 2012. [Jaf90] S. Jaffard. Contrˆ ole interne exact des vibrations d’ une plaque rectangulaire. (Internal exact control for the vibrations of a rectangular plate).Port. Math. , 47(4):423–429, 1990. [Leb96] G. Lebeau. Equation des ondes amorties. In Algebraic and Geometric Methods in Mathematical Physics: Proceedings of the Kaciveli Summe r School, Crimea, Ukraine, 1993 , pages 73–109. Springer Netherlands, Dordrecht, 1996. [LL17] M. L´ eautaud and N. Lerner. Energy decay for a locally undamped wave equation. Annales de la facult des sciences de Toulouse Sr.6 , 26(1):157–205, 2017. [LR97] G. Lebeau and L. Robbiano. Stabilisation de l´ equati on des ondes par le bord. Duke Math. J. , 86(3):465–491, 1997. [LR05] Z. Liu and B. Rao. Characterization of polynomial dec ay rate for the solution of linear evolution equation. Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP, 56(4):630–644, 2005. [LS90] L.H. Loomis and S. Sternberg. Advanced Calculus . Jones and Bartlett Publishers, Inc., 1990. [Mac10] F. Maci` a. High-frequency propagation for the Schr ¨ odinger equation on the torus. J. Funct. Anal. , 258(3):933–955, 2010. [Ral69] J. Ralston. Solutions of the wave equation with loca lized energy. Communications on Pure and Applied Mathematics , 22(6):807–823, 1969. [RT75] J. Rauch and M. Taylor. Exponential decay of solution s to hyperbolic equations in bounded domains. Indiana Univ. Math. J. , 24:7986, 1975. [Sta17] R. Stahn. Optimal decay rate for the wave equation on a square with constant damping on a strip. Zeitschrift f¨ ur angewandte Mathematik und Physik , 68(2):36, 2017. [Zwo12] M. Zworski. Semiclassical analysis , volume 138. American Mathematical Soc., 2012. 20

2018-05-16

We study the decay rate of the energy of solutions to the damped wave equation in a setup where the geometric control condition is violated. We consider damping coefficients which are $0$ on a strip and vanish like polynomials, $x^{\beta}$. We prove that the semigroup cannot be stable at rate faster than $1/t^{(\beta+2)/(\beta+3)}$ by producing quasimodes of the associated stationary damped wave equation. We also prove that the semigroup is stable at rate at least as fast as $1/t^{(\beta+2)/(\beta+4)}$. These two results establish an explicit relation between the rate of vanishing of the damping and rate of decay of solutions. Our result partially generalizes a decay result of Nonnemacher in which the damping is an indicator function on a strip.

Stabilization rates for the damped wave equation with Hölder-regular damping

1805.06535v3

Using rf voltage induced ferromagnetic resonance to study the spin-wave density of states and the Gilbert damping in perpendicularly magnetized disks Thibaut Devolder Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Universit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France (Dated: September 18, 2018) We study how the shape of the spinwave resonance lines in rf-voltage induced FMR can be used to extract the spin-wave density of states and the Gilbert damping within the precessing layer in nanoscale magnetic tunnel junctions that possess perpendicular magnetic anisotropy. We work with a field applied along the easy axis to preserve the cylindrical symmetry of the uniaxial perpendicularly magnetized systems. We first describe the experimental set-up to study the susceptibility contributions of the spin waves in the field-frequency space. We then identify experimentally the maximum device size above which the spinwaves confined in the free layer can no longer be studied in isolation as the linewidths of their discrete responses make them overlap into a continuous density of states. The rf-voltage induced signal is the sum of two voltages that have comparable magnitudes: a first voltage that originates from the linear transverse susceptibility and rectification by magneto-resistance and a second voltage that arises from the non-linear longitudinal susceptibility and the resultant time-averaged change of the exact micromagnetic configuration of the precessing layer. The transverse and longitudinal susceptibility signals have different dc bias dependences such that they can be separated by measuring how the device rectifies the rf voltage at different dc bias voltages. The transverse and longitudinal susceptibility signals have different lineshapes; their joint studies in both fixed field-variable frequency, or fixed frequency-variable field configura- tions can yield the Gilbert damping of the free layer of the device with a degree of confidence that compares well with standard ferromagnetic resonance. Our method is illustrated on FeCoB-based free layers in which the individual spin-waves can be sufficiently resolved only for disk diameters below 200 nm. The resonance line shapes on devices with 90 nm diameters are consistent with a Gilbert damping of 0:011. A single value of the damping factor accounts for the line shape of all the spin-waves that can be characterized. This damp- ing of 0.011 exceeds the value of 0.008 measured on the unpatterned films, which indicates that device-level measurements are needed for a correct evaluation of dissipation. The frequencies of the magnetization eigenmodes of magnetic body reflect the energetics of the magnetization. As a result the frequency-based methods – the ferromag- netic resonances (FMR)1and more generally the spin-wave spectroscopies– are particularly well designed for the metrol- ogy of the various magnetic interactions. In particular, mea- suring the Gilbert damping parameter that describes the coupling of the magnetization dynamics to the thermal bath, specifically requires high frequency measurements. There are two main variants of these resonance techniques. The so- called conventional FMR and its modern version the vector network analyzer2(VNA)-FMR are established technique to harness the coupling of microwave photons to the magneti- zation eigenmodes to measure to anisotropy fields1, demag- netizing fields, exchange stiffness3, interlayer exchange4and spin-pumping5, most often at film level. More recent meth- ods, like the increasingly popular spin-transfer-torque-(STT)- FMR, are developed6to characterize the magnetization dy- namics of magnetic bodies embodied in electrical devices pos- sessing a magneto-resistance of some kind. In conventional FMR or VNA-FMR, the community is well aware that the line shape of a resonance is more complicated than simple arguments based on the Landau-Lifshitz-Gilbert equation would tell. There are for instance substantial contri- butions from microwave shielding effects7(”Eddy currents”) for conductive ferromagnetic films8or ferromagnetic films in contact with (or capacitively coupled to) a conductive layers. A hint to these effect is for instance to compare the lineshapes8 for the quasi-uniform precession mode and the first perpen- dicular standing spin wave modes that occur in different res-onance conditions. Note that the experimental lineshapes are already complex in VNA-FMR despite the fact that the dy- namics is induced by simple magnetic fields supposedly well controlled. In contrast, STT-FMR methods rely on torques [spin-orbit torques (SOT)9or STT] that have less hindsight that magnetic fields or that are the targeted measurements. These torques are related to the current across the device and the experimental analysis generally assumes that this current is in phase with the applied voltage. This implicitly assumes that the sam- ple is free of capacitive and inductive responses, even at the microwave frequencies used for the measurement. A careful analysis is thus needed when the STT-FMR methods analyze the phase of the device response to separate the contribution of the different torques6,10,11. Besides, the quasi-uniform mode is often the sole to be analyzed despite that fact that the line shapes of the higher frequency modes can be very different10. Finally, an external field is generally applied in a direction that is not a principal direction of the magnetization energy functional12. While this maximizes the signal, this unfortu- nately makes numerical simulation unavoidable to model the experimental responses. With the progress in MTJ technologies, much larger magneto-resistance are now available13, such that signals can be measured while maintaining sample symmetries, for in- stance with a static field applied collinearly to the magne- tization. In addition, high anisotropy materials can now be incorporated in these MTJs. This leads to a priori much more uniform magnetic configurations in which analytical de- scriptions are more likely to apply. In this paper, we revisitarXiv:1703.07310v2 [cond-mat.mtrl-sci] 4 Sep 20172 rf-voltage induced FMR in a situation where the symmetry is chosen so that all torques should yield a priori the same canonical lineshape for all spinwaves excited in the system. We use PMA MTJ disks of sizes 500 nm, on which a quasi- continuum of more that 20 different spin-wave modes can be detected, down to sizes of 60 nm where only a few discrete spinwave modes can be detected. We discuss the lineshapes of the spin-wave signals with the modest objective of deter- mining if at least the Gilbert damping of the dynamically ac- tive magnetic layer can be reliably extracted. We show that the linear transverse susceptibility and the non-linear longitu- dinal susceptibilities must both be considered when a finite dc voltage is applied through the device. We propose a method- ology and implement it on a nanopillars made with a stan- dard MgO/FeCoB/MgO free layer system in which we obtain a Gilbert damping of 0:0110:0003 . This exceeds the value of 0.008 measured on the unpatterned film, which indicates that device-level measurements are needed for a correct eval- uation of dissipation. The paper is organized as follows: The first section lists the experimental considerations, includ- ing the main properties of the sample, the measurement set- up and the mathematical post-processing required for an in- creased sensitivity. The second section discusses the origins of the measured resonance signals and their main properties. The third section describes how the device diameter affects the spin-wave signals in rf-voltage-induced ferromagnetic reso- nance. The last section describes how the voltage bias depen- dence of the spinwave resonance signals can be manipulated to extract the Gilbert damping of the dynamically active mag- netic layer. After the conclusion, an appendix details the main features of the spectral shapes expected in ideal perpendicu- larly magnetized systems. I. EXPERIMENTAL CONSIDERATIONS A. Magnetic tunnel junctions samples We implement our characterization technique on the sam- ples described in detail in ref. 14. They are tunnel junctions with an FeCoB-based free layer and a hard reference system based on a well compensated [Co/Pt]-based synthetic antifer- romagnet. All layers have perpendicular magnetic anisotropy (PMA). The perpendicular anisotropy of the thick ( t= 2nm) free layer is ensured by a dual MgO encapsulation and an iron-rich composition. After annealing, the free layer has an areal moment of Mst1:8mA and an effective perpendic- ular anisotropy field 0(HkMs)= 330 mT. Before pat- tering, standard ferromagnetic resonance measurements in- dicated a Gilbert damping parameter of the free layer being = 0:008. Depending on the size of the patterned device, the tunnel magnetoresistance (TMR) is 220 to 250%, for a stack resistance-area product is RA = 12 :m2. The de- vices are circular pillars with diameters varied from 60 to 500 nm. The materials, processing and device rfcircuitry were optimized for fast switching14spin-transfer-torque magnetic random access memories (STT-MRAM15) ; the quasi-static dcRF50 ΩPulse modulation ac, 50 kHz 50 ΩVac ~ 40 mVLI75AamplifierMSG3697synthetiser K2400sourcemeter SR830lock-inamplifier×100-10 dB attenuation~ 0 dBm~ 14 µAdc10 mVdcMTJ device300 nm~700 ΩFIG. 1. (Color online). Sketch of the experimental set-up with an 300300m2optical micrograph of the device circuitry. The given numbers are the typical experimental parameters for a 300 nm diam- eter junction. Inset: resistance versus out-of-plane field hysteresis loop for a device with 300 nm diameter. dcswitching voltage is 600mV . In the present report, the applied voltages shall never exceed 100 mV to minimize spin- transfer-torque effects. The fields will always be applied along (z) which is the easy magnetization axis. The sample will be maintained in the antiparallel (AP) state. B. Measurement set-up The pillars are characterized in a set-up (Fig. 1) inspired from spin-torque diode experiments6but an electrical band- width increased to 70 GHz. The objective is to identify the regions in theffrequency, fieldgspace in which the magneti- zation is responding in a resonant manner. The device is at- tacked with an rfvoltageVrf. A 10 dB attenuator is inserted at the output port of the synthesizer to improve its impedance matching so as to avoid standing waves in the circuit. This im- proves the frequency flatness of the amplitude of the stimulus arriving at the device. To ease the detection of the sample’s re- sponse, the rfvoltage is pulse-modulated at an acfrequency !ac=(2) = 50 kHz (Fig. 1). The current passing through the MTJ has thus frequency components at the two sidebands !rf!ac. The acvoltage which appears across the device is amplified and analyzed by a lock-in amplifier. We shall discuss the origin of this acvoltage in section II. Optionally, the device is biased using a dcsourcemeter supplying Vdcand measuringIdc. Figure 2 shows a representative map of thedVac dHzresponse obtained on a pillar of diameter 300 nm with Vdc= 10 mV. As positive fields are parallel to the free layer magnetization, the spin waves of the free layer appear with a positive fre- quency versus field slope, expected to be the gyromagnetic3 ⦰ 300 nm FIG. 2. Field derivative of the rectified voltagedVac dHzin the ffrequency-fieldgparameter space for a 300 nm diameter device in the AP state when the field is parallel to the free layer magnetization. The linear features with positive (resp. negative) slopes correspond to free layer (resp. reference layers) confined spin-wave modes. Black and white colors correspond to signals exceeding 0:01V/T. The one-pixel high horizontal segments are experimental artefacts due to transient changes of contact resistances. ratio 0of the free layer material (see appendix). Conversely, the reference layer eigenmodes appear with a negative slope, expectedly 0, where this time 0is gyromagnetic ratio of the reference layer material combination. Working in the AP state is thus a convenient way to easily distinguish between the spinwaves of the free layer and of the reference layers. Note that the gyromagnetic ratios 0of the free layer mode and the reference layer modes differ slightly owing to their difference chemical nature. The free layer has a Land ´e factor g= 2:0850:015where the error bar is given by the precision of the field calibration; the reference layer modes are consis- tent with a 1.2% larger gyromagnetic ratio. The accuracy of this latter number is limited only by the signal-to-noise ratio in the measurement of the reference layer properties. Looking at Fig. 2, one immediately notices that the linewidths of the reference layer modes are much broader than that of the free layer. While the linewidh of the reference layer modes will not be analyzed here, we mention that this increased linewidth is to be expected for reference layers that contain heavy metals (Pt, Ru) with large spin-orbit couplings, hence larger damping factors16.C. Experimental settings In practice, we choose an applied field interval of [110;110mT]that is narrow enough to stay in a state whose resistance is very close to that of the remanent AP state. The frequency!rf=(2)is varied from 1 to 70 GHz; we gener- ally could not detect signals above 50 GHz. The practical frequency range 250GHz= 01:6T is much wider that our accessible field range. For wider views of the experimen- tal signals (for instance when the spin-wave density of states is the studied thing), we shall thus prefer to plot them versus frequency than versus field. The response is recorded pixel by pixel in in theffrequency, fieldgspace. The typical pixel size isfHzfg=f1 mT50 MHzg. The field and frequency resolutions are thus comparable (indeed 2f= 0= 1:7 mT). D. Signal conditioning 1. Mathematical post-treatments Finally, despite all our precautions to suppress the rectify- ing phenomena that do not originate from magnetization dy- namics, we have to artificially suppress the remaining ones. This was done by mathematical differentiation, and we gener- ally plotdVac dfordVac dHzin the experimental figures (Figs. 2-5). 2. Dynamic range improvement by self-conformal averaging A special procedure (Fig. 3) is applied when a better signal to noise ratio is desired while the exact signal lineshape and amplitudes are not to meant to be looked at. This procedure harnesses the fact that the normalized shape of the sample’s response is essentially self-conformal when moving across a line withd! dHz= 0in theffrequency, fieldgparameter space (see appendix). The procedure consists in calculating the fol- lowing primitive: s(f0) =1 2 0HmaxzZ contourdVac dHzdf ; (1) in which the integration contour is the segment linking the points (Hmax z;f0 0Hmax z) and (Hmax z;f0+ 0Hmax z) in theffield, frequencygparameter space. Such contours ap- pear as pixel columns in Fig. 3(b). This primitive (eq. 1) is efficient to reveal the free layer spin-wave modes that yield an otherwise too small signal. For instance when only 7 modes can be detected in single field spectra [Fig. 3(a)], the aver- aging procedure can increase this number to typically above 25. The averaging procedure is also effective in suppressing the signals of the reference layer as these laters average out over a contour designed for the free layer mode when in the AP state. However as the linewidth of the free layer modes is proportional to the frequency, it is not constant across the contour; the higher signal to noise ratio is thus unfortunately4 f0Hz2⇡Hz(b)(a) FIG. 3. (Color online). Illustration of the dynamic range improve- ment by self-conformal averaging (section I D 2). The procedure is implemented on a 300 nm diameter device to evidence the free layer modes. Bottom panel: field derivative of the acsignal in the rotated frame in which the modes withdf dHz= 0 2should appear as vertical lines. Top panel: comparison of a single field frequency scan (red) with the average over all scans as performed in the != 0Hzdi- rection. Note that the signal of the lowest frequency mode (which corresponds to the quasi-uniform precession) disappears near zero field, at 5 mT (see the apparent break in the middle of the most left line in the bottom panel). obtained at the expense of a distorted (and unphysical) line- shape. Note also that this procedure can not be applied to the quasi-uniform precession mode as will be explained in section II D 2). II. ORIGIN AND NATURE OF THE RECTIFIED SIGNAL Let us now discuss the origin of the demodulated acvolt- age. In this section, we assume that the reference layer mag- netization is static but not necessarily uniformly magnetized. We can thus express any change of the resistance by writing R=R MM wherehas to be understood as a functional derivative with respect to the free layer magnetization distri- bution. A. The two origins of the rectified signals Theacsignal can contain two components V1;acandV2;ac of different physical origins17. The first component is the ’standard’ STT-FMR signal: the pulse-modulated rfcurrent is at the frequency sidebands !rf!acand it rectifies to acany oscillation of the resistance Rrfoccurring at the frequency !rf. We simply have V1;ac=Rrfi!rf!ac. The second acsignal (V2;ac) is related to the change of the time-averaged resistance due to the population of spinwavescreated when the rfcurrent is applied12. Indeed the time- averaged magnetization distribution is not the same when the rfisonoroff. This change of resistance Raccan revealed by the (optional) dccurrentIdcpassing through the sample, i.e. V2;ac=RacIdc. Note that a third rectification channel18can be obtained by a combination of spin pumping and inverse spin Hall effect in in-plane magnetized systems19. This third rectification chan- nel yields symmetric lorentzian lines when applied to PMA systems in out-of-plane applied fields (see eq. 23 in ref. 18). Besides, the spin-pumping is known to be largely suppressed by the MgO tunnel barrier20, such that we will consider that we can neglect this third rectification channel from now on. In summary, we have: V1;ac=Vrf R+ 50R MMrf and (2) V2;ac=Vdc R+ 50R MMac (3) This has important consequences. B. Compared signal amplitudes in the P and AP states The first important consequence of Eq. 2 and 3 is that the signal amplitude depends on the nature of the micromagnetic configuration. As intuitive, both V1;acandV2;acscale with how much the instantaneous device resistance depends on its instantaneous micromagnetic configuration. This is expressed by the sensitivity factorR Mwhich is essentially a magneto- resistance. We expect no signal when the resistance is insen- sitive to the magnetization distribution at first order (i.e. when R M0). In our samples, the shape of the hysteresis loop (Fig. 1) seems to indicate that the free layer magnetization is very uni- form when in the Parallel state. Consistently, the experimental rectified signal were found to be weak signals when in the P state. Conversely, there is a pronounced curvature in the AP branch of the R(Hz)hysteresis loop (see one example in the inset of Fig. 1). This indicates that the resistance is much de- pendent on the exact magnetization configuration when in the AP state. Consistently, this largerR Min the AP state is proba- bly the reason why the rectified signal is much easier to detect in the AP state for our samples. C. Bias dependence of the rectified signals The second important consequence of Eqs. 2-3 concerns the dependence of the rectified acsignalsV1;acandV2;acon the dcandrfstimuli. AsMrfscales with the applied rftorque according to a linear transverse susceptibility ( <e(xx), see appendix),V1;acis expected to scale with the rfpowerVrf2 (see Eq. 2) independently from the dcbias, i.e. we have V1;ac/Vrf2:5 In contrast,Macis related to a longitudinal susceptibility and is thus quadratic with the rftorques (see appendix). Using Eq. 3, we thus expect the following bias dependence V2;ac/Vrf2Vdc: D. Peculiarities of the quasi-uniform precession (QUP) mode The last important consequence of Eqs. 2-3 concerns specifically the quasi-uniform precession (QUP) mode that shows a peculiar acsignal. 1. Quasi-absence of STT-FMR like signal for the quasi-uniform precession mode In the idealized macrospin case (see appendix) the uniform precession is perfectly circular with no rfvariation of Mzat any order. If the fixed layer was uniformly magnetized along exactly (z), this would lead V1;ac= 0 such that the signal of the QUP mode would be given by purely V2;ac. This qual- itatively ’pure V2;accharacter’ is confirmed experimentally by the fact that the signal of the QUP mode systematically changes sign with Vdcin our sample series (not shown). 2. Strong dependence of the QUP signal amplitude with the applied field In addition, the experimental signal of the quasi-uniform mode is found to disappear at low fields [see Figs. 2, 3(b) and 4(a, b)] exactly at the apex of the AP branch of the hysteresis loop (Fig. 1), i.e. for the field leading specifically todR dHz= 0. Moreover, the amplitude of the acsignal of the QUP mode ap- pears to be essentially linearly correlated with the loop slope dR dHz(not shown). For instance, the V2;acof the QUP mode changes sign when the applied field crosses the apex of the R(Hz)loops (Fig. 1). The reason stems probably from the sensitivity factorR M and its correlation with the loop slopedR dHz; in some sense, a large loop slope should translate in a large sensitivity factor. While a numerical evaluation of this correlation goes beyond the scope of this paper, we stress that if the magnetization was perfectly uniform there would be a one-to-one correlation be- tween loop slopedR dHzand magnetoresistance sensitivity fac- torR M. This trend remains qualitatively true for the QUP mode. Indeed as the hysteresis loop is monitoring the spatial average of the magnetization, it is more insightful for the uni- form mode than for any other (higher order) modes whose dy- namic profiles spatially average to essentially zero21; the cor- relation betweendR dHzandR Mis thus expected to be maximal for quasi-uniform changes of the magnetization configuration. While this property – the disappearance of the QUP mode signal whendR dHz= 0 – can be used to distinguish the QUP mode from the higher order spin waves, the pronounced field dependence of the QUP signal complicates the analysis, as it prevents to conveniently analyze the field derivative of the acsignal (I D 1). In the remainder of this paper we shall focus on only higher order modes to avoid such difficulties. E. Signals for non-uniform spin-waves Before analyzing the spin-wave density of states (section III), let us comment on the amplitude of the STT-FMR-like signalV1;acfor the non-uniform spin-waves. In the perpen- dicular magnetization state, these spin-waves have a circular precession22. By symmetry, the resistance is not expected to change during a period of circular precession when in the per- fect collinear cases and for radial spin waves maintaining the cylindrical symmetry of the system. In other words, when the dynamical magnetization of the eigenmode maintains the cylindrical symmetry and when the free and reference layers equilibrium magnetizations follow ~Mfree~Mref=~0every- where in the (xy) plane, with being the conventional vec- tor product) the device resistance is not expected to oscillate. While we can not identify to what extent we depart from this ideal situation, we speculate that this perfect collinearity does not happen in practice at least because of finite thermal fluctu- ations. The effect of thermal fluctuations on the device resis- tance is not averaged out for non-uniform spin-waves, while it could be essentially averaged out for the QUP mode ana- lyzed earlier. In practice a finite variation of the resistance Rrf6= 0 is always present during a precession period for a non-uniform spin-wave. This provides a finite sensitivity to any spin-wave mode. This resistance variation at !rfhas the spectral shape of a transverse susceptibility term <e(xx)(see appendix). III. SPIN WA VE DENSITY OF STATES AGAINST LATERAL CONFINEMENT Any reliable analysis of a spectral lineshape or linewidth re- quires to determine priorly how many spin-waves contribute to the lineshape under study. Therefore, before discussing the lineshapes of the individual spin-wave modes, let us determine how the lateral confinement influences the measured rectified signal. The impact of the device diameter on the spectral sig- nals is reported in Fig. 4. A. Spin-waves within the references layers Fig. 4 indicates that the modes of the reference layers have frequencies that are almost not affected by the device diam- eter. This fact is related to the well compensated character of the synthetic antiferromagnet that composes the reference layers. Indeed the internal demagnetizing fields compensate to some extent, such that they do not influence the frequency of the acoustical mode of a SAF as much as the anisotropies and the interlayer exchange couplings do.6 Signal Spectral Peak-to-peak Full Width Zero crossings shape separation at Half Maximum separation Expected signals and their stimulus dependence: V1;ac/V2 rf <e(xx) 2! - - V2;ac/V2 rfVdc
Mz - 2! - Signal extraction procedure from experiments: d dHzVexp 1;acestimated fromdVexp ac dHz Vdc=0d<e(xx) d!- - 2! d dHzVexp 2;acestimated from" dVexp ac dHz Vdc6=0dVexp ac dHz Vdc=0# d=m(xx) d!2p 3! - - TABLE I. Summary of the expected lineshapes and linewidths for the different signals that can be encountered in rf-voltage-induced FMR experiments. An hyphen is inserted when the concept is not applicable. B. Spin-waves within the free layer Conversely, the frequencies of the modes of the free layer are strongly affected by the device diameter (Fig. 4). First, the modes are pushed to higher frequencies as the device is shrunk. At remanence, the lowest frequency mode is at fQUP= 12:3GHz for a diameter of 500 nm; it reaches 19.5 GHz for 60 nm devices (not shown). Second, the frequency spacing between the free layer modes increases substantially when downsizing the device. The first effect – increase of fQUPat downscaling – is in- dicative of a dependence of some effective fields with the device diameter. Among the effective fields, the only ones that vary with the diameter are the exchange fields (positive contribution to the frequency fQUPif magnetization is non- uniform), the demagnetizing fields (positive contribution to the frequency fQUPat downscaling) and the local effective anisotropy fields in case some process damages alter locally the interface anisotropy at the perimeter of the free layer (neg- ative contribution to the frequency fQUPat downscaling) or alter the local magnetization of the rim of the free layer (pos- itive contribution to the frequency fQUPat downscaling). The exchange fields are related to the non uniformities of either the static configuration –the fact that the AP state is not perfectly uniform as inferred previously from the loop in Fig. 1– or non uniformities of the dynamic magnetization –i.e. the fact that the quasi-uniform mode is not a strictly uniform mode–. If the frequency increase was due to the sole demagnetizing ef- fects, it could be estimated from the demagnetizing factors of disks23which areNz1(3=8)t=a, wheretandaare the thickness and radius of the free layer. However a fQUP against 1=aplot (not shown) has a perceivable curvature near all sizes; an unwise linear fit through fQUPagainst the ex- pected Nzwould give a slope of 2:5T, which is obvi- ously too large for the magnetization of the free layer. This indicates that the sole change of the global shape anisotropy with the device diameter is insufficient to account for the in-crease offQUPat downscaling: exchange contributions or non uniformities induced by process damages also contribute to the frequencies. Exchange contributions should not contribute for the largest devices, however even for those devices the ex- perimental frequencies are larger than the ones expected from global shape anisotropy only, which argues for some process damages. Since the TMR is almost independent of the de- vice size14, we can reasonably assume that the MgO/FeCoB interface is not substantially affected by the patterning and that consequently the interface anisotropy is essentially pre- served at the rim of the free layer. We conclude that part of the increase of fQUPat downscaling is due to a reduced mag- netization (magnetically ”dead” or weak zone) near the edges of the free layer. This interpretation is probably very much stack and process technology dependent, hence it should not be considered as general. The second effect – the increased frequency spacing be- tween the modes at small diameters – is the expected effect of the confinement of the spin waves and the resulting in- crease of the exchange contribution to the mode frequencies17. The eigenmodes of perpendicularly magnetized circular disks are well understood and can be described analytically in a semi-quantitative manner21,24–28. The frequency spacing be- tween the lowest frequency modes scales with 0HJwhere HJ=2Ak2 0MSis a generalized exchange field with Athe ex- change stiffness. The effective wavevector kis reminiscent of the lateral confinement and reads k2= (u2 2u2 1)=a29=a2 whereu1andu2are the first zeros of the first and second Bessel functions21. The lowest frequency spinwave modes can be resolved only if their frequency spacing is comparable or greater than their linewidth 2 0(Hz+HkMs+HJ) (see appendix). This condition can be used to define a critical device diam- eter: a2 crit=9A 0MS(Hz+HkMs)(4) For large devices with aacritwe expect to observe a quasi-7 continuum of overlapping modes above fQUP, while discrete non-overlapping modes are anticipated in the opposite limit. Typical parameters of an FeCoB-based free layer include a magnetization of 0Ms= 1:2T and a Gilbert damping of29 = 0:01. From the quasi-uniform mode frequency, we can get our effective anisotropy which is HkMs= 330 kA/m. If the exchange stiffness of the free layer was bulk-like (i.e. A= 22 pJ/m) like in ref. 17, the critical diameter would be2acrit= 444 nm. In practice the small frequency spac- ings between the modes of our samples indicates that the ex- change stiffness of our free layer is in the range of 6-7 pJ/m i.e. well below the bulk value. This estimate of the exchange stiffness was deduced assuming perfectly pinned boundary conditions for the spin-waves at the device edge, which is a questionable30assumption. However the exchange stiffness is anyway weak in the free layer and this can be also qualita- tively seen directly from the spin wave spectroscopy: indeed the frequency spacing of the lowest modes of the reference layer system is typically twice larger that the frequency spac- ing of the lowest modes of the free layer [see for instance Fig. 3(b)]. While the reason for this small value of the free layer exchange stiffness is not entirely clear, we emphasize that having such a small exchange stiffness is not uncommon in magnetic systems that comprise only a small number of atomic layers, starting for instance from 2 pJ/m for a single layer of iron31. Anyway with these parameters, we expect a clear separation of the lowest frequency modes at rema- nence provided that the device diameter is much smaller than 2acrit= 250 nm. In practice for 300 and 500 nm devices a fine structure can still be detected in the spin-wave density of states [see Fig. 4(c)] but it is hard to count the modes and guess their frequencies out of this fine structure. In the remainder of this paper, we shall thus only consider devices of diameter less than 200 nm, in which the different spin-wave modes can be unambiguously resolved [see Fig. 4(e-f)]. IV . LINESHAPE EVOLUTIONS WITH BIAS AND EXTRACTION OF THE GILBERT DAMPING Let us now compare the shapes of the experimental rectified signal with those expected (see appendix). For that purpose we harness the different bias dependences (Table I) of the rec- tified signals V1;acandV2;acto isolate each of them in the experimental signal Vac. We identify V1;acto the experimen- tal curveVacmeasured at Vdc= 0, and we construct an esti- mate ofV2;acby subtracting Vacmeasured at Vdc= 0 from that measured at Vdc6= 0 (see Table I). Note because of this subtraction, any dependence of the spin-wave frequency with thedcvoltage will prevent the measurement of the voltage de- pendence of the damping factor or of the voltage dependence of the exciting torques. We illustrate this procedure in Fig. 5 in which we plot the field derivatives of the so-calculated V1;acandV2;acrectified signals in both fixed field or fixed frequency experimental con- ditions for a device of diameter 90 nm. We center the curves on the second lowest frequency eigenmode since it provides (a)(b) (c)(e)(f)(d)FIG. 4. (Color online). Dependence of the field derivative of the ac voltage over the device diameter. Top panels: full spectral depen- dence in the window [6;40GHz][90;90mT]for 90 nm (left) and 500 nm (right) diameter devices. Bottom panels: frequency de- pendence of the field derivative of the acvoltage after self-conformal averaging for various device sizes. The signal amplitudes have been normalized to ease their comparison. the largest signals and it is reasonably separated from both the quasi-uniform precession mode and from the other high order modes. The obtained experimental V1;accurves [see Fig. 5 (a) and (d)] have the expected line shapes (see appendix) with a negative peak surrounded by two tiny positive halos (areas shaded in red). The separation between the two zero crossings is9:50:5mT or 28510MHz. The obtained experimental V2;accurves also have the expected line shape of the deriva- tive of a Lorentzian distribution (see appendix). As expected, the sign of the response changes with the dcbias voltage. The separation between the positive and negative maxima of the distribution are 6:10:5mT and 17010MHz. These four different ways of measuring the linewidths [Fig. 5 (a, d, b, e)] are consistent with a free layer damping of= 0:0110:0003 . Indeed this value of damping would predict linewidths of 2f= 295 MHz and 20!= 0= 9:54mT (materialized as black bars in [Fig. 5 (a, d) and (d)]) and1:15f= 171 MHz or 1:15!= 0= 6:21mT (material- ized as blue bars in [Fig. 5 (b, c, e, f)]). This proposed value of damping is also consistent with the linewidths of higher order spin waves that appear at larger fre- quencies but with a lower signal. This is illustrated in fig. 6 where a comparison is drawn between the values of the damp- ing estimated for each applied field from the second (and most intense) mode and from the third mode for a device of diame- ter 80 nm. The used procedure is a direct fit of the experimen-8 tal lineshapes to the derivative of Eq. 7 with the damping, the resonance frequency and the signal amplitude as free parame- ters. For this specific device, the estimates of the damping pa- rameter are subjected to a random error of standard deviation 0.0018 around a mean value of = 0:0119 . It is interesting to note that the non-local contributions32,33to the damping ex- pected for the relatively large wavevectors of the second and third spin-wave modes seem to be too small to be observed in our samples. Note that as mentioned earlier, the same pro- cedure can in principle not be applied to the quasi-uniform mode since it exhibits a strong dependence of the mode am- plitude with the field which invalidates the procedure to some extent. For the sake of completeness of this paper, we have anyway fitted the experimental QUP lineshapes with the field derivative of Eq. 8; this is not possible near zero field, as the corresponding signal vanishes. The value of the Gilbert damp- ing that would be illegitimately deduced would be 0:01, i.e. 20% lower than the correct value. Besides, the estimates from the QUP mode would exhibit a substantially larger spread in the fit results [compare the histograms in in fig. 6(b)]. For these two reasons, we consider that the reliable estimate of the damping is the one extracted from the non-uniform modes. Above 100 mV of dcbias, the amplitude of the constructed experimental V2;acstart to depart from proportionality with Vdcand a frequency shift is observed, as expected when dc field-like spin torques are applied. This comes with by a dis- tortion of the line shape, probably linked to the modification of the spin waves lifetimes by spin-transfer torque as commonly observed in in-plane magnetized MTJs34. V . SUMMARY AND CONCLUSIONS In this paper, we have studied how to use rf-voltage- induced ferromagnetic resonance to study the spin-wave den- sity of states and the Gilbert damping in perpendicularly mag- netized disks embodied in magnetic tunnel junctions. We have applied the field along the easy axis to preserve the cylindrical symmetry of the magnetization energy functional. The inter- est of this configuration is that all the current-induced torques that potentially excite the dynamics yield the same type of susceptibility spectral shape. Additionally, this configuration is the sole in which the applied field and the frequency play similar roles near FMR so that consistency crosschecks be- tween variable-field and variable-frequency experiments can be performed to reveal and suppress potential experimental artefacts. Working in a situation in which the fixed layer and the free layer are oppositely magnetized in a convenient way to clas- sify the spin-waves according to their hosting layer, as the two sub-systems have opposite eigenmode frequency-versus-field slopes. The dcbias dependence of the signal of the quasi- uniform mode is peculiar and can be used to ambiguously identify the free layer quasi-uniform mode in the manifold of spin-waves. The non-uniform (higher order) spin-waves are easier to analyze, as their amplitudes weakly depend on the applied field so that field differentiation can be used safely for background subtraction. Optionally, the dynamic range of theexperiment can be improved by self-conformal averaging of the resonance spectra. The unambiguous identification of the spin-wave frequen- cies requires devices that are sufficiently small to avoid that the spinwave modes overlap into a quasi-continuous density of states. The critical device size is set by the exchange stiffness, the damping, the magnetization and the effective anisotropy field. In practice, device diameters below 200 nm are needed in our low-damped FeCoB-based PMA system. For each spin-wave mode, the rf-voltage-induced spin- wave spectra contain contributions from two different phys- ical mechanisms. The first one is the standard STT-FMR-like signal, whose spectral shape is a linear transverse susceptibil- ity term. It is independent from the dcvoltage applied across the MTJ. The second one is a variation of the time-averaged magnetic configuration when the rfvoltage is applied. It is proportional to the dcvoltage applied across the MTJ and it has the spectral shape of a non-linear longitudinal susceptibil- ity. The bias dependence can be used to separate these two signals. The analysis of their spectral shape yields the Gilbert damping within the precessing layer. A single value of the damping factor is found to account for the lineshapes of all studied spin-waves. The spectra of rf-voltage-induced rectified voltages for a vanishing dcvoltage bias are in principle sufficient to get the Gilbert damping of the dynamically active layer. However as microwave methods are prone to artefacts, a consistency check exploiting the bias dependence of the resonance spectra is useful for a consolidation of the numerical estimation of the damping. This work was supported in part by the Samsung Global MRAM Innovation Program, who provided also the samples. Critical discussions with Vladimir Nikitin, Jean-Paul Adam, Joo-V on Kim and Paul Bouquin are acknowledged. VI. APPENDIX: SUSCEPTIBILITIES IN AN IDEALIZED PMA FILM In this appendix, our aim is to determine the transverse and longitudinal microwave susceptibility versus frequency fand static fieldHzfor a PMA film as a response to an harmonic transverse field hxcos(!t)~ ex. We shall write the equations with this transverse field hxbut any other effect that yields a torque possessing a component transverse to the static mag- netization will yield similar lineshapes. This includes current- induced Oersted-Ampere fields but also Slonczewski STT and field-like STT as soon as the reference layer magnetization ~Mrefis not strictly collinear with that of the free layer ~Mfree. The susceptibility tensor will be used to deduce the shape of the line expected in rf-voltage-induced FMR, as summarized in Table I. Throughout this appendix, we assume a dcfield Hzperfectly perpendicular to the plane and a free layer mag- netization~M=Mz~ ez+mx~ ex+my~ ey, where the transverse terms are assumed small and written as complex numbers in the frequency space. For conveniency, we will use the nota- tionH0=Hz+HkMs. We shall also write the frequencies in field units and define !0=!= 0. This is meant to empha-9 ddHzVexp2,a cforVdc= 100 mVddHzVexp1,a cforVdc=0 ddHzVexp2,a cforVdc=100 mV FIG. 5. (Color online). Rectified acsignals versus frequency at fixed applied field (left panels) or versus field at fixed frequency (right panels). The plotted data ared dHzV1;ac(black, panels a and d) and d dHzV2;ac(blue, panels b, c, e and f) as estimated according to the formulas of Table I. The arbitrary vertical scale is the same for all panels. The dotted black lines are separated by 9.5 mT and 270 MHz. The dotted blue lines are separated by 6.1 mT or 170 MHz. These dotted lines correspond to the expected linewidth for a damping of 0.011. The panel (b) is measured for a device different from that of the other panels. size the fact that the generalized field H0and the generalized frequency!0play very similar roles in the FMR of perpendic- ularly magnetized macrospin when near resonnance. We shall systematically assume that 1and only keep the lowest order of the damping terms in the equations. In sections VI A -VI C we make the macrospin approximation. This approxi- mation is discussed in section VI D. A. Transverse linear susceptibility Following the usual procedure, we project the linearized Landau-Lifshitz-Gilbert equation along ~ exet~ ey:  H0(my+mx) +imx!0=hxMs H0(mxmy)imy!0=hxMs We then invert this system of equations to get the susceptibil- itiesmx=xxhxandmy=yxhx. They are: xx=MsH0 H02!02+ 2iH0!0(5) and yx=iMs!0 H02!02+ 2iH0!0(6) -100- 500 5 01 000.0000.0050.0100.0150.0200 .0080 .0100 .0120 .014Third modeSecond modeOccurrence (a. u.)D amping parameterQuasi-uniform mode( b)DampingF ield (mT)(a)FIG. 6. (Color online). Statistical view of the Gilbert damping parameters obtained from the fitting of the three lowest spin-wave resonances of a device of 90 nm diameter. For each applied field, the spectral line shapes of the quasi-uniform mode is fitted using d dHzV2;acfunction (red symbols) while the lineshapes of the two next modes are fitted withd dHzV1;acfunction (blue and black). The panel (a) gathers the values of the damping parameters that best fit each spectrum recorded at a given applied field. Panel (b) displays the histogram of the distribution of these estimates of the Gilbert damping for the non-uniform modes (dashed-dotted line histogram and its blue Gaussian guide to the eye, of half width 0.0009) and a Gaussian fit (red curve) of the distribution for the quasi-uniform mode, of half width 0.0015. Several points are worth to remind: (i) The dctransverse susceptibilities are dc xx=Ms=H0and dc yx= 0. (ii) The in-phase transverse susceptibility xxis peaked at the FMR condition !0=H0. It reachesFMR xx =1 2idc xx. (iii) When near the FMR condition, we have yxixx. Hence the two transverse components of the magnetization are in quadrature and the forced precession is essentially circular. Note that this holds true despite the fact that the pumping field is linearly polarized (i.e. along (x)only). It would also remain true for other (e.g STT) pumping torques. From Eq. 5 we deduce the classical expressions for the real and imaginary parts of the transverse susceptibility: <e(xx) =MsH0(H02!02) 42H02!02+ (H02!02)2(7) =m(xx) =2Ms!0H02 42H02!02+ (H02!02)2(8) The lineshapes given by the above expressions are shown in Fig. 7. Their main properties are summarized in Table I.10 B. Longitudinal non-linear susceptibility Let us now express the non-linear change of the longitu- dinal magnetization
Mzthat occurs due to the precession. This can be viewed as an rf-induced reduction of the rema- nence. Using the circularity of the precession near the FMR resonance and the conservation of the magnetization norm to second order in mx;y, one gets:
Mzjjhxjj2 2MSM2 sH02 [H02!02]2+ 42H02!02(9) wherejjhxjjis the (constant) amplitude of the applied rffield. It is worth noticing that the longitudinal loss of the magneti- zation is stationary (constant in time) despite the fact that the magnetization precesses continuously. C. Lineshapes and linewidths of the susceptibiliies and their derivatives The different susceptibility expressions are plotted in Fig. 7. The lineshape of the functions
Mzandxxare essentially determined by their denominator which are the fast varying functions of eqs. 8 and 9. As xxand
Mzare two signatures of the same resonance process, their denominators are equal (see eqs. 8 and 9) and a simple algebra confirms that =m(xx) and
Mzlead to the same frequency or field linewidths (Half Width at Half Maximum) which are:
!= 2!or equivalently,
Hz= 2H0(10) Note that
!(resp.
Hz) is also the frequency (resp. field) spacing between the positive maximum and negative maxi- mum of<e(xx)about the FMR condition. We stress that this linewidth is different from that obtained by conventional FMR in which people examine the derivative of the absorption signald=m(xx) dHzversusHz. The peak-to- peak separation of the conventional FMR signal is
Hz= 2p 3H0=2p 3!0. The factor 2=p 3is 1.1547. The spectral shapes of<e(xx)and
Mzas deduced above are to be used in the main part of the paper to describe respectively V1;ac andV2;ac(Table I). D. Linewidth beyond the macrospin approximation Some words of caution are needed as the calculations done so far the appendix are for the uniform precession mode of a macrospin, while most of our experimental results were ob- tained on the higher order (non-uniform) spin-waves. When modeling non-uniform spin-waves, one needs to take into ac- count additional exchange and dipole-dipole terms. For non-uniform spin-waves in perpendicularly magnetized system, the exchange fields related to the non uniformity of the dynamical magnetizations mxandmycan be added to H0 to form a new generalized field ~H=H+HkMs+2Ak2 0MS, /s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s45/s49/s48/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48 /s77 /s122 /s82/s101 /s40 /s41 /s112/s112/s32/s82/s101/s115/s111/s110/s97/s110/s99/s101/s45/s108/s105/s107/s101 /s71/s101/s110/s101/s114/s97/s108/s105/s122/s101/s100/s32/s102/s105/s101/s108/s100/s32/s111/s114/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40 /s39/s32/s111/s114/s32/s72/s39/s41/s70/s87 /s72/s77/s100/s100/s73/s109 /s40 /s41 /s100/s100 /s77 /s122 /s111/s114/s100/s100/s82/s101 /s40 /s41 /s73/s109 /s40 /s41/s32 /s32/s76/s111/s115/s115/s45/s108/s105/s107/s101/s111/s114FIG. 7. (Color online). Transverse susceptibilities, longitudinal sus- ceptibility and their derivatives in the PMA macrospin model. The curves are plotted for = 0:02and a resonance condition of unity. The responses amplitudes have been normalized to ease the compar- ison between the different lineshapes. The black horizontal segment in the bottom panel is the FWHM of =m(xx)and
Mzor equiva- lently the peak-to-peak separation of <e(xx)(also sketched as the black square dots). The blue segment sketches the peak-to-peak sep- aration ofd=m(xx) d!0 (also sketched as the empty blue square dots). The area shaped in red is the positive halo that surrounds the main (negative) peak ind<e(xx) d!0. wherekis a generalized wavevector21,26,28. The additional exchange contributions act on mxandmyon equal footing, hence they maintain the circularity of the precession. The Gilbert linewidth for a non-uniform spin-wave is simply35,36:
!=!k@!k 0@~H(11) Where this equation holds as ~His a circular term. As a result of Eq. 11 the proportionality of an eigemode linewidth to its frequency (Eq. 10) is not broken by the exchange contribu- tions in non-uniform spin-waves. Conversely, the directional nature of the dipole-dipole in- teraction is such that the related effective fields act differently on the two dynamical magnetizations mxandmyfor spin- waves having a non-radial character. As a result the dynamic demagnetizing fields can not be simply added to the circu- lar precession H0term and they induce some ellipticity of the precession of the non-uniform spin-waves. Dipole-dipole interactions make the eigenmode frequency non-linear with the field (see Eq. 52 in ref. 22). For the lowest lying non- uniform spin-wave the dipole-dipole stiffness field is MSkt=2 withk=a. It is negligible against the generalized field ~Hfor the thickness used in practice in PMA systems meant for spin-torque applications. 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2017-03-21

We study how the shape of the spinwave resonance lines in rf-voltage induced FMR can be used to extract the spinwave density of states and the damping within the precessing layer in nanoscale tunnel junctions that possess perpendicular anisotropy. We work with a field applied along the easy axis to preserve the uniaxial symmetry of the system. We describe the set-up to study the susceptibility contributions of the spin waves in the field-frequency space. We then identify the maximum device size above which the spinwaves can no longer be studied in isolation as the linewidths of their responses make them overlap. The rf-voltage induced signal is the sum of two voltages that have comparable magnitudes: a first voltage that originates from the transverse susceptibility and rectification by magnetoresistance and a second voltage that arises from the non-linear longitudinal susceptibility and the resultant time-averaged change of the micromagnetic configuration. The transverse and longitudinal susceptibility signals have different dc bias dependences such that they can be separated by measuring how the device rectifies the rf voltage at different dc bias voltages. The transverse and longitudinal susceptibility signals have different lineshapes; their joint studies can yield the Gilbert damping of the free layer of the device with a degree of confidence that compares well with standard FMR. Our method is illustrated on FeCoB-based free layers in which the individual spin-waves can be sufficiently resolved only for disk diameters below 200 nm. The resonance line shapes on devices with 90 nm diameters are consistent with a Gilbert damping of 0.011. This damping of 0.011 exceeds the value of 0.008 measured on the unpatterned films, which indicates that device-level measurements are needed for a correct evaluation of dissipation.

Using rf voltage induced ferromagnetic resonance to study the spin-wave density of states and the Gilbert damping in perpendicularly magnetized disks

1703.07310v2

1 Low Gilbert Damping Constant in Perpendicularly Magnetized W/CoFeB/MgO Films with High Thermal Stability Dustin M. Lattery1†, Delin Zhang2†, Jie Zhu1, Jian-Ping Wang2*, and Xiao jia Wang1* 1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA 2Department of Electrical and Computer Engineering , University of Minnesota, Minneapolis, MN 55455, USA †These authors contributed equally to this work. *Corres ponding a uthor s: wang4940@umn.edu & jpwang@umn.edu Abstract: Perpendicular magnetic materials with low damping constant and high thermal stability have great potential for realizing high -density, non -volat ile, and low -power consumption spintronic devices, which can sustain operation reliability for high processing temperatures. In this work, we study the Gilbert damping constant ( α) of perpendicularly magnetized W/CoFeB/MgO films with a high perpendicular magnetic anisotropy (PMA) and superb thermal stability. The α of these PMA films annealed at different temperatures ( Tann) is determined via an all -optical T ime- Resolved Magneto -Optical Kerr Effect method. We find that α of these W/CoFeB/MgO PMA films decreases with increasing Tann, reaches a minimum of α = 0.016 at Tann = 350 °C, and then increases to 0.024 after post -annealing at 400 °C. The minimum α observed at 350 °C is rationalized by two competing effects as Tann becomes higher : the enhanced crystallization of CoFeB and dead -layer growth occurring at the two interfaces of the CoFeB layer. We further demonstrate that α of the 400 °C -annealed W/CoFeB/MgO film is comparable to that of a reference Ta/CoFeB/MgO PMA fil m annealed at 300 °C , justif ying the enhanced thermal stability of the W - seeded CoFeB films. 2 I. INTRODUCTION Since the first demonstration of perpendicular magnetic tunnel junctions with perpendicular magnetic anisotropy (PMA) Ta/CoFeB/MgO stacks [1], interfacial PMA materials have been extensively studied as promising candidates for ultra -high-density and low -power consumption spintronic devices, including spin -transfer -torque magnetic rando m access memory (STT -MRAM) [2,3] , electrical -field induced magnetization switching [4-6], and spin -orbit torque (SOT) devices [7-9]. An interfacial PMA stack typically consists of a thin ferromagnetic layer (e.g., CoFeB) sandwiched between a heavy metal layer ( e.g., Ta) and an oxide layer ( e.g., MgO). The heavy metal layer interface with the ferromagnetic layer is responsible for the spin Hall effect , which is favorable for SOT and skyrmion devices [10,11] . The critical switching current ( Jc0) should be minimized to decrease the power consumption of perpendicular STT -MRAM and SOT devices . Reducing Jc0 requires the exploration of new materials with low Gilbert damping constant (α), large spin Hall angle ( θSHE), and large effective anisotropy ( Keff) [12,13] . In addition, spintronic devices need to sustain operation reliability for processing temperatures as high as 400 °C for their integration with existing CMOS fabrication technologies, providing the standard back -end-of-line process compatibility [14]. Based on this requirement, the magnetic properties of a PMA material shou ld be thermally stable at annealing temperature s (Tann) up to 400 °C. Unfortunately, Ta/CoFeB/MgO PMA films commonly used in spintronic devices cannot survive with Tann higher than 350 °C, due to Ta diffusion or CoFeB oxidation at the interfaces [15,16] . The diffusion of Ta atoms can act as scattering sites to increase the spin -flip probability [17] and lead to a higher Gilbert Damping constant ( α), a measure of the energy dissipation from the magnetic precession into phonons or magnons [18]. 3 Modifying the composition of thin-film stack s can prevent heavy metal diffusion , which is beneficial to both lowering α and improving thermal stability [19]. Along this line , new interfacial PMA stacks have been developed, such as Mo/CoFeB/MgO, to circumvent the limitation on device processing temperatures [20,21] . While Mo/CoFeB/MgO films can indeed exhibit PMA at temperatures higher than 400 °C , they cannot be used for SOT devices due to the weak spin Hall effect of the Mo layer [20,21] . Recently, W/CoFeB/MgO PMA thin films have been proposed because of their PMA property at high post -annealing temperature [22], and the large spin Hall angle of the W laye r (θSHE ≈ 0.30) [23], which is twice that of a Ta layer (θSHE ≈ 0.12 ~ 0.15) [9]. While there have been a few scattered studies demonstrating the promise of fabricating SOT devices using the W/CoFeB/MgO stacks, special attention has been given to their PMA properties and functionalities as SOT devic es [24]. A systematic investigation is lacking on the effect of Tann on α of W/CoFeB/MgO PMA thin films. II. SAMPLE PREPERATION AND MAGNETIC CHARACTERIZATION In this work, we grow a series of W(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) thin films on Si/SiO 2(300) substrates (thickness in nanometers) with a magnetron sputtering system (<5×1 08 Torr). These films are then post -annealed at varying temperatures (Tann = 250 ~ 400 °C) within a high -vacuum furnace (<1×106 Torr) and their magnetic properties and damping constants as a function of Tann are systematically investigated . For comparison, a reference sample of Ta(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) is also prepared to examine the effect of seeding layer to the damping constant of these PMA films. The saturation magnetization ( Ms) and anisotropy of these films are measured with the Vibrating Sample Magnetometer (VSM) module of a Physical 4 Property Measurement System . Figure 1 plots the magnetic hysteresis loops and associated magnetic properties extracted from VSM measurements. Figure 1. Room temperature magnetic hysteresis loops of W/CoFeB/MgO PMA thin films post - annealed at ( a) 250 °C, (b) 300 °C, (c) 350 °C, and ( d) 400 °C. Black and red curves denote external magnetic field ( Hext) applied along and perpendicular to the film plane, res pectively . (e-g) Plots of the saturation magnetization ( Ms), effective interfacial anisotropy ( Keff × t), and interfacial anisotropy ( Ki) as functions of Tann. With the increase of Tann, Ms for the W/ CoFeB /MgO films decreases from ~780 to ~630 emu/cm3 [Fig. 1 (e)]. The effective interfacial anisotropy [(Keff × t) depicted in Fig. 1 (f)] shows a n increasing trend with Tann (from ~0.18 to ~0.34 erg/cm2 when Tann increases from 250 to 350 °C) and saturates at Tann = 350 °C. The positive values of Keff × t suggest that these W/CoFeB /MgO films maintain high PMA properties at elevated temperatures including 400 °C, demonstrating their enhanced thermal stability compared to Ta/ CoFeB /MgO films that can only sustain PMA up to 350 ° C. Removing the influence of the demagnetization energy from Keff × t 5 results in the interfacial anisotropy ( Ki), which changes from 0.6 to 0.7 erg/cm2 with the increase of Tann up to 350 °C and then decreases to ~0.6 erg/cm2 at Tann = 400 °C [Fig. 1(f)]. Details about the determination of Ki and Keff are provided in Section S1 of the Supp lemental Material (SM). Figure 2 . The dead -layer extraction results. ( a), (b), (c), and ( d) represent the series of samples annealed at 250, 300, 350, and 400 C respectively. The tdead value is the extrapolated x-axis intercept from the linear fitting of the thickness -dependent saturation magnetization area product (Ms×t). We attribute the decrease of Ki at high Tann to the growth of a dead layer at the CoFeB interfaces, which becomes prominent at higher Tann. To quantitatively determine the thickness of the dead layer as Tann increases, we prepare four sets of PMA stacks of W(7)/CoFeB( t)/MgO(2)/Ta( 3). One set contains five stacks with varying thicknesses for the CoFeB layer ( t = 1.2, 1.5, 1.8, 2.2, and 2.5 nm) and is post -annealed at a fixed Tann. Four Tann of 6 250, 300, 350, and 400 °C are used for four sets of the PMA stacks, respectively. The anne aling conditions are the same as those for the W(7)/CoFeB(1.2)/MgO(2)/Ta(3) samples for discussed previously. We measure the magnetic hysteresis loops of these samples using VSM and plot their saturation magnetization area product ( sMt ) as a function of film thickness ( t) in Fig. 2. Linear extrapolati on of the sMt data provides the dead -layer thickness, at which the magnetization reduces to zero as illustrate d by the x-axis intercept in Fig. 2. III. TR-MOKE MEASUREM ENTS The magnetization dynamics of these PMA thin films are determined using the all-optical Time-Resolved Magneto -Optical Kerr Effect (TR -MOKE) method [25-29]. This pump -probe method utilizes ultra -short laser pulses to thermally demagnetize the sample and probe the resulting Kerr rotation angle ( θK). In the polar -MOKE configuration, θK is proportional to the change of the out-of-plane component of magnetization [Mz in Fig. 3(a)] [30]. Details of the TR - MOKE setup are provided in Section S2 of the SM. The TR-MOKE signal is fitted to the equation // K sin 2t C tA Be D ft e , where A, B, and C are the offset, amplitude, and exponential decaying constant of the thermal background , respectively . D denotes the amplitude of oscillations , f is the resonance frequency , φ is a phase shift (related to the demagnetization process), and is the relaxation time of magnetization precession. Directly from TR -MOKE measurements, an effective damping constant (αeff) can be extracted based on the relationship αeff = 1/(2πf). However, αeff is not an intrinsic material property; rather, it depends on measurement conditions, such as the applied field direction [θH in Fig. 3(a)], the magnitude of the applied field ( Hext), and inhomogeneities of the sample ( e.g. local variation in the magnetic properties of the sample) [31,32] . 7 Figure 3. (a) Definition of the parameters and angles used in TR -MOKE experiments. The red circle indicates the magnetization precession. θ is the equilibrium direction of the magnetization . θK is measured by the probe beam at a given time delay (Δ t). (b) The TR -MOKE data (open symbols) and model fitting of θK (black curves) for the 400 °C sample at 76° , for varying Hext from 2.0 to 20 kOe. To obtain the Gilbert damping constant, the inhomogene ous contribution needs to be removed from αeff, such that the remaining value of damping is a n intrinsic material property and independent of the measurement conditions. To determine the inhomogeneous broadening in the 8 sample, the effective anisotropy field ( k,eff eff s 2/ H K M ) needs to be pre -determined from either (1) the magnetic hysteresis loops; or (2) the fitting results of f vs. Hext obtained from TR -MOKE. The resonance frequency, f, can be related to Hext through the Smit -Suhl approach by identifying the second derivatives of the total magnetic free energy, which combines a Zeeman energy, an anisotropy energy, and a demagnetization energy [33-35]. For a perpen dicularly magnetized thin film, f is defined by Eqs. ( 1-4) [35]. 12 f H H , (1) 2 1 ext H k,eff cos cos H H H , (2) 2 ext H k,eff cos cos 2 H H H , (3) ext H k,eff2 sin sin 2HH . (4) This set of equations permits calculat ion of f with the material gyromagnetic ratio ( ), Hext, θH, Hk,eff, and the angle between the equilibrium magnetization di rection and the surface normal [θ, determined by Eq. ( 4)]. The measured values of f as a function of Hext can be fitted to Eq. (1) by treating and Hk,eff as fitting parameters. To minimize the fitting errors resulting from the inhomogeneous broadening effect that is pronounced at the low fields, we use measured frequencies at high fields ( Hext > 10 kOe) to determine Hk,eff. With a known value of Hk,eff , the Gilbert damping constant of the sample can be determined through a fitting of the inverse relaxation time (1/ ) to Eq. (5). The two terms of Eq. (5) take into account , respectively, contributions from the intrinsic Gilbert damping of the materials (first term) and inhomogeneous broadening (second term) [31]: 1 2 k,eff k,eff1 1 1 22dH H HdH , (5) 9 where H1 and H2 are related to the curvature of the magnetic free energy surface as defined by Eqs. (2) and (3) [35,36] . The second term on the right side of Eq. (5) capture s the inhomogeneous effect by attributing it to a spatial variation in the magnetic properties (Δ Hk,eff), analogous to the linewidth broadening effect in F erromagnetic Resonance measurements [37]. The magnitude of k,eff/d dH can be calculated once the relationship of ω vs. Hext is determined with a numerical method. Both α and Δ Hk,eff (the inhomogeneous term related to the amount of spatial variation in Hk,eff) are determined via the fitting of the measured 1/ based on Eq. ( 5). In this way, we can uniquely extract the field -independent α, as an intrinsic material property, from the ef fective damping ( αeff), which is directly obtained from TR -MOKE and dependent on Hext. It should be noted here that the inhomogeneous broadening of the magnetization precession is presumably due to the multi -domain structure of the materials, which becomes negligible in the high-field regime ( Hext >> Hk,eff) as the magnetization direction of multiple magnetic domains becomes uniform. This is also reflected by the fact that the derivative in the second term of Eq. (5) approaches zero for the high -field regim e [38]. IV. RESULTS AND DISCUSSION The measurement method is validated by measuring the Tann = 400 C at multiple angles (θH) of the external magnetic field direction. By repeating this meas urement at varying θH, we can show that α is an intrinsic material property , independent of θH. Figure 4(a) plots the resonance frequenc ies derived from TR -MOKE and model fittings for the 400 °C sample at two field directions ( θH = 76° and 89° ). For the data acquired at θH = 89°, a minimum f occurs at Hext ≈ Hk,eff. This corresponds to the smallest amplitude of magnetization precession, when the equilibrium direction of the magnetization is aligned with the applied field direction at the magnitud e of Hk,eff 10 [35]. The dip at this local minimum diminishes when θH decreases, as reflected by the comparison between the red ( θH = 89°) and blue ( θH = 76°) lines in Fig. 4(a). With the Hk,eff extracted from the fitting of frequency data with θH = 89°, we generate the plot of theoretically predicted f vs. Hext [θH = 76° theory, blue line in Fig. 4(a)], which agrees well with experimental data [open square s in Fig. 4(a)]. Figure 4. (a) Measured f vs. Hext results for the 400 C sample at θH = 89° (open circles) and θH = 76° (open squares) and corresponding modeling at θH = 89° (red line) and θH = 76° (blue line) . (b) The measured inverse of relaxation time (1/) at θH = 89° (open symbols) and the fitting of 1/ based on Eq. (5) (dotted line). For reference, the first term of 1/ in Eq. (5) is also plotted (solid line), which accounts for the contribution from the Gilbert damping only. (c) αeff as a function of Hext for θH = 89° (red circles). Black circles are the extracted Gilb ert damping, which is independent of Hext. The black dotted line shows the average of this extracted damping; ( d) and ( e) depict similar plots of 1/ and damping constants for θH = 76°. Error bars in ( b) through ( e) come from the uncertainty in the mathematical fitting. 11 The inverse relaxation time (1/ should also have a minimum value near Hk,eff for θH = 89° if the damping was purely from Gilbert damping [as shown by the solid lines in Figs. 4(b) and 4(d)]; however, the measured data do not follow this trend. Adding the inhomogeneous term [dotted lines in Figs. 4(b) and 4(d)] more accurately describes the field dependen ce of the measured 1/[open symbols in Figs. 4(b) and 4(d)] It should be noted that the dip of the predicted 1/ occurs when the frequency derivative term in Eq. (5) approaches zero; however, this is not captured by the measurement due to the finite interval over which we vary Hext. Figures 4(c) and 4(e) depict the field -dependent effective damping ( αeff) and the Gilbert damping ( α) as the intrinsic material’s property obtained from fitting the measured 1/. With the knowledge that the value of α extracted with this method is the intrinsic material property, we repeat this data reduction technique for the annealed W/CoFeB /MgO samples discussed in Fig. 1. The symbols in Fig. 5 represent the resonance frequency and damping constants (both effective damping and Gilbert damping) for all samples measured at θH ≈ 90°. The fittings for the resonance frequency [red lines , from Eq. (1)] are also shown to demonstrate the good agreement between our TR -MOKE measurement and theoretical prediction. The uncertainties of f, , and Hk,eff are calculated from the least-square s fitting uncertainty and the uncertainty of measuring Hext with the Hall sensor. 12 Figure 5 . Results for f (a-d) and αeff (e-h, on a log scale) for individual samples. For comparison, the Gilbert damping constant α is also plotted by subtracting the inhomogeneous terms from αeff. The dashed line in (e -h) indicat es the average α. All samples are measured at θH = 90° except for the 400 C sample ( θH = 89°). The summary of the anisotropy and damping measured via TR -MOKE is shown in Fig. 6. Figure 6(a) plots Hk,eff obtained from VSM (black open circle s) and TR -MOKE (blue open squares) , both of which exhibit a monotonic increasing trend as Tann becomes higher. Discrepancies in Hk,eff from these two methods can be attributed to the difference in the size of the probing region , which is highly localized in TR -MOKE but sample -averaged in VSM. Since Hk,eff determined from TR-MOKE is obtained from fitting the measured frequency for a localized region, we expect these values more consistently describ e the magnetization precession th an those obtained from VSM. The increase in Hk,eff with Tann can be partially attributed to the crystallization 13 of the CoFeB layer [32]. For temperatures higher than 350 °C, this increasing trend of Hk,eff begins to lessen, presumably due to the diffusion of W atoms into the CoFeB layer , which is more pronounced at higher Tann. The W diffusion process is also responsible for the decrease in Ms of the CoFeB layer as Tann increases [Fig. 1 (e)]. Subsequently, the decrease in Ms leads to a further - reduced demagnetizing energy and thus a larger Hk,eff. Similar observation of Ms has been reported in literature for Ta/CoFeB/MgO PMA structures and attribute d to the growth of a dead layer at the heavy metal/CoFeB inter face [1]. Figure 6(b) summarizes tdead as a function of Tann with tdead increas ing from 0.17 to 0.53 nm as Tann changes fr om 250 to 400 C, as discussed in Section II. Figure 6. Summary of the magnetic properties of W -seeded CoFeB as a function of Tann. (a) The dependence of Hk,eff on Tann obtained from both the VSM (black open circles) and TR-MOKE fitting (blue open squares ). (b) The dependence of dead -layer thickness on Tann. (c) Damping constants as a function of Tann. The minim um damping constant of α = 0.016 occurs at 350°C. The values for the all samples are obtained from measurements at θH = ~90°. For comp arison, α of the reference Ta/CoFeB/MgO PMA sample annealed at 300 °C is also shown as a red triangle in ( c). 14 Figure 6(c) depicts the dependence of α on Tann, which first decreases with Tann, reaches a minimum of 0.016 at 350 °C, and then increases as Tann rises to 400 °C. Similar trends have been observed for Ta/CoFeB/MgO previously (minimum α at Tann = 300 °C) [32]. We speculate that this dependence of damping on Tann is due to two competing effects: (1) the inc rease in crystallization in the CoFeB layer with Tann which reduces the damping, and (2) the growth of a dead layer, which results from the diffusion of W and B atoms and is prominent at higher Tann. At Tann = 400 °C, the dead -layer formation leads to a la rger damping presumably due to an increase in scattering sites (diffused atoms) that contribute to spin -flip events, as described by the Elliot - Yafet relaxation mechanisms [17]. The observation that our W -seeded samples still sustain excellent PMA properties at Tann = 400 °C confirms their enhanced thermal stability, compared with Ta/CoFeB/MgO stacks which fail at Tann = 350 °C or higher. The damping constants are comparable for the W/CoFeB/MgO and Ta/CoFeB/MgO films annealed at 300 °C, both of which are higher than that of the W/ CoFeB /MgO PMA with the optimal Tann of 350 °C. Nevertheless, our work focuses on the enhanced thermal stability of W - seeded CoFeB PMA films while still maintaining a relatively low damping constant. Such an advantage enables W -seeded CoFeB layers to be viable and promising alternatives to Ta/CoFeB/MgO , which is currently widely used in spintronic devices. V. CONCLUSION In summary, we deposit a series of W -seeded CoFeB PMA films with varying annealing temperatures up to 400 °C and conduct ultrafast all -optical TR -MOKE measurements to study their magnetization precession dynamics. The Gilbert damping, as a n intrinsic material property, is proven to be independent of meas urement conditions, such as the amplitudes and directions of 15 the applied field. The damping constant varies with Tann, first decreasing and then increasing, leading to a minimum of α = 0.016 for the sample anneale d at 350 °C. Due to the dead -layer growth , the damping constant slightly increases to α = 0.024 at Tann = 400 °C, comparable to the reference Ta/ CoFeB /MgO PMA film annealed at 300°C, which demonstrates the improved enhanced thermal stability of W/ CoFeB /MgO over the Ta/ CoFeB /MgO structures. This strongly suggests the great potential of W/ CoFeB /MgO PMA material systems for future spintronic device integration that requires materials to sustain a processing temperature as high as 400 °C. 16 Acknowledgements This work is supported by C -SPIN (award #: 2013 -MA-2381) , one of six centers of STARnet, a Semiconductor Research Corporation program, sponsored by MARCO and DARPA. The authors would like to thank Prof. Paul Crowell and Dr. Changjiang Liu for valuable discussions. Supplemental Materials Available: Complete description of the TR -MOKE mea surement method, the angular dependent result summary , and the description for determining the interface anisotropy. 17 References [1] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, A perpendicular -anisotropy CoFeB -MgO magnetic tunnel junction, Nat. Mater. 9, 721 (2010). [2] H. Sato, E. C. I. Enobio, M. Yamanouchi, S. Ik eda, S. Fukami, S. Kanai, F. 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2017-09-21

Perpendicular magnetic materials with low damping constant and high thermal stability have great potential for realizing high-density, non-volatile, and low-power consumption spintronic devices, which can sustain operation reliability for high processing temperatures. In this work, we study the Gilbert damping constant ({\alpha}) of perpendicularly magnetized W/CoFeB/MgO films with a high perpendicular magnetic anisotropy (PMA) and superb thermal stability. The {\alpha} of these PMA films annealed at different temperatures is determined via an all-optical Time-Resolved Magneto-Optical Kerr Effect method. We find that {\alpha} of these W/CoFeB/MgO PMA films decreases with increasing annealing temperature, reaches a minimum of {\alpha} = 0.016 at an annealing temperature of 350 {\deg}C, and then increases to 0.024 after post-annealing at 400 {\deg}C. The minimum {\alpha} observed at 350 {\deg}C is rationalized by two competing effects as the annealing temperature becomes higher: the enhanced crystallization of CoFeB and dead-layer growth occurring at the two interfaces of the CoFeB layer. We further demonstrate that {\alpha} of the 400 {\deg}C-annealed W/CoFeB/MgO film is comparable to that of a reference Ta/CoFeB/MgO PMA film annealed at 300 {\deg}C, justifying the enhanced thermal stability of the W-seeded CoFeB films.

Low Gilbert Damping Constant in Perpendicularly Magnetized W/CoFeB/MgO Films with High Thermal Stability

1709.07483v1

arXiv:2212.01029v4 [math.AP] 23 Aug 2023EQUIVALENCE BETWEEN THE ENERGY DECAY OF FRACTIONAL DAMPED KLEIN–GORDON EQUATIONS AND GEOMETRIC CONDITIONS FOR DAMPING COEFFICIENTS KOTARO INAMI AND SOICHIRO SUZUKI Abstract. We consider damped s-fractional Klein–Gordon equations on Rd, wheresdenotes the order of the fractional Laplacian. In the one-di mensional cased= 1, Green (2020) established that the exponential decay for s≥2 and the polynomial decay of order s/(4−2s) hold if and only if the damping coeffi- cient function satisfies the so-called geometric control co ndition. In this note, we show that the o(1) energy decay is also equivalent to these conditions in the case d= 1. Furthermore, we extend this result to the higher-dimens ional case: the logarithmic decay, the o(1) decay, and the thickness of the damp- ing coefficient are equivalent for s≥2. In addition, we also prove that the exponential decay holds for 0 < s <2 if and only if the damping coefficient function has a positive lower bound, so in particular, we can not expect the exponential decay under the geometric control condition. 1.Introduction We consider the following fractional damped Klein–Gordon equations onRd: (1.1) utt(t,x)+γ(x)ut(t,x)+(−∆+1)s/2u(t,x) = 0,(t,x)∈R≥0×Rd, wheres >0, and 0 ≤γ∈L∞(Rd). Here we note that γutrepresents the damping force and the operator ( −∆+1)s/2is defined by the Fourier transform on L2(Rd); (−∆+1)s/2u:=F−1(|ξ|2+1)s/2Fu, ξ∈Rd. We recast the equation ( 1.1) as an abstract first-order equation for U= (u,ut): Ut=AγU,Aγ=/parenleftbigg0 I −(−∆+1)s/2−γ(x)/parenrightbigg , (1.2) thenAγgenerates a C0-semigroup ( etAγ)t≥0onHs/2(Rd)×L2(Rd) (see [4]). Here the Sobolev space Hr(Rd) is defined by Hr(Rd):=/braceleftbigg u∈L2(Rd) :/bardblu/bardbl2 Hr=/integraldisplay Rd(|ξ|2+1)r|Fu(ξ)|2dξ <∞/bracerightbigg . In this paper, we discuss the decay rate of the energy E(t):=/bardbletAγ(u(0),ut(0))/bardblHs/2×L2=/parenleftbigg/integraldisplay Rd(|(−∆+1)s/4u(t,x)|2+|ut(t,x)|2)dx/parenrightbigg1/2 . 2020Mathematics Subject Classification. 35L05, 42A38. The first author was supported by JST SPRING Grant Number JPMJ SP2125 , the Inter- disciplinary Frontier Next-Generation Researcher Progra m of the Tokai Higher Education and Research System . The second author was supported by Japan Society for the Prom otion of Science (JSPS) KAKENHI Grant Number JP20J21771 and JP23KJ1939. 12 K. INAMI AND S. SUZUKI By standard calculus, we have E(t) =E(0) ifγ≡0 and the exponential energy decay ifγ≡C >0. In recent works, the intermediate case, that is, the case that γ= 0 on a large set is studied: Definition 1.1. We say that Ω ⊂Rdsatisfies the Geometric Control Condition (GCC) if there exist L >0 and 0< c≤1 such that for any line segments l∈Rdof lengthL, the inequality H1(Ω∩l)≥cL holds, where H1denotes the one-dimensional Hausdorff measure. Burq and Joly [ 2] proved that if γis uniformly continuous and {γ≥ε}satisfies (GCC) for some ε >0, then we have the exponential energy decay in the non- fractional case s= 2. After that, Malhi and Stanislavova [ 6] pointed out that (GCC) is also necessary for the exponential decay in the one-dimen sional case d= 1: Theorem 1.2 ([6, Theorem 1]) .Letd= 1, lets= 2, and let 0≤γ∈L∞(R)be continuous. Then the following conditions are equivalent: (1.3)There exists ε >0such that the upper level set {γ≥ε}satisfies (GCC). (1.4)There exist C,ω >0such that whenever (u(0),ut(0))∈H1(R)×L2(R), E(t)≤Cexp(−ωt)E(0) holds for any t≥0.1 (1.5) lim t→+∞/bardbletAγ/bardblH2×H1→H1×L2= 0. Note that for 0 ≤γ∈L∞(R), the condition ( 1.3) is also equivalent to that there existsR >0 such that inf a∈R/integraldisplaya+R a−Rγ(x)dx >0. In another paper [ 7], Malhi and Stanislavova introduced the fractional equation (1.1) and showed that if γis periodic, continuous and not identically zero, then we have the exponential decay for any s≥2 and the polynomial decay of order s/(4−2s) for any 0 < s <2 in the case d= 1. Remark. Nonzero periodic functions satisfy (GCC) in the case d= 1, but it is not true in the higher-dimensional case d≥2. Wunsch [ 10] showed that continuous periodic damping gives the polynomial energy decay of order 1 /2 for the non- fractional equation in the case d≥2. In addition, recently another proof and an extension to fractional equations of Wunsch’s result were obta ined by T¨ aufer [9] and Suzuki [ 8], respectively. Note that these results for periodic damping are established by reducing to estimates on the torus Td. Indeed, there are numerous studies on bounded domains; see references in [ 2] and [3], for example. Green [4] improved results of Malhi and Stanislavova as follows: Theorem 1.3 ([4, Theorem 1]) .Letd= 1, lets >0and let0≤γ∈L∞(R). Then the following conditions are equivalent: 1To be precise, the exponential decay estimate given in [ 6, Theorem 1] is a little weaker: E(t)≤Cexp(−ωt)/bardbl(u(0),ut(0))/bardblH2×H1. However, this is because the Gearhart–Pr¨ uss theorem in their paper ([ 6, Theorem 2]) is stated incorrectly. Using the theorem corre ctly (see Theorem 3.1), one can obtain the exponential decay estimate E(t)≤Cexp(−ωt)E(0) as in ( 1.4).ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 3 (1.3)There exists ε >0such that the upper level set {γ≥ε}satisfies (GCC). (1.6)There exist C,ω >0such that whenever (u(0),ut(0))∈Hs(R)×Hs/2(R), E(t)≤/braceleftBigg (1+t)−s 4−2s/bardbl(u(0),ut(0))/bardblHs×Hs/2if0< s <2, Cexp(−ωt)E(0) ifs≥2 holds for any t≥0. In comparison with the result of [ 7], which states that ( 1.6) holds if γis peri- odic, continuous and not identically zero, Theorem 1.3refines this result by giving a necessary and sufficient condition for ( 1.6). Furthermore, Theorem 1.3also im- proves the ( 1.3)⇐⇒(1.4) part of Theorem 1.2by extending it to fractional equations and removing the continuity of γ, but on the other hand, it lacks the (1.5) =⇒(1.3),(1.4) part. One of our goal is to recover this part for fractional equations: Theorem 1.4. Letd= 1, lets >0, and let 0≤γ∈L∞(R). Then the following conditions are equivalent: (1.3)There exists ε >0such that the upper level set {γ≥ε}satisfies (GCC). (1.6)There exist C,ω >0such that whenever (u(0),ut(0))∈Hs(R)×Hs/2(R), E(t)≤/braceleftBigg C(1+t)−s 4−2s/bardbl(u(0),ut(0))/bardblHs×Hs/2if0< s <2, Cexp(−ωt)E(0) ifs≥2 holds for any t≥0. (1.7) lim t→+∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0. We also give the following result, which says that we cannot expect th e expo- nential decay for 0 < s <2 under (GCC). Theorem 1.5. Letd≥1, let0< s <2, and let 0≤γ∈L∞(Rd). Then there exist C,ω >0such that whenever (u(0),ut(0))∈Hs/2(Rd)×L2(Rd), E(t)≤Cexp(−ωt)E(0) holds for any t≥0if and only if essinfRdγ >0. Note that the “if” part easily follows by reducing to the constant da mping case, so we will prove the “only if” part. Furthermore, we extend Theore m1.4to the higher-dimensional case d≥2 using a notion of thickness , which is equivalent to (GCC) in the case d= 1: Definition 1.6. We say that a set Ω ⊂Rdisthickif there exists R >0 such that inf a∈Rdmd(Ω∩(a+[−R,R]d))>0 holds, where mddenotes the d-dimensional Lebesgue measure. Then we have the following result: Theorem 1.7. Letd≥2, lets≥2, and let 0≤γ∈L∞(Rd). Then the following conditions are equivalent: (1.8)There exists ε >0such that the upper level set {γ≥ε}is thick.4 K. INAMI AND S. SUZUKI (1.9)There exists C >0such that whenever (u(0),ut(0))∈Hs(Rd)×Hs/2(Rd), E(t)≤C log(e+t)/bardbl(u(0),ut(0))/bardblHs×Hs/2 holds for any t≥0. (1.10) lim t→+∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0. The implication ( 1.8) =⇒(1.9) is a generalization of the result given by Burq and Joly [ 2]. They established ( 1.9) under the so-called network control condition, which is stronger than ( 1.8). Also, similarly to the case d= 1, the condition ( 1.8) is equivalent to that there exists R >0 such that inf a∈R/integraldisplay a+[−R,R]dγ(x)dx >0. Finally, we explain the organizationof this paper. In Sections 2,3, and4, we will give proofs of Theorems 1.4,1.5, and1.7, respectively. To prove these theorems, we use a kind of uncertainty principle and results of the C0semigroup theory. 2.Proof of Theorem 1.4 To prove this theorem, we use the following result by Batty, Boriche v, and Tomilov [ 1]: Theorem 2.1 ([1, Theorem 1.4]) .LetAbe a generator of a bounded C0-semigroup (etA)t≥0on a Banach space X, andλ∈ρ(A). Then the following are equivalent: (2.1)σ(A)∩iR=∅, (2.2) lim t→∞/bardbletA(λ−A)−1/bardblB(X)= 0. In the case A=Aγ, forλ∈ρ(Aγ), the map ( λ−Aγ)−1:Hs/2(R)×L2(R)→ Hs(R)×Hs/2(R) is surjective. Thus, we have: Lemma 2.2 ([6, Corollary 2]) .For the semigroup etAγof the Cauchy problem (1.2), the following are equivalent: (2.3)σ(Aγ)∩iR=∅, (2.4) lim t→∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0. Proof of Theorem 1.4.It is enough to show that ( 1.7) =⇒(1.3), since (1.3)⇐⇒ (1.6) is already known by Green [ 4] (Theorem 1.3) and (1.6) =⇒(1.7) is triv- ial. Suppose that ( 1.7) holds, that is, lim t→+∞/bardbletAγ/bardblHs×Hs/2→Hs/2×L2= 0. By Lemma2.2, we have iR⊂ρ(Aγ). This implies that for each λ∈R, there exists somec0>0 such that c0/bardblU/bardblHs/2×L2≤ /bardbl(Aγ−iλI)U/bardblHs/2×L2 holds for any U∈Hs(R)×Hs/2(R). Letting u∈L2(Rd) andU= ((−∆ + 1)−s/4u,iu), we obtain 2c0/bardblu/bardbl2 L2≤ /bardbl((−∂xx+1)s/4−λ)u/bardbl2 L2+/bardbl((−∂xx+1)s/4−λ+iγ)u/bardbl2 L2 ≤3/bardbl((−∂xx+1)s/4−λ)u/bardbl2 L2+2/bardblγu/bardbl2 L2.ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 5 Now we consider the case λ= 1. Let u∈Hs/2(R) satisfy supp /hatwideu⊂[−D,D] for someD >0, which is chosen later. For such u, we have /bardbl((−∂xx+1)s/4−1)u/bardbl2 L2=/integraldisplayD −D/bracketleftBig (|ξ|2+1)s/4−1/bracketrightBig2 |/hatwideu(ξ)|2dξ ≤/bracketleftBig (D2+1)s/4−1/bracketrightBig2 /bardblu/bardbl2 L2. Hence, taking D >0 small enough, we get some c >0 such that c/bardblu/bardblL2≤ /bardblγu/bardblL2 holds for any u∈Hs/2(R) satisfying supp /hatwideu⊂[−D,D]. Fixf∈ S(R)\ {0} such that supp /hatwidef⊂[−D,D] and write fa(x):=f(x−a) for each a∈R, so that /hatwidefa(ξ) =eiaξ/hatwidef(ξ). Then, for each a∈RandR >0, we have 0< c/bardblf/bardblL2=c/bardblfa/bardblL2≤ /bardblγfa/bardblL2=/parenleftBigg/integraldisplay [a−R,a+R]+/integraldisplay [a−R,a+R]c/parenrightBigg |γ(x)fa(x)|2dx. The second integralgoesto 0 as R→+∞sinceγis bounded and |fa|2is integrable, andthisconvergenceisuniformwithrespectto a. Furthermore,forthefirstintegral, we have/integraldisplaya+R a−R|γ(x)fa(x)|2dx≤ /bardblγ/bardblL∞/bardblf/bardbl2 L∞/integraldisplaya+R a−Rγ(x)dx, sinceγandfare bounded and /bardblfa/bardblL∞=/bardblf/bardblL∞. Thus, there exists R >0 such that inf a∈R/integraldisplaya+R a−Rγ(x)dx >0 holds, which is equivalent to ( 1.3). /square 3.Proof of Theorem 1.5 This section is based on the proof of Theorem 2 in Green [ 4]. To prove this theorem, we use the classical semigroup result by Gearhart, Pr¨ u ss, and Huang: Theorem 3.1 (Gearhart–Pr¨ uss–Huang) .LetXbe a complex Hilbert space and let (etA)t≥0be a bounded C0-semigroup on Xwith infinitesimal generator A. Then there exist C,ω >0such that /bardbletA/bardbl ≤Cexp(−ωt) holds for any t≥0if and only if iR⊂ρ(A)andsupλ∈R/bardbl(iλ−A)−1/bardblB(X)<∞. Proof of Theorem 1.5.We will prove the contraposition of the “only if” part of Theorem 1.5, that is, if the energy decays exponentially and essinf x∈Rdγ(x) = 0 holds, then s≥2. By the Gearhart–Pr¨ uss–Huang theorem and the exponential decay, there exists c0>0 such that c0/bardblU/bardbl2 Hs/2×L2≤ /bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2 holds for any U∈Hs/2(Rd)×L2(Rd) and any λ∈R. Letting u∈L2(Rd) and U= ((−∆+1)−s/4u,iu), we obtain 2c0/bardblu/bardbl2 L2≤ /bardbl((−∆+1)s/4−λ)u/bardbl2 L2+/bardbl((−∆+1)s/4−λ+iγ)u/bardbl2 L2 ≤3/bardbl(−∆+1)s/4−λ/bardbl2 L2+2/bardblγu/bardbl2 L2.6 K. INAMI AND S. SUZUKI Now letu∈L2(Rd) satisfy supp/hatwideu⊂ {ξ∈Rd:|(|ξ|2+1)s/4−λ| ≤K}=:Aλ(K) for some K, which is chosen later. For such u, we have /bardbl((−∆+1)s/4−λ)u/bardbl2 L2=/integraldisplay Aλ(K)[(|ξ|2+1)s/4−λ]2|/hatwideu(ξ)|2dξ ≤K2/bardblu/bardbl2 L2. Hence, taking K >0 small enough, we get some c >0 such that (3.1) c/bardblu/bardbl2 L2≤ /bardblγu/bardbl2 L2 holds for any u∈L2(Rd) satisfying supp /hatwideu⊂Aλ(K) with some λ∈R. We prove s≥2 by contradiction. Assume that s <2. In this case, the thickness of the annulus Aλ(K) is unbounded with respect to λ: lim λ→∞/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalBig (λ+K)4/s−1−/radicalBig (λ−K)4/s−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim λ→∞λ4/s−1 λ2/s=∞. Thus, the inequality ( 3.1) holds for any u∈L2(Rd) such that supp /hatwideuis compact. To see this, notice that there exist a∈Rdandλ∈Rsatisfying a+supp/hatwideu⊂Aλ(K) for such u. Therefore, letting ua(x):=eia·xu(x), we have c/bardblu/bardbl2 L2=c/bardblua/bardbl2 L2≤ /bardblγua/bardbl2 L2=/bardblγu/bardbl2 L2 since supp/hatwiderua=a+supp/hatwideu⊂Aλ(K). Now note that Eε:={x∈Rd:γ(x)< ε}has a positive measure for any ε >0, since essinf x∈Rdγ(x) = 0. For each ε >0, we take a subset Fε⊂Eεsuch that 0< md(Fε)<∞. TakeR,ε >0 arbitrarily and set fε:=χFε//radicalbig md(fε), gR,ε:=F−1χB(0,R)Ffε, whereχΩdenotes the indicator function of Ω ⊂Rd. By the definition, we have supp/hatwidestgR,ε⊂B(0,R) andgR,ε→fεasR→ ∞inL2(Rd). Therefore, applying the inequality ( 3.1) togR,ε, we get c/bardblgR,ε/bardblL2≤ /bardblγgR,ε/bardblL2 ≤ /bardblγfε/bardblL2+/bardblγ(gR,ε−fε)/bardblL2 =/parenleftbigg1 md(Fε)/integraldisplay Fε|γ(x)|2dx/parenrightbigg1/2 +/bardblγ(gR,ε−fε)/bardblL2 ≤ε+/bardblγ(gR,ε−fε)/bardblL2. Taking the limit as R→+∞, we obtain 0< c=c/bardblfε/bardblL2≤ε. This is a contradiction since ε >0 is arbitrary. /square 4.Proof of Theorem 1.7 The proof of ( 1.10) =⇒(1.8) is similar to that of ( 1.7) =⇒(1.3) in Section 2, and the implication ( 1.9) =⇒(1.10) is trivial. Therefore, we will show that (1.8) =⇒(1.9). We use a kind of the uncertainty principle to obtain a certain resolvent estimate for the fractional Laplacian:ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 7 Theorem 4.1 ([5, Theorem 3]) .LetΩ⊂Rdbe thick. Then there exist a constant C >0such that for each R >0, the inequality /bardblf/bardblL2(Rd)≤Cexp(CR)/bardblf/bardblL2(Ω) holds for any f∈L2(Rd)satisfying supp/hatwidef⊂B(0,R). In orderto obtain the logarithmic energydecay ( 1.9), we use the following result. Theorem 4.2 ([2, Theorem 5.1]) .LetAbe a maximal dissipative operator (and hence generate the C0-semigroup of contractions (etA)t≥0) in a Hilbert space X. Assume that iR⊂ρ(A)and there exists C >0such that /bardbl(A−iλI)−1/bardblB(X)≤CeC|λ| holds for any λ∈R. Then, for each k >0, there exists Ck>0such that /bardbletA(I−A)−k/bardblB(X)≤Ck (log(e+t))k holds for any t≥0. 4.1.Resolvent estimate. The proof of these propositions are based on [ 4]. Proposition 4.3. Lets≥1andΩ⊂Rdbe thick. Then there exist C,c >0such that for all f∈L2(Rd)and allλ≥0, cexp(−Cλ)/bardblf/bardbl2 L2(Rd)≤ /bardbl((−∆+1)s/2−λ)f/bardbl2 L2(Rd)+/bardblf/bardbl2 L2(Ω). Proof of Proposition 4.3.LetAλ:={ξ∈Rd:|(|ξ|2+ 1)1/2−λ1/s| ≤1}. Since Aλ⊂B(0,λ+2) and Ω is thick, Theorem 4.1implies that there exists C >0 such that (4.1) /bardblf/bardblL2(Rd)≤Cexp(Cλ)/bardblf/bardblL2(Ω) holds for any λ≥0 and any f∈L2(Rd) satisfying supp /hatwidef⊂Aλ. Next, we set a projection Pλ:=F−1χAλF, whereχAλdenotes the indicator function of Aλ. Then, sincePλfsatisfies the inequality ( 4.1) for each f∈L2(Rd), we obtain /bardblf/bardbl2 L2(Rd)=/bardblPλf/bardbl2 L2(Rd)+/bardbl(I−Pλ)f/bardbl2 L2(Rd) ≤Cexp(Cλ)/bardblPλf/bardbl2 L2(Ω)+/bardbl(I−Pλ)f/bardbl2 L2(Rd) =Cexp(Cλ)/bardblf−(I−Pλ)f/bardbl2 L2(Ω)+/bardbl(I−Pλ)f/bardbl2 L2(Rd) ≤2Cexp(Cλ)/bardblf/bardbl2 L2(Ω)+2Cexp(Cλ)/bardbl(I−Pλ)f/bardbl2 L2(Ω)+/bardbl(I−Pλ)f/bardbl2 L2(Rd) ≤2Cexp(Cλ)/bardblf/bardbl2 L2(Ω)+(2Cexp(Cλ)+1)/bardbl(I−Pλ)f/bardbl2 L2(Rd). Also, by Lemma 1 in [ 4], we have c/bardbl(I−Pλ)f/bardbl2 L2(Rd)≤ /bardbl((−∆+1)s/2−λ)f/bardbl2 L2(Rd) for some c >0 independent with λ. Therefore, we conclude that /bardblf/bardbl2 L2(Rd)≤Cexp(Cλ)/bracketleftBig /bardbl((−∆+1)s/2−λ)f/bardbl2 L2(Rd)+/bardblf/bardbl2 L2(Ω)/bracketrightBig . /square Proposition 4.4. Lets≥2and assume that Ω⊂Rdis thick. Then there exist C,c >0such that for all U= (u1,u2)∈Hs(Rd)×Hs/2(Rd)and allλ∈R, cexp(−C|λ|)/bardblU/bardbl2 Hs/2(Rd)×L2(Rd)≤ /bardbl(A0−iλI)U/bardbl2 Hs/2(Rd)×L2(Rd)+/bardblu2/bardbl2 L2(Ω).8 K. INAMI AND S. SUZUKI Proof of Proposition 4.4.ForU= (u1,u2)∈Hs(Rd)×Hs/2(Rd), we set /parenleftbigg w1 w2/parenrightbigg =/parenleftbigg (−∆+1)s/4−i (−∆+1)s/4i/parenrightbigg/parenleftbigg u1 u2/parenrightbigg . By the parallelogram law, we obtain /bardblw1/bardbl2 L2(Rd)+/bardblw2/bardbl2 L2(Rd)= 2/bardblU/bardbl2 Hs/2(Rd)×L2(Rd). Moreover, we have /bardbl(A0−iλI)U/bardbl2 Hs/2×L2=/bardbl(−∆+1)s/2(−iλu1+u2)/bardbl2 L2+/bardbl−(−∆+1)s/2u1−iλu2/bardbl2 L2 =/bardbl−λw1+w2 2+(−∆+1)s/2w1−w2 2/bardbl2 L2 +/bardbl−(−∆+1)s/2w1+w2 2+λw1−w2 2/bardbl2 L2 =/bardblλw1−(−∆+1)s/2w1/bardbl2 L2+/bardblλw2+(−∆+1)s/2w2/bardbl2 L2. Forλ≥0, applying Proposition 4.3tow1withs/2, we have 2cexp(−Cλ)/bardblU/bardbl2 Hs/2×L2 =cexp(−Cλ)(/bardblw1/bardbl2 L2+/bardblw2/bardbl2 L2) ≤ /bardbl((−∆+1)s/4−λ)w1/bardbl2 L2+/bardblw1/bardbl2 L2(Ω)+cexp(−Cλ)/bardblw2/bardbl2 L2 ≤ /bardbl((−∆+1)s/4−λ)w1/bardbl2 L2+2/bardblw1−w2/bardbl2 L2(Ω)+c/bardblw2/bardbl2 L2 ≤ /bardbl((−∆+1)s/4−λ)w1/bardbl2 L2+c/bardbl((−∆+1)s/4+λ)w2/bardbl2 L2+8/bardblu2/bardbl2 L2(Ω) ≤c/bardbl(A0−iλI)U/bardbl2 Hs/2×L2+8/bardblu2/bardbl2 L2(Ω). Forλ <0, we get the same inequality replacing the role of w1withw2. /square 4.2.Energy decay. Finally we prove ( 1.8) =⇒(1.9). By the assumption ( 1.8), Ω ={γ≥ε}is thick for some ε >0. Therefore, by Proposition 4.4, we have cexp(−C|λ|)/bardblU/bardbl2 Hs/2×L2≤ /bardbl(A0−iλI)U/bardbl2 Hs/2×L2+/bardblu2/bardbl2 L2(Ω) ≤2/bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2+(2+ε−2)/bardblγu2/bardbl2 L2(Ω). SinceA0is skew-adjoint, we obtain Re/a\}bracketle{t(Aγ−iλI)U,U/a\}bracketri}ht= Re/a\}bracketle{t(A0−iλI)U,U/a\}bracketri}ht−/a\}bracketle{tγu2,u2/a\}bracketri}ht=−/bardbl√γu2/bardbl2 L2. By the Cauchy–Schwarz inequality, we have D/bardblγu2/bardbl2 L2≤ /bardblγ/bardblL∞/bardbl√γu2/bardbl2 L2≤D2/bardblγ/bardbl2 L∞/bardbl(Aγ−iλ)U/bardbl2 Hs/2×L2 δ+δ/bardblU/bardbl2 Hs/2×L2. for anyD,δ >0. Taking D= 2+ε−2andδ=cexp(−C|λ|)/2, we obtain cexp(−C|λ|)/bardblU/bardbl2 Hs/2×L2 ≤2/bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2+(2+ε−2)/bardblγu2/bardbl2 L2(Ω) ≤2/bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2+(2+ε−2)2/bardblγ/bardbl2 L∞ cexp(−C|λ|)/bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2 +1 2cexp(−C|λ|)/bardblU/bardbl2 Hs/2×L2.ENERGY DECAY OF DAMPED KLEIN–GORDON EQUATIONS 9 By this inequality, we have cexp(−C|λ|)/bardblU/bardbl2 Hs/2×L2≤ /bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2, here the constants c,Cmay differ from the previous ones. Applying Theorem 4.2 withk= 1, we conclude that ( 1.9) holds. Acknowledgment The authors would like to thank Professor Mitsuru Sugimoto for valu able dis- cussions. References [1] C. J. K. Batty, A. Borichev, and Y. Tomilov, Lp-tauberian theorems and Lp-rates for en- ergy decay , J. Funct. Anal. 270(2016), no. 3, 1153–1201, DOI 10.1016/j.jfa.2015.12.003. MR3438332 [2] N. Burq and R. Joly, Exponential decay for the damped wave equation in unbounded domains , Commun. Contemp. Math. 18(2016), no. 6, 1650012, 27, DOI 10.1142/S0219199716500127 . MR3547102 [3] R. Chill, D. Seifert, and Y. Tomilov, Semi-uniform stability of operator semigroups and energy decay of damped waves , Philos. Trans. Roy. Soc. A 378(2020), no. 2185, 20190614, 24. MR4176383 [4] W. Green, On the energy decay rate of the fractional wave equation on Rwith relatively dense damping , Proc. Amer. Math. Soc. 148(2020), no. 11, 4745–4753, DOI 10.1090/proc/15100. MR4143391 [5] O. Kovrijkine, Some results related to the Logvinenko-Sereda theorem , Proc. Amer. Math. Soc.129(2001), no. 10, 3037–3047, DOI 10.1090/S0002-9939-01-059 26-3. MR1840110 [6] S. Malhi and M. Stanislavova, When is the energy of the 1D damped Klein-Gordon equa- tion decaying? , Math. Ann. 372(2018), no. 3-4, 1459–1479, DOI 10.1007/s00208-018-1725- 5. MR3880304 [7] ,On the energy decay rates for the 1D damped fractional Klein- Gordon equation , Math. Nachr. 293(2020), no. 2, 363–375, DOI 10.1002/mana.201800417. MR406 3985 [8] S. Suzuki, The uncertainty principle and energy decay estimates of the fractional Klein- Gordon equation with space-dependent damping , arXiv:2212.02481 (2022). [9] M. T¨ aufer, Controllability of the Schr¨ odinger equation on unbounded domains without geo- metric control condition , ESAIM Control Optim. Calc. Var. 29(2023), Paper No. 59, 11, DOI 10.1051/cocv/2023037. MR4621418 [10] J. Wunsch, Periodic damping gives polynomial energy decay , Math. Res. Lett. 24(2017), no. 2, 571–580, DOI 10.4310/MRL.2017.v24.n2.a15. MR36852 85 (Kotaro Inami) Graduate School of Mathematics, NagoyaUniversity, Furocho, Chikusa- ku, Nagoya, Aichi, 464-8602, Japan Email address :m21010t@math.nagoya-u.ac.jp (SoichiroSuzuki) Department of Mathematics, Chuo University, 1-13-27, Kasug a, Bunkyo- ku, Tokyo, 112-8551, Japan Email address :soichiro.suzuki.m18020a@gmail.com

2022-12-02

We consider damped $s$-fractional Klein--Gordon equations on $\mathbb{R}^d$, where $s$ denotes the order of the fractional Laplacian. In the one-dimensional case $d = 1$, Green (2020) established that the exponential decay for $s \geq 2$ and the polynomial decay of order $s/(4-2s)$ hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the $o(1)$ energy decay is also equivalent to these conditions in the case $d=1$. Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the $o(1)$ decay, and the thickness of the damping coefficient are equivalent for $s \geq 2$. In addition, we also prove that the exponential decay holds for $0 < s < 2$ if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition.

Equivalence between the energy decay of fractional damped Klein-Gordon equations and geometric conditions for damping coefficients

2212.01029v4

Self-similar shrinkers of the one-dimensional Landau–Lifshitz–Gilbert equation Susana Gutiérrez1and André de Laire2 Abstract The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere S2, at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles. Keywords and phrases: Landau–Lifshitz–Gilbert equation, self-similar expanders, backward self-similar solutions, blow up, asymptotics, ferromagnetic spin chain, heat flow for harmonic maps, quasi-harmonic sphere. 2010Mathematics Subject Classification: 82D40; 35C06; 35B44; 35C20; 53C44; 35Q55; 58E20; 35K55. 82D40;35C06;35B44; 35C20;53C44;35Q55;58E20;35K55 1 Introduction 1.1 The Landau–Lifshitz–Gilbert equation: self-similar solutions In this paper we continue the investigation started in [32, 33] concerning the existence and prop- erties of self-similar solutions for the Landau–Lifshitz–Gilbert equation (LLG). This equation describes the dynamics for the magnetization or spin in ferromagnetic materials [43, 27] and is given by the system of nonlinear equations ∂tm=βm×∆m−αm×(m×∆m), (LLG) wherem= (m 1,m2,m3) :RN×I−→S2is the spin vector, I⊂R,β≥0,α≥0,×denotes the usual cross-product in R3, and S2is the unit sphere in R3. This model for ferromagnetic materials constitutes a fundamental equation in the magnetic recording industry [53]. The parameters β≥0andα≥0are, respectively, the so-called exchange constant and Gilbert damping, and take into account the exchange of energy in the system and the effect of damping on the spin chain. By considering a time-scaling, one can assume without loss of generality that the parameters αandβsatisfy α∈[0,1]andβ=/radicalbig 1−α2. 1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom. E-mail: s.gutierrez@bham.ac.uk 2Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul Painlevé, F-59000 Lille, France. E-mail: andre.de-laire@univ-lille.fr 1arXiv:2002.06858v2 [math.AP] 21 May 2020From a purely mathematical point of view, the LLG equation is extremely interesting since it interpolates between two fundamental geometric evolution equations, the Schrödinger map equation and the heat flow for harmonic maps, via specific choices of the parameters involved. Precisely, we recall that in the limit case α= 1(and, consequently, β= 0), (LLG) reduces to the heat flow for harmonic maps onto S2, ∂tm−∆m=|∇m|2m, (HFHM) and, ifα= 0(no damping), it reduces to the Schrödinger map equation ∂tm=m×∆m. (SM) When 0<α< 1, (LLG) is of parabolic type. We refer the reader to [40, 29, 32, 33, 12, 14, 15, 13] and the references therein for more details and surveys on these equations. A natural question, that has proved relevant to the understanding of the global behavior of solutions and formation of singularities, is whether or not there exist solutions which are invariant under scalings of the equation. In the case of the LLG equation it is straightforward to see that the equation is invariant under the following scaling: If mis a solution of (LLG), thenmλ(t,x) =m(λx,λ2t), for any positive number λ, is also a solution. Associated with this invariance, a solution mof (LLG) defined on I=R+orI=R−is called self-similar if it is invariant under rescaling, that is m(x,t) =m(λx,λ2t),∀λ>0,∀x∈RN,∀t∈I. FixingT∈Rand performing a translation in time, this definition leads to two types of self- similar solutions: A forward self-similar solution or expander is a solution of the form m(x,t) =f/parenleftbiggx√ t−T/parenrightbigg ,for (x,t)∈RN×(T,∞), (1.1) and a backward self-similar solution or shrinker is a solution of the form m(x,t) =f/parenleftbiggx√ T−t/parenrightbigg ,for (x,t)∈RN×(−∞,T), (1.2) for some profile f:RN−→S2. In this manner, expanders evolve from a singular value at time T, while shrinkers evolve towards a singular value at time T. Self-similar solutions have received a lot of attention in the study of nonlinear PDEs because they can provide important information about the dynamics of the equations. While expanders are related to non-uniqueness phenomena, resolution of singularities and long time description of solutions, shrinkers are often related to phenomena of singularity formation (see e.g. [26, 18]). On the other hand, the construction and understanding of the dynamics and properties of self- similar solutions also provide an idea of which are the natural spaces to develop a well-posedness theory that captures these often very physically relevant structures. Examples of equations for whichself-similarsolutionshavebeenstudiedinclude,amongothers,theNavier–Stokesequation, semilinear parabolic equations, and geometric flows such as Yang–Mills, mean curvature flow and harmonic map flow. We refer to [37, 48, 36, 51, 5] and the references therein for more details. Although the results that will be presented in this paper relate to self-similar shrinkers of the one-dimensional LLG equation (that is, to solutions m:R×I−→S2of LLG), for the sake of context we describe some of the most relevant results concerning maps from RN×IintoSd, withN≥2andd≥2. In this setting one should point out that the majority of the works in the 2literature concerning the study of self-similar solutions of the LLG equation are confined to the heat flow for harmonic maps equation, i.e. α= 1. In the case when α= 1, the main works on the subject restrict the analysis to corotational maps taking values in Sd, which reduces the analysis of (HFHM) to the study of a second order real-valued ODE. Then tools such as the maximum principle or the shooting method can be used to show the existence of solutions. We refer to [19, 21, 23, 7, 8, 6, 22] and the references therein for more details on such results for maps taking values in Sd, withd≥3.Recently, Deruelle and Lamm [17] have studied the Cauchy problem for the harmonic map heat flow with initial data m0:RN→Sd, withN≥3andd≥2, where m0is a Lipschitz 0-homogeneous function, homotopic to a constant, which implies the existence of expanders coming out of m0. When 0<α≤1, the existence of self-similar expanders for the LLG equation was recently established by the authors in [33]. This result is a consequence of a well-posedness theorem for the LLG equation considering an initial data m0:RN→S2in the space BMO of functions of bounded mean oscillation. Notice that this result includes in particular the case of the harmonic map heat flow. As mentioned before, in the absence of damping ( α= 0), (LLG) reduces to the Schrödinger map equation (SM), which is reversible in time, so that the notions of expanders and shrinkers coincide. For this equation, Germain, Shatah and Zeng [24] established the existence of ( k- equivariant) self-similar profiles f:R2→S2. 1.2 Goals and statements of main results The results of this paper aim to advance our understanding of self-similar solutions of the one- dimensional LLG equation. In order to contextualize and motivate our results, we continue to provide further details of what is known about self-similar solutions in this context. In the 1d-case, when α= 0, (SM) is closely related to the Localized Induction Approximation (LIA), and self-similar profiles f:R→S2were obtained and analyzed in [34, 35, 41, 10]. In the context of LIA, self-similar solutions constitute a uniparametric family of smooth solutions that develop a singularity in the shape of a corner in finite time. For further work related to these solutions, including the study of their continuation after the blow-up time and their stability, we refer to the reader to [4, 3]. At the level of the Schrödinger map equation, these self-similar solutions provide examples of smooth solutions that develop a jump singularity in finite time. In the general case α∈[0,1], the analytical study of self-similar expanders of the one- dimensional (LLG) was carried out in [32]. Here, it was shown that these solutions are given by a family of smooth profiles {fc,α}c,α, and that the corresponding expanders are associated with a discontinuous (jump) singular initial data. We refer to [32, 33] for the precise statement of this result, and the stability of these solutions, as well as the qualitative and quantitative analysis of their dynamics with respect to the parameters candα. It is important to notice that in the presence of damping ( α > 0), since the LLG equation is not time-reversible, the notion of expander is different from that of shrinker. It is therefore natural to ask the following question: What can be said about shrinker solutions for the one- dimensional LLG equation? Answering this question constitutes the main purpose of this paper. Precisely, our main goals are to establish the classification of self-similar shrinkers of the one-dimensional LLG equation of the form (1.2) for some profile f:R→S2, and the analytical study of their properties. In particular, we will be especially interested in studying the dynamics of these solutions as ttends to the time of singularity T, and understanding how the dynamical behavior of these solutions is affected by the presence of damping. Since, as it has been already mentioned, the case α= 0 3has been previously considered in the literature (see [4, 31]), in what follows we will assume that α∈(0,1]. In order to state our first result, we observe that if mis a solution to (LLG) of the form (1.2) for some smooth profile f, thenfsolves the following system of ODEs xf/prime 2=βf×f/prime/prime−αf×(f×f/prime/prime),onR, (1.3) which recasts as αf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime 2= 0,onR, (1.4) due to the fact that ftakes values in S2. In the case α∈(0,1), it seems unlikely to be able to find explicit solutions to (1.4), and even their existence is not clear (see also equation (1.16)). Nevertheless, surprisingly we can establish the following rigidity result concerning the possible weak solutions to (1.4) (see Section 2 for the definition of weak solution). Theorem 1.1. Letα∈(0,1]. Assume that fis a weak solution to (1.4). Thenfbelongs to C∞(R;S2)and there exists c≥0such that|f/prime(x)|=ceαx2/4, for allx∈R. Theorem 1.1 provides a necessary condition on the possible (weak) solutions of (1.4): namely the modulus of the gradient of any solution mustbeceαx2/4, for somec≥0. We proceed now to establish the existence of solutions satisfying this condition for any c>0(notice that the case whenc= 0is trivial). To this end, we will follow a geometric approach that was proven to be very fruitful in similar contexts (see e.g. [41, 46, 42, 34, 16]), including the work of the authors in the study of expanders [32]. As explained in Subsection 3.1, this approach relies on identifying fas the unit tangent vectorm:=fof a curveXminR3parametrized by arclength. Thus, assuming that fis a solution to (1.4) and using the Serret–Frenet system associated with the curve Xm, we can deduce that the curvature and the torsion are explicitly given by k(x) =ceαx2/4,andτ(x) =−βx 2, (1.5) respectively, for some c≥0(see Subsection 3.1 for further details). In particular, we have |m/prime(x)|=k(x) =ceαx2/4, in agreement with Theorem 1.1. Conversely, given c≥0and denoting mc,αthe solution of the Serret–Frenet system m/prime(x) =k(x)n(x), n/prime(x) =−k(x)m(x) +τ(x)b(x), b/prime(x) =−τ(x)n(x),(1.6) with curvature and torsion as in (1.5), and initial conditions (w.l.o.g.) m(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1), (1.7) we obtain a solution to (1.4). Moreover, we can show that the solutions constructed in this manner provide, up to symmetries, all the solutions to (1.3). The precise statement is the following. Proposition 1.2. The set of nonconstant solutions to (1.3)is{Rmc,α:c >0,R∈SO(3)}, whereSO(3)is the group of rotations about the origin preserving orientation. 4The above proposition reduces the study of self-similar shrinkers to the understanding of the familyofself-similarshrinkersassociatedwiththeprofiles {mc,α}c,α. Thenextresultsummarizes the properties of these solutions. Theorem 1.3. Letα∈(0,1],c > 0,T∈Randmc,αbe the solution of the Serret–Frenet system(1.6)with initial conditions (1.7), k(x) =ceαx2/4andτ(x) =−/radicalbig 1−α2x 2. Define mc,α(x,t) =mc,α/parenleftbiggx√ T−t/parenrightbigg , t<T. (1.8) Then we have the following statements. (i) The function mc,αbelongs toC∞(R×(−∞,T);S2), solves(LLG)fort∈(−∞,T), and |∂xmc,α(x,t)|=c√ T−teαx2 4(T−t), for all (x,t)∈R×(−∞,T). (ii) The components of the profile mc,α= (m1,c,α,m2,c,α,m3,c,α)satisfy that m1,c,αis even, whilem2,c,αandm3,c,αare odd. (iii) There exist constants ρj,c,α∈[0,1], Bj,c,α∈[−1,1],andφj,c,α∈[0,2π),forj∈{1,2,3}, such that we have the following asymptotics for the profile mc,α= (m1,c,α,m2,c,α,m3,c,α) and its derivative: mj,c,α(x) =ρj,c,αcos(cΦα(x)−φj,c,α)−βBj,c,α 2cxe−αx2/4 +β2ρj,c,α 8csin(cΦα(x)−φj,c,α)/integraldisplay∞ xs2e−αs2/4ds+β α5c2O(x2e−αx2/2),(1.9) and m/prime j,c,α(x) =−cρj,c,αsin(cΦα(x)−φj,c,α)eαx2/4 +β2ρj,c,α 8cos(cΦα(x)−φj,c,α)eαx2/4/integraldisplay∞ xs2e−αs2/4ds+β α5cO(x2e−αx2/4),(1.10) for allx≥1, where Φα(x) =/integraldisplayx 0eαs2 4ds. Moreover, the constants ρj,c,α,Bj,c,α, andφj,c,αsatisfy the following identities ρ2 1,c,α+ρ2 2,c,α+ρ2 3,c,α= 2, B2 1,c,α+B2 2,c,α+B2 3,c,α= 1andρ2 j,c,α+B2 j,c,α= 1, j∈{1,2,3}. (iv) The solution mc,α= (m 1,c,α,m2,c,α,m3,c,α)satisfies the following pointwise convergences lim t→T−(mj,c,α(x,t)−ρj,c,αcos/parenleftbigcΦα/parenleftbigx√ T−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx>0, lim t→T−(mj,c,α(x,t)−ρ− j,c,αcos/parenleftbigcΦα/parenleftbig−x√ T−t/parenrightbig−φj,c,α/parenrightbig= 0,ifx<0,(1.11) forj∈{1,2,3}, whereρ− 1,c,α=ρ1,c,α,ρ− 2,c,α=−ρ2,c,αandρ− 3,c,α=−ρ3,c,α. 5(v) For anyϕ∈W1,∞(R;R3), we have lim t→T−/integraldisplay Rmc,α(x,t)·ϕ(x)dx= 0. In particular, mc,α(·,t)→0ast→T−, as a tempered distribution. It is important to remark that Theorem 1.3 provides examples of (smooth) solutions to the 1d- LLG equation that blow up in finite time. In order to see this, let us first recall that the existence of smooth solutions to (LLG) on short times can be established as in the case of the heat flow for harmonic maps [45], using that (LLG) is a strongly parabolic system [30, 2]. In particular, in the one-dimensional case, for any initial condition m0∈C∞(R,S2), there exists a maximal time 0< T max≤∞such that (LLG) admits a unique, smooth solution m∈C∞(R×[0,Tmax);S2). Moreover, if Tmax<∞, then lim t→T− max/bardbl∂xm(·,t)/bardblL∞(R)=∞. Next, observe that for any c>0andT∈R, the solution of the initial value problem associated with (LLG) and with initial condition mc,α(·)at timeT−1is given by mc,αin Theorem 1.3, fort∈[T−1,T), and blows up at time T. Indeed, from (i)in Theorem 1.3 , we have that lim t→T−|∂xmc,α(x,t)|= lim t→T−c√ T−teαx2 4(T−t)=∞, forc>0and for allx∈R. Notice also that from the asymptotics in part (iii)and the symmetries of the profile estab- lished in part (ii), we obtain a precise description of the fast oscillating nature of the blow up of the solution (1.8) given in Theorem 1.3. In this setting, we observe that part (iii)of the above theorem provides the asymptotics of the profile mc,αat infinity, in terms of a fast oscillating principal part, plus some exponentially decaying terms. Notice that for the integral term in (1.9), we have (see e.g. [1]) /integraldisplay∞ xs2e−αs2/4ds=2xe−αx2/4 α/parenleftBig 1 +2 αx2−4 α2x4+···/parenrightBig ,asx→∞, and that using the asymptotics for the Dawson’s integral [1], we also get Φα(x) =2eαx2/4 αx/parenleftBig 1 +2 αx2+12 α2x4+···/parenrightBig ,asx→∞. It is also important to mention that the big- Oin the asymptotics (1.9) does not depend on the parameters, i.e. there exists a universal constant C > 0, such that the big- Oin (1.9) satisfies |O(x2e−αx2/2)|≤Cx2e−αx2/2,for allx≥1. In this manner, the constants multiplying the big- Oare meaningful and in particular, big- O vanishes when β= 0(i.e.α= 1). In Figure 1 we have depicted the profile mc,αforα= 0.5andc= 0.5, where we can see their oscillating behavior. Moreover, the plots in Figure 1 suggest that the limit sets of the trajectories are great circles on the sphere S2whenx→±∞. This is indeed the case. In our last result we establish analytically that mc,αoscillates in a plane passing through the origin whose normal vector is given by B+ c,α= (B1,c,α,B2,c,α,B3,c,α), andB− c,α= (−B1,c,α,B2,c,α,B3,c,α)asx→+∞ andx→−∞, respectively. 6m1m2m3 B+ c,α m1m2m3 -1.0-0.50.00.51.0-1.0-0.50.00.51.0 m1m2 Figure 1: Profile mc,αforc= 0.5andα= 0.5. The figure on the left depicts profile for x∈R+ and the normal vector B+ c,α≈(−0.72,−0.3,0.63). The figure on the center shows the profile for x∈R; the angle between the circles C± c,αisϑc,α≈1.5951. The figure on the right represents the projection of limit cycles C± c,αon the plane. Theorem 1.4. Using the constants given in Theorem 1.3, let P± c,αbe the planes passing through the origin with normal vectors B+ c,αandB− c,α= (−B1,c,α,B2,c,α,B3,c,α), respectively. Let C± c,αbe the circles in R3given by the intersection of these planes with the sphere, i.e. C± c,α=P± c,α∩S2. Then the following statements hold. (i) For all|x|≥1, we have dist(mc,α(x),C± c,α)≤30√ 2β cα2|x|e−αx2/4. (1.12) In particular lim t→T−dist(mc,α(x,t),C+ c,α) = 0,ifx>0, lim t→T−dist(mc,α(x,t),C− c,α) = 0,ifx<0.(1.13) (ii) Letϑc,α= arccos(1−2B2 1,c,α)be the angle between the circles C± c,α. Forc≥β√π/√α, we have ϑc,α≥arccos/parenleftBigg −1 +2πβ2 c2α/parenrightBigg . (1.14) In particular limc→∞ϑc,α=π,for allα∈(0,1], and lim α→1ϑc,α=π,for allc>0.(1.15) The above theorem above establishes the convergence of the limit sets of the trajectories of the profile mc,αto the great circles C± c,αas shown in Figure 1. Moreover, (1.12) gives us an exponential rate for this convergence. In terms of the solution mc,αto the LLG equation, Theorem 1.4 provides a more precise geometric information about the way that the solution blows up at time T, as seen in (1.13). The existence of limit circles for related ferromagnetic models have been investigated for instance in [52, 9] but to the best of our knowledge, this is the first time that this type of phenomenon has been observed for the LLG equation. In Figure 1 can see that ϑc,α≈1.5951forα= 0.5andc= 0.5, where we have chosen the value of csuch that the angle is close to π/2. Finally, (1.14) and (1.15) in Theorem 1.4 provide some geometric information about behavior of the limit circles with respect to the parameters candα. In particular, formulae (1.15) states 7that the angle between the limiting circles C+ c,αandC− c,αisπasc→∞, for fixedα∈(0,1], and the same happens as α→1, for fixedc>0. In other words, in these two cases the circles C± c,α are the same (but differently oriented). 1.3 Comparison with the limit cases α= 0andα= 1 It is well known that the Serret–Fenet system can be written as a second-order differential equation. Forinstance, if (m,n,b) = (mj,nj,bj)3 j=1isasolutionof (1.5)–(1.6), usingLemma3.1 in [32], we have that new variable gj(s) =e1 2/integraltexts 0k(σ)ηj(σ)dσ,withηj(x) =nj(x) +ibj(x) 1 +mj(x), satisfies the equation, for j∈{1,2,3}, g/prime/prime j(x)−x 2(α+iβ)g/prime j(x) +c2 4eαx2/2gj(x) = 0. (1.16) Then, in the case α= 1, it easy to check (see also Remark 3.3) that the profile is explicitly given by the plane curve mc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0). (1.17) In particular, we see that the asymptotics in Theorem 1.3 are satisfied with ρ1,c,1= 1, ρ 2,c,1= 1, ρ 3,c,1= 0, φ 1,c,1= 0, φ 2,c,1= 3π/2, φ 3,c,1∈[0,2π). The caseα= 0is more involved, but using (1.16), the solution {mc,0,nc,0,bc,0}of the system (1.6)canstillbeexplicitlydeterminedintermsofconfluenthypergeometricfunctions. Thisleads to the asymptotics [34, 32, 20] mc,0(x) =Ac−2c xBccos/parenleftBigg x2 4+c2ln(x) +π 2/parenrightBigg +O/parenleftbigg1 x2/parenrightbigg , (1.18) asx→∞, for some vectors Ac∈S2andBc∈R3. In particular, we see that mc,0(x)converges to some vector Ac, asx→∞. Hence, there is a drastic change in the behavior of the profile in the cases α= 0andα > 0: In the first case mc,0converges to a point at infinity, while in the second case (1.12) tells us that mc,αconverges to a great circle. In this sense, there is a discontinuity in the behavior of mc,αatα= 0. Also, from equation (1.16), we can formally deduce that the difference between the expanders and shrinkers corresponds to flipping the sign in the parameters α→−αandβ→−β. Notice that the exponential coefficient in (1.16) is proportional to the square of the curvature, given by ce−αx2/4for the skrinkers, and ceαx2/4for the expanders. We used equation (1.16) (with flipped signs) to obtain the asymptotics of the expanders in [32], relying on the fact the exponential term in equation vanishes as x→∞. However, the exponential grow in the case of skrinkers in (1.16) changes the behavior of the solution and we cannot use the methods introduced in [32]. Going back to Theorem 1.3, it is seems very difficult to get asymptotics for the constants in (1.9). Our strategy for the constants appearing in the asymptotics for the expanders in [32] relied on obtaining uniform estimates and using continuity arguments. In particular, using the fact that the constants in (1.18) are explicit, we were able to get a good information about the constants in the asymptotics when αwas close to 0. Due to the above mentioned discontinuity 8ofmc,αatα= 0, it seems unlikely that the use of continuity arguments will provide information for the constants in the asymptotics for the shrinkers. Finally, let us also remark that we cannot use continuation arguments to find the behavior of the circles for csmall. This is expected since m0,α(x) = (1,0,0)for allx∈R, whenc= 0 (see (4.6)). In Section 4 we give some numerical simulations for csmall. Structure of the paper. The outline of this paper is the following. In Section 2, we study (1.4) as an elliptic quasilinear system and prove the rigidity result Theorem 1.1. By using the Serret–Frenet system, we prove there existence and uniqueness of solution, up to a rotation, in Section 3. We also use this system to obtain the asymptotics of the self-similar profiles. Finally, Section 4 is devoted to the proof of Theorem 1.4. 2 Rigidity result. Theorem 1.1 The purpose of this section is to prove the rigidity result stated in Theorem 1.1 concerning (weak) solutions of the system xf/prime 2=βf×f/prime/prime−αf×(f×f/prime/prime),onR. (2.1) We start by introducing the notion of weak solution of the above system. To this end, we first observe that the system (2.1) recasts as αf/prime/prime+α|f/prime|2f+β(f×f/prime)/prime−xf/prime 2= 0, (2.2) using the following vector identities for a (smooth) function fwith|f|= 1: f×f/prime/prime= (f×f/prime)/prime, −f×(f×f/prime/prime) =f/prime/prime+|f/prime|2f.(2.3) We prefer to use the formulation (2.2) since it is simpler to handle in weak sense. Indeed, we say thatf= (f1,f2,f3)∈H1 loc(R,S2)is aweak solution to the system (2.2) if /integraldisplay R/parenleftBig −αf/prime·ϕ/prime+α|f/prime|2f·ϕ−β(f×f/prime)·ϕ/prime−x 2f/primeϕ/parenrightBig dx= 0, (2.4) for allϕ= (ϕ1,ϕ3,ϕ3)∈C∞ 0(R). Using (2.3), we can recast (2.2) as, αf/prime/prime 1+α|f/prime|2f1+β(f2f/prime/prime 3−f3f/prime/prime 2)−x 2f/prime 1= 0, (2.5a) αf/prime/prime 2+α|f/prime|2f2+β(f3f/prime/prime 1−f1f/prime/prime 3)−x 2f/prime 2= 0, (2.5b) αf/prime/prime 3+α|f/prime|2f3+β(f1f/prime/prime 2−f2f/prime/prime 1)−x 2f/prime 3= 0. (2.5c) Thus we see that the weak formulation (2.4) can be written as /integraldisplay RA(f(x))f/prime(x)·ϕ/prime(x) =/integraldisplay RG(x,f,f/prime)ϕ(x),for allϕ∈C∞ 0(R), (2.6) with A(u) = α−βu3βu2 βu3α−βu1 −βu2βu1α, andG(x,u,p) = αu1|p|2−xp1 2 αu2|p|2−xp2 2 αu3|p|2−xp3 2 , 9whereu= (u1,u2,u3)andp= (p1,p2,p3). We want now to invoke the regularity theory for quasilinear elliptic system (see [39, 25]). To verify that the system is indeed uniformly elliptic, we can easily check that A(u)ξ·ξ=α|ξ|2,for allξ,u∈R3. In addition, Ghas quadratic growth on bounded domains, i.e. |G(x,u,p)|≤√ 3(M|p|2+R|p|), for all|u|≤Mand|x|≤R. Since a weak solution fto (2.6) belongs by definition to H1 loc(R;S2), we have by the Sobolev embedding theorem that fis Hölder continuous with |f(x)|= 1. Therefore we can apply the results in Theorem 1.2 in [25] (see also Lemma 8.6.3 in [38] or Theorem 2.4.3 in [49] for detailed proofs), to conclude that f∈H2 loc(R)∩W1,4 loc(R), and so thatf∈C1,γ loc(R), for someγ∈(0,1). We get that G(x,f(x),f/prime(x))belongs toC0,γ loc(R), which allows us to invoke the Schauder regularity theory (see e.g. Theorem A.2.3 in [38]) to infer that f∈C2,γ loc(R). This implies that G(x,f(x),f/prime(x))belongs toC1,γ loc(R), as well as the coefficients of A(u), so the Schauder estimates yield that f∈C3,γ loc(R). By induction, we this argument shows thatf∈C∞(R). We are now in position to complete the proof of Theorem 1.1. Indeed, let first remark that differentiating the relation |f|2= 1, we have the identities f·f/prime= 0, (2.7) f·f/prime/prime=−|f/prime|2. (2.8) By taking the cross product of fand (2.2), and using (2.3), we have βf/prime/prime+β|f/prime|2f−α(f×f/prime)/prime+x 2f×f/prime= 0. (2.9) Thus, by multiplying (2.2) by α, (2.9) byβ, and recalling that α2+β2= 1, we get f/prime/prime+|f/prime|2f−x 2(αf/prime−βf×f/prime) = 0. Taking the scalar product of this equation and f/prime, the identity (2.7) allow us to conclude that 1 2(|f/prime|2)/prime−αx 2|f/prime|2= 0. (2.10) Integrating, we deduce that there is a constant C≥0such that|f/prime|2=Ceαx2/2. This completes the proof of Theorem 1.1. We conclude this section with some remarks. Remark 2.1. A similar result to the one stated in Theorem 1.1 also holds for the expanders solutions. Precisely, any weak solution to (2.1), withxf/prime/2replaced by−xf/prime/2in the l.h.s., is smooth and there exists c≥0such that|f/prime(x)|=ce−αx2/4, for allx∈R. Remark 2.2. Let us mention that in the case α= 1, a nonconstant solution u:RN→Sdto equation ∆u+|∇u|2u−x·∇u 2= 0,onRN, (2.11) 10is usually called quasi-harmonic sphere , since it corresponds to the Euler–Lagrange equations of a critical point of the (so-called) quasi-energy [44] Equasi(u) =/integraldisplay RN|∇u(y)|2e−|y2|/4dy. It has been proved in [19] the existence of a (real-valued) function hsuch that u(x) =/parenleftBigx |x|sin(h(|x|)),cos(h(|x|)/parenrightBig is a solution to (2.11)with finite quasi-energy for 3≤N=d≤6. In addition, there is no solution of this form if d≥7[8]. Both results are based on the analysis of the second-order ODE associated with h. We refer also to [21] for a generalization of the existence result for N≥3of other equivariant solutions to (2.11). In the case N= 1andd= 2, the solution to (2.11)is explicitly given by (1.17), and its associated quasi-energy is infinity, as remarked in [54]. 3 Existence, uniqueness and properties 3.1 Existence and uniqueness of the self-similar profile. Proposition 1.2 In the previous section we have shown that any solution to the profile equation αm/prime/prime+α|m/prime|2m+β(m×m/prime)/prime−xm/prime 2= 0, (3.1) is smooth and that there is c≥0such that |m/prime(x)|=ceαx2/4,for allx∈R. (3.2) We want to give now the details about how to construct such a solution by using the Serret– Frenet frame, which will correspond to the profile mc,αin Theorem 1.3. The idea is to identify mas the tangent vector to a curve in R3, so we first recall some facts about curves in the space. Givenm:R→S2a smooth function, we can define the curve Xm(x) =/integraldisplayx 0m(s)ds, (3.3) so thatXmis smooth, parametrized by arclenght, and its tangent vector is m. In addition, if|m/prime|does not vanish on R, we can define the normal vector n(x) =m/prime(x)/|m/prime(x)|and the binormal vector b(x) =m(x)×n(x). Moreover, we can define the curvature and torsion of Xm ask(x) =|m/prime(x)|andτ(x) =−b/prime(x)·n(x). Since|m(x)|2= 1,for allx∈R, we have that m(x)·n(x) = 0, for allx∈R, that the vectors {m,n,b}are orthonormal and it is standard to check that they satisfy the Serret–Frenet system m/prime=kn, n/prime=−km+τb, b/prime=−τn.(3.4) Let us apply this construction to find a solution to (3.1). We define curve Xmas in (3.3), and remark that equation (3.1) rewrites in terms of {m,n,b}as x 2kn=β(k/primeb−τkn)−α(−k/primen−kτb). 11Therefore, from the orthogonality of the vectors nandb, we conclude that the curvature and torsion ofXmare solutions of the equations x 2k=αk/prime−βτkandβk/prime+αkτ= 0, that is k(x) =ceαx2 4andτ(x) =−βx 2, (3.5) for somec≥0. Of course, the fact that k(x) =ceαx2/4is in agreement with the fact that we must have|m/prime(x)|=ceαx2/4. Now, given α∈[0,1]andc>0, consider the Serret–Frenet system (3.4) with curvature and torsion function given by (3.5) and initial conditions m(0) = (1,0,0),n(0) = (0,1,0),b(0) = (0,0,1). (3.6) Then, by standard ODE theory, there exists a unique global solution {mc,α,nc,α,bc,α}in (C∞(R;S2))3, and these vectors are orthonormal. Also, it is straightforward to verify that mc,αis a solution to (3.1) satisfying (3.2). The above argument provides the existence of solutions in the statement of Proposition 1.2. We will now complete the proof of Proposition 1.2 showing the uniqueness of such solutions, up to rotations. To this end, assume that ˜mis a weak nontrivial solution to (3.1). By Theorem 1.1, ˜mis inC∞(R,S2)and there exists c >0such that|˜m/prime(x)|=ceαx2/4, for allx∈R. Following the above argument, the curve X˜m(defined in (3.3)), has curvature ceαx2/4and torsion−βx/2. Since the curve Xmc,αassociated with mc,α, andX˜mhave the same curvature and torsion, using fundamental theorem of the local theory of space curves (see e.g. Theorem 1.3.5 in [47]), we conclude that both curves are equal up to direct rigid motion, i.e. there exist p∈R3and R∈SO(3)such thatX˜m(x) =R(Xmc,α(x))+p, for allx∈R3. By differentiating this identity, we finally get that ˜m=Rmc,α, which proves the uniqueness of solution, up to a rotation, as stated in Proposition 1.2. 3.2 Asymptotics of the self-similar profile The rest of this section is devoted to establish properties of the family of solutions {mc,α}c,α, for fixedα∈(0,1]andc >0. Due to the self-similar nature of these solutions, this analysis reduces to study the properties of the associated profile mc,α, or equivalently, of the solution {mc,α,nc,α,bc,α}of the Serret–Frenet system (3.4) with curvature and torsion given in (3.5), and initial conditions (3.6). It is important to mention that the recovery of the properties of the trihedron {m,n,b}, and in particular of the profile m, from the knowledge of its curvature and torsion is a difficult question. This can be seen from the equivalent formulations of the Serret–Frenet equation in terms of a second-order complex-valued highly non-linear EDO, or in terms of a complex-valued Riccati equation (see e.g. [11, 50, 42, 32]). For this reason, the integration of the trihedron can often only be done numerically, rather than analytically. Since the Serret–Frenet equations are decoupled, we start by analyzing the system for the 12scalarfunctionsmc,α,nc,αandbc,α m/prime c,α(x) =ceαx2 4nc,α(x), n/prime c,α(x) =−ceαx2 4mc,α(x)−βx 2bc,α(x), b/prime c,α(x) =βx 2nc,α(x),(3.7) withinitialconditions (mc,α,bc,α,nc,α)(0),thatwesupposeindependentof candα,andsatisfying mc,α(0)2+bc,α(0)2+nc,α(0)2= 1. Then by ODE theory, the solution is smooth, global and satisfies mc,α(x)2+bc,α(x)2+nc,α(x)2= 1,for allx∈R. (3.8) Moreover, the solution depends continuously on the parameters c>0andα∈(0,1]. To study the behavior of the solution of the system (3.7), we need some elementary bounds for the non-normalized complementary error function. Lemma 3.1. Letγ∈(0,1]. The following upper bounds hold for x>0 /integraldisplay∞ xe−γs2ds≤1 2γxe−γx2and/integraldisplay∞ xse−γs2ds=1 2γe−γx2. (3.9) Also, forγ∈(0,1]andx≥1, /integraldisplay∞ xs2e−γs2ds≤x γ2e−γx2,and/integraldisplay∞ xs3e−γs2ds≤x2 γ2e−γx2. (3.10) Proof.We start recalling some standard bounds the complementary error function (see e.g. [1, 28]) xe−x2 2x2+ 1≤/integraldisplay∞ xe−s2ds≤e−x2 2x,forx>0. (3.11) The first formula in (3.9) follows by scaling this inequality. The second formula in (3.9) follows by integration by parts. To prove the first estimate in (3.10), we use integration by parts and (3.9) to show that /integraldisplay∞ xs2e−γs2ds=xe−γx2 2γ+1 2γ/integraldisplay∞ xe−γs2ds≤e−γx2/parenleftbiggx 2γ+1 4γ2x/parenrightbigg ≤xe−γx2/parenleftbigg1 2γ+1 4γ2/parenrightbigg ,∀x≥1. Sinceγ∈(0,1], we haveγ2≤γand thus we conclude the estimate for the desired integral. The second inequality in (3.10) easily follows from the identity /integraldisplay∞ xs3e−γs2ds=1 +γx2 2γ2e−γx2,∀x∈R, noticing that 1 +γx2≤x2(1 +γ)≤2x2, sincex≥1andγ∈(0,1]. Now we can state a first result on the behavior of {mc,α,nc,α,bc,α}. 13Proposition 3.2. Letα∈(0,1]andc>0, and define Φα(x) =/integraldisplayx 0eαs2 4ds. Then the following statements hold. i) For allx∈R, bc,α(x) =Bc,α+βx 2ce−αx2/4mc,α(x) +β 2c/integraldisplay∞ x/parenleftBigg 1−αs2 2/parenrightBigg e−αs2/4mc,α(s)ds,(3.12) where Bc,α=bc,α(0)−β 2c/integraldisplay∞ 0/parenleftBigg 1−αs2 2/parenrightBigg e−αs2/4mc,α(s)ds. (3.13) In particular, for all x≥1 |bc,α(x)−Bc,α|≤6β cαxe−αx2/4. (3.14) ii) Setting wc,α=mc,α+inc,α, for allx∈R, we have wc,α(x) =e−icΦα(x)/parenleftBig Wc,α−βx 2ceicΦα(x)−αx2/4bc,α(x) −β 2c/integraldisplay∞ xeicΦα(s)−αs2/4/parenleftbigβs2 2nc,α(s) +/parenleftbig1−αs2 2/parenrightbigbc,α(s)/parenrightbigds/parenrightBig ,(3.15) where Wc,α=wc,α(0) +β 2c/integraldisplay∞ 0eicΦα(s)−αs2/4/parenleftbigβs2 2nc,α(s) +/parenleftbig1−αs2 2/parenrightbigbc,α(s)/parenrightbigds.(3.16) In particular, for all x≥1, |wc,α(x)−e−icΦα(x)Wc,α|≤10β cα2xe−αx2/4. (3.17) Furthermore, the limiting values Bc,αandWc,αare separately continuous functions of (c,α)for (c,α)∈(0,∞)×(0,1]. Proof.For simplicity, we will drop the subscripts candαif there is no possible confusion. From (3.7), we get b(x)−b(0) =/integraldisplayx 0b/prime(s)ds=β 2c/integraldisplayx 0se−αs2 4m/prime(s)ds =β 2c/parenleftBig xe−αx2 4m(x)−/integraldisplayx 0/parenleftbig1−αs2 2/parenrightbige−αs2 4m(s)ds/parenrightBig ,(3.18) where we have used integration by parts. Notice that/integraltext∞ 0(1−αs2/2)e−αs2/4m(s)dsis well- defined, since α∈(0,1]andmis bounded. Therefore, the existence of B:= limx→∞b(x)follows from (3.18). Moreover, B:=b(0)−β 2c/integraldisplay∞ 0/parenleftBigg 1−αs2 2/parenrightBigg e−αs2/4m(s)ds. Formula (3.12) easily follows from integrating b/primefromx∈Rto∞and arguing as above. 14To prove (3.14), it is enough to observe that by Lemma 3.1, for x≥1and0<α≤1, /integraldisplay∞ xe−αs2/4ds≤2 αxe−αx2 4≤2 αxe−αx2 4,and/integraldisplay∞ xs2e−αs2/4ds≤16 α2xe−αx2 4.(3.19) Settingw=m+inand using (3.7), we obtain that wsatisfies the ODE w/prime+iceαx2/4w=−iβx 2b(x), (3.20) or, equivalently,/parenleftBig eicΦα(x)w/parenrightBig/prime =−iβx 2b(x)eicΦα(x). (3.21) Integrating (3.21) from 0tox>0, and writing eicΦα(x)=−i c/parenleftBig eicΦα(x)/parenrightBig/prime e−αx2/4, integrating by parts, and using once again (3.7), we get eicΦα(x)w(x) =w(0)−β 2cxb(x)eicΦα(x)−αx2/4 +β 2c/integraldisplayx 0eicΦα(s)−αs2/4/parenleftBigβ 2s2n(s) + (1−αs2 2)b(s)/parenrightBig ds. Sinceα∈(0,1], from the above identity it follows the existence of W:= limx→∞eicΦα(x)w(x), and formula (3.16) for W. Formula (3.15) now follows from integrating (3.21) from x >0to∞and arguing as in the previous lines. The estimate in (3.17) can be deduced as before, since the bounds in (3.19) imply that |wc,α(x)−e−icΦα(x)Wc,α|≤β 2cxe−αx2/4/parenleftbigg 1 +16(α+β) 2α2+2 α/parenrightbigg ≤10β cα2xe−αx2/4, where we used that α+β≤2andα≤1. To see that the limiting values Bc,αandWc,αgiven by (3.13) and (3.16) are continuous functions of (c,α), for (c,α)∈(0,∞)×(0,1], we recall that by standard ODE theory, the func- tionsmc,α(x),nc,α(x)andbc,α(x)are continuous functions of x,candα. Then, the dominated convergence theorem applied to the formulae (3.13) and (3.16) yield the desired continuity. Remark 3.3. As mentioned before, the shrinkers of the 1d-harmonic heat flow can be computed explicitly, because if α= 1, the system (1.5)-(1.6)-(1.7)can be solved easily. Indeed, in this case β= 0, so that we obtain mc,1(x) = (cos(cΦ1(x)),sin(cΦ1(x)),0), nc,1(x) = (−sin(cΦ1(x)),cos(cΦ1(x)),0), bc,1(x) = (0,0,1), for allx∈R. In order to obtain a better understanding of the asymptotic behavior of {mc,α,nc,α,bc,α}, we need to exploit the oscillatory character of the function eicΦα(s)in the integrals (3.12) and (3.15). In our arguments we will use the following two lemmas. 15Lemma 3.4. Let0<α≤1. Forσ∈R\{0}andx∈R, the limit /integraldisplay∞ xseiσΦα(s)ds:= limy→∞/integraldisplayy xseiσΦα(s)ds exists. Moreover, for all x≥1, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞ xseiσΦα(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤11x |σ|αe−αx2/4, (3.22) and/integraldisplay∞ xseiσΦα(s)ds=ix σeiσΦα(x)−αx2/4+O/parenleftBigg x2 σ2e−αx2/2/parenrightBigg . (3.23) Proof.Letx∈Rand takey≥x. Then, integrating by parts, /integraldisplayy xseiσΦα(s)ds=1 iσ/integraldisplayy xs(eiσΦα(s))/primee−αs2/4ds =s iσeiσΦα(s)−αs2/4/vextendsingle/vextendsingle/vextendsingle/vextendsingley x−1 iσ/integraldisplayy xeiσΦα(s)−αs2/4/parenleftbig1−αs2 2/parenrightbigds. (3.24) The existence of the improper integral/integraltext∞ xseiσΦα(s)dsfollows taking the limit as ygoes to∞ in the above formula, and bearing in mind that α>0. The estimate (3.22) follows from (3.19) and the fact that x≥1and0<α≤1. Finally, integrating by parts once more, we have iσ/integraldisplay∞ xeiσΦα(s)−αs2/4/parenleftbig1−αs2 2/parenrightbigds=−eiσΦα(x)−αx2/2/parenleftbig1−αx2 2/parenrightbig−/integraldisplay∞ xeiσΦα(s)−αs2/2/parenleftbigα2s3 2−2αs/parenrightbigds. Hence, using Lemma 3.1 and (3.24), we obtain (3.23). Lemma 3.5. Letσ∈R\{0},γ∈R,α>0and set ˜γ=γ+α/4. If0<˜γ≤1, then forx≥1, /integraldisplay∞ xeiσΦα(s)−γs2ds=O/parenleftBigg e−˜γx2 |σ|/parenrightBigg ,/integraldisplay∞ xseiσΦα(s)−γs2ds=O/parenleftBigg xe−˜γx2 |σ|˜γ)/parenrightBigg , /integraldisplay∞ xs2eiσΦα(s)−γs2ds=O/parenleftBigg x2e−˜γx2 |σ|˜γ/parenrightBigg .(3.25) Proof.Forn∈{0,1,2}, we set In=/integraldisplay∞ xsneiσΦα(s)−γs2ds. In=1 iσ/parenleftbigg −xneiσΦα(x)−˜γx2−/integraldisplay∞ xeiσΦα(s)−˜γs2/parenleftBig nsn−1−2˜γsn+1/parenrightBig ds/parenrightbigg . Then the desired asymptotics follow from Lemma 3.1. Using previous lemmas, we can now improve the asymptotics in Proposition 3.2 and obtain explicitly the term decaying as e−αx2/4(multiplied by a polynomial). Corollary 3.6. With the same notation as in Proposition 3.2, the following asymptotics hold forx≥1 bc,α(x) =Bc,α+βx 2ce−αx2/4Re(e−icΦα(x)Wc,α) +β c2α3O(x2e−αx2/2), (3.26) wc,α(x) =e−icΦα(x)/parenleftBig Wc,α−βBc,α 2cxeicΦα(x)−αx2/4+iβ2Wc,α 8c/integraldisplay∞ xs2e−αs2/4ds/parenrightBig (3.27) +β c2α5O(x2e−αx2/2). 16Proof.As usual, we drop the subscripts candαin the rest of the proof. Recalling that w= m+in, we have from (3.17), m= Re(e−icΦα(x)W) +β cα2O(xe−αx2/4). Thus, replacing in (3.12), b(x) =B+βx 2ce−αx2/4Re(e−icΦα(x)W) +β2 c2α2O(x2e−αx2/2) +Rb(x),(3.28) with Rb(x) =β 2cRe/parenleftBigg W/integraldisplay∞ x/parenleftbig1−αs2 2/parenrightbige−icΦα(s)−αs2/4ds+/integraldisplay∞ x/parenleftbig1−αs2 2/parenrightbigO/parenleftbigse−αs2/2 cα2/parenrightbigds/parenrightBigg . ByusingLemmas 3.1and3.5toestimatethefirstandsecondintegrals, respectively, weconclude that Rb(x) =β c2α3O/parenleftbigx2e−αx2/2/parenrightbig. (3.29) By putting together (3.28) and (3.29), we obtain (3.26). To establish (3.27) we integrate (3.21) fromx≥1and∞, and use (3.26) and Lemma 3.1 to get eicΦw(x)−W=I1(x) +I2(x) +I3(x) +β2 c2α5O(xe−αx2/2), (3.30) with I1(x) =iβB 2/integraldisplay∞ xseicΦα(s)ds, I 2(x) =iβ2W 8c/integraldisplay∞ xs2e−αs2/4ds,and I3(x) =iβ2¯W 8c/integraldisplay∞ xs2e2icΦα(s)−αs2/4ds, where we have used that Re(z) = (z+ ¯z)/2. The conclusion follows invoking again Lemmas 3.1, 3.4 and 3.5. In Figure 2 we depict the first components of the trihedron {mc,α,nc,α,bc,α}forc= 0.5and α= 0.5, andx>0. As described in Corollary 3.6 (recall that wc,α=mc,α+inc,α), in the plots in Figure 2 one can observe that, while both m1,c,αandb1,c,αoscillate highly for large values of x>0, the component b1,c,αconverges to a limit B1,c,α≈−0.72asx→+∞. 246810 -1.0-0.50.51.0 (i)m1,c,α (ii)n1,c,α 246810 -1.0-0.8-0.6-0.4-0.2 (iii)b1,c,α Figure 2: Functions m1,c,α,n1,c,αandb1,c,αforc= 0.5andα= 0.5onR+. The limit at infinity in (iii) isB1,c,α≈−0.72. 173.3 Proof of Theorem 1.3 For simplicity, we will drop the subscripts candαin the proof of Theorem 1.3. Proof of Theorem 1.3. Let{m,n,b}be the solution of the Serret–Frenet system (3.4)–(3.5) with initial condition (3.6). By ODE theory, we have that the solution {m,n,b}is smooth, is global, and satisfies |m(x)|=|n(x)|=|b(x)|= 1,for allx∈R, (3.31) and the orthogonality relations m(x)·n(x) =m(x)·b(x) =n(x)·b(x) = 0,for allx∈R. (3.32) Define m(x,t) =m/parenleftbiggx√ T−t/parenrightbigg , t<T. Then, it is straightforward to check that mis a smooth solution of the profile equation (3.1), and consequently msolves (LLG) for t∈(−∞,T). Moreover, using the Serret–Frenet system (3.4)–(3.5), we have |∂xm(x,t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1√ T−tm/prime/parenleftbiggx√ T−t/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle=c√ T−teαx2 4(T−t). This shows part (i). Notice also that, since {mj,nj,bj}forj∈{1,2,3}solves (3.7) with initial condition (1,0,0), (0,1,0)and(0,0,1)respectively, the relation (3.8) is satisfied, i.e. m2 j(x) +n2 j(x) +b2 j(x) = 1,for allx∈R, j = 1,2,3. (3.33) Part(ii)follows from the uniqueness given by the Cauchy–Lipschitz theorem for the solution of (3.1)–(3.5) with initial condition (3.6), and the invariance of (3.1)–(3.5)–(3.6) under the transformations m(x) = (m1(x),m2(x),m3(x))→(m1(−x),−m2(−x),−m3(−x)), n(x) = (n1(x),n2(x),n3(x))→(−n1(−x),n2(−x),n3(−x)), b(x) = (b1(x),b2(x),b3(x))→(−b1(−x),b2(−x),b3(−x)), SettingWj=ρjeiφj, withρj≥0andφj∈[0,2π], the asymptotics for mjand it derivative m/prime j, forj∈{1,2,3}in part(iii)are a direct consequence of the asymptotic behavior of the profile established in Proposition 3.2 and Corollary 3.6 (recall also that m/prime(x) =ceαx2 4n(x)and w=m+in). In particular the following limits exist: Wj:= limx→∞eicΦα(x)(mj+inj)(x), Bj:= limx→∞bj(x). Notice also that the relations in (3.33) implies that ρ2 j+B2 j= 1, and the fact that bis unitary implies that B2 1+B2 2+B2 3= 1, and thatρ2 1+ρ2 2+ρ2 3= 2. The convergence in part (iv)is an immediate consequence of the definition of min terms of the profilem, and the asymptotics established in part (iii). The relations between ρ− jandρj, forj∈{1,2,3}follow from the parity relations for the profile mestablished in part (ii). It remains to prove part (v). Forϕ∈W1,∞(R), bearing in mind (3.15) and (3.17), it suffices to show that lim t→0+/integraldisplay Re−icΦα(x/√ t)ϕ(x)dx= 0,and lim t→0+/integraldisplay R/parenleftbig1 +|x|√ t/parenrightbige−αx2 4t|ϕ(x)|dx= 0.(3.34) 18The second limit is a direct consequence of following the explicit computations: /integraldisplay Re−αx2 4tdx=/radicalbiggπt α,and/integraldisplay R|x|e−αx2 4tdx=4t α. (3.35) For the first limit in (3.34), we integrate by parts to obtain ic/integraldisplay Re−icΦα(x/√ t)ϕ(x)dx=−√ t/integraldisplay R/parenleftbige−icΦα(x/√ t))/primee−αx2 4tϕ(x)dx =/integraldisplay Re−icΦα(x/√ t)−αx2 4t/parenleftbigg√ tϕ/prime(x)−αx 2√ tϕ(x)/parenrightbigg dx. Since|ϕ|and|ϕ/prime|are bounded on R, the conclusion follows again by using (3.35). 4 Limiting behaviour of the trajectories of the profiles In this section we prove Theorem 1.4. In what follows we denote C± c,αthe great circleCc,α= P± c,α∩S2, withP± c,αbeing the planes passing through the origin with (unitary) normal vectors B+ c,α= (B1,c,α,B2,c,α,B3,c,α)andB− c,α= (−B1,c,α,B2,c,α,B3,c,α), given by Theorem 1.3. We will show that the trajectories of the profiles mc,αconverge to the great circles C± c,αas x→±∞, and study the behavior of the limit circles C± c,αwith respect to the parameters cand α, analyzing the angle ϑc,α∈[0,π]between their normal vectors B+ c,αandB− c,α. We start this section stating a corollary of Theorem 1.3 that will be used in what follows. Precisely, recalling that w=m+in, from (3.17) and using the constants defined in Theorem 1.3, we have the following Corollary 4.1. Forx≥1andj∈{1,2,3}, we have mj,c,α(x) =ρj,c,αcos(cΦα(x)−φj) +Rj(x), nj,c,α(x) =−ρj,c,αsin(cΦα(x)−φj) +˜Rj(x), for some functions Rjand ˜Rjsatisfying the bounds |Rj(x)|,|˜Rj(x)|≤10β/(cα2)xe−αx2/4. We will also use the two lemmas below in the proof of Theorem 1.4. The first one establishes some relations between the constants appearing in the asymptotics of the profile mc,αthat can be deduced by using some geometric properties of the Serret–Frenet system. Lemma 4.2. The constants given by Theorem 1.3 satisfy the following identities B1,c,α=ρ2,c,αρ3,c,αsin(φ3,c,α−φ2,c,α), B2,c,α=ρ1,c,αρ3,c,αsin(φ1,c,α−φ3,c,α), B3,c,α=ρ1,c,αρ2,c,αsin(φ2,c,α−φ1,c,α),(4.1) and B1,c,αρ1,c,αeiφ1,c,α+B2,c,αρ2,c,αeiφ2,c,α+B3,c,∞ρ3,c,αeiφ3,c,α= 0, (4.2) ρ2 1,c,αe2iφ1,c,α+ρ2 2,c,αe2iφ2,c,α+ρ2 3,c,αe2iφ3,c,α= 0. (4.3) 19Proof.Dropping the subscripts candα, using the relation b=m×nand the asymptotics in Corollary 4.1, we get for x≥1, b1(x) =−ρ2ρ3/parenleftbigcos(cΦα(x)−φ2) sin(cΦα(x)−φ3)−cos(cΦα(x)−φ3) sin(cΦα(x)−φ2)/parenrightbig+o(1), b2(x) =−ρ3ρ1/parenleftbigcos(cΦα(x)−φ3) sin(cΦα(x)−φ1)−cos(cΦα(x)−φ1) sin(cΦα(x)−φ3)/parenrightbig+o(1), b3(x) =−ρ1ρ2/parenleftbigcos(cΦα(x)−φ1) sin(cΦα(x)−φ2)−cos(cΦα(x)−φ2) sin(cΦα(x)−φ1),/parenrightbig+o(1), whereo(1)is a function of x, which depends on the parameters αandc, that converges to 0as x→∞. Noticing that, for k,j∈{1,2,3}, −cos(cΦα(x)−φj) sin(cΦα(x)−φk) + cos(cΦα(x)−φk) sin(cΦα(x)−φj) = sin(φk−φj), we obtain b1(x) =ρ2ρ3sin(φ3−φ2) +o(1), b2(x) =ρ1ρ3sin(φ1−φ3) +o(1), b3(x) =ρ1ρ2sin(φ2−φ1) +o(1). Lettingx→∞, in the above identities we obtain (4.1). To establish the other identities, we first recall that the vectors m,nandbsatisfy the following relations m·n=n·b=b·m= 0and|m|=|n|=|b|= 1. Now, from the asymptotics in Corollary 4.1 and the identity (m+in)·b= 0, we have b1(x)ρ1e−i(cΦα(x)−φ1)+b2(x)ρ2e−i(cΦα(x)−φ2)+b3(x)ρ3e−i(cΦα(x)−φ3)=o(1), fori∈{1,2,3}. Dividing by e−icΦα(x)and letting x→∞, we obtain formula (4.2). Finally, formula (4.3) follows easily using a similar argument, bearing in mind the orthogonality relation m·n= 0,and that 2 cos(y) sin(y) = Im(e2iy), fory∈R. Remark 4.3. Although we do not use (4.3)in this work, this relation could be helpful in estab- lishing further properties of the solutions. Next we study the angle between the normal vectors to the great circles C± c,α, given by ϑc,α= arccos(2B2 1,c,α−1),withϑc,α∈[0,π]. We have the following. Lemma 4.4. Forc≥β√π/√α, we have ϑc,α≥arccos/parenleftBigg −1 +2πβ2 c2α/parenrightBigg . (4.4) Proof.Using the formula (3.12) in Proposition 3.2 for b1,c,αwithx= 0, we get |b1,c,α(0)−B1,c,α|≤β 2c/integraldisplay∞ 0/parenleftBigg 1 +αs2 2/parenrightBigg e−αs2/4. 20Noticing that b1,c,α(0) = 0, and that /integraldisplay∞ 0/parenleftBigg 1 +αs2 2/parenrightBigg e−αs2/4=2√π√α, we conclude that 2B2 1,c,α≤2πβ2 c2α. Sincec≥β√π/√α, we have−1 + 2πβ2/(c2α)∈[−1,1], so we can use that the function arccos is decreasing to obtain (4.4). We continue to prove Theorem 1.4. Proof of Theorem 1.4. As usual, we omit the subscripts candαwhen there is no confusion. In view of the symmetries established in Theorem 1.3, it is enough to prove the theorem for x→∞. We start noticing that |Bc,α|= 1, so that the distance between mandP+is given by dist(m(x),P+) =|m1(x)B1+m2(x)B2+m3(x)B3|. To compute the leading term, we notice that using (4.2), we have 3/summationdisplay j=1ρjBjcos(cΦα(x)−φj) = Re/parenleftBigg eicΦα(x)/parenleftBigg3/summationdisplay j=1ρjBje−iφj/parenrightBigg/parenrightBigg = 0. Thus the estimates in Corollary 4.1 give us dist(m(x),P+))≤30β cα2xe−αx2/4,forx≥1, (4.5) Let us fix x≥1and letQ(x)be the orthogonal projection of m(x)on the planeP+, so that dist(m(x),P+) = dist(m(x),Q(x)). We also set C(x)∈C+such that dist(m(x),C+) = dist(m(x),C(x)). By the Pythagorean theorem, |Q(x)|2= 1−dist(m(x),Q(x))2 and dist(m(x),C(x))2= dist(m(x),Q(x))2+ (1−|Q(x)|)2. Therefore dist(m(x),C(x))2= 2−2(1−dist(m(x),Q(x))2)1/2, and using the elementary inequality√1−y≥1−y, fory∈[0,1], we get dist(m(x),C(x))≤√ 2 dist(m(x),Q(x)). Combing this estimate with (4.5), we conclude that dist(m(x),C+)) = dist(m(x),C(x)))≤√ 2 dist(m(x),P+))≤30√ 2β cαxe−αx2/4. The limits in (1.13) follow at once using the definition of mc,αin (1.8) (recall that ϑc,α∈[0,π]). The statement in (ii)is an immediate consequence of Lemma 4.4. This finishes the proof of Theorem 1.4. 21The limit limc→∞ϑc,α=πin part (ii) of Theorem 1.4 helps us to understand the angle between the great circles as c→∞. However, the behavior of the limit circles C± c,αforcsmall is much more involved. We conclude this section with some reflections on the behavior of the limit circlesC± c,αwhencis small. Let us remark that when c= 0, the explicit solution to (1.6)–(1.7) is given by m0,α(x) = (1,0,0), n0,α(x) = (0,cos(βx2/4),−sin(βx2/4)), b0,α(x) = (0,sin(βx2/4),cos(βx2/4)).(4.6) Thus we see that in this limit case, there is a change in the behavior of the solution: There is no limit circle and the vector b0,αdoes not have a limit at infinity. On the other hand, we know that (mc,α(x),nc,α(x),bc,α(x))are continuous with respect to the to c,αandx. Therefore, lim c→0b1,c,α(x) =b1,0,α(x) = 0,for allx∈R. (4.7) Of course, we cannot conclude from (4.7) estimates for B1,c,α, ascgoes to 0. For this reason, we performed some numerical simulations for different values of csmall. In Figures 3 and 4, we show one of these simulations, in the case α= 0.5andc= 0.01, where we see that m1,c,α≈1 andb1,c,α≈0on[0,2]in agreement with (4.6) and the continuous dependence on c. On the other hand, for x≥8.5, we haveb1,c,α(x)≈−1. However, it seems difficult to infer from our simulations the behavior of Bc,αascgoes to zero. For instance, we have computed numerically Bc,α, and it is not clear that this quantity converges forcsmall. Forinstance, wehaveobtained B1,c,α=−0.99215, forc= 10−12,B1,c,α=−0.992045, forc= 10−14, andB1,c,α=−0.991965, forc= 10−16. 2468100.20.40.60.81.0 (i)m1,c,α 246810 -1.0-0.8-0.6-0.4-0.2 (ii)b1,c,α Figure 3: Functions m1,c,αandb1,c,αforc= 0.01andα= 0.5. The limit at infinity in (ii) is B1,c,α≈−0.996417. Acknowledgements. S. Gutiérrez was partially supported by ERCEA Advanced Grant 2014 669689 - HADE. The Université de Lille also supported S. Gutiérrez’s research visit during July 2018 through their Invited Research Speaker Scheme. 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Ser. B , 17(2):164–170, 2002. 26

2020-02-17

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere $\mathbb{S}^2$, at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

Self-similar shrinkers of the one-dimensional Landau-Lifshitz-Gilbert equation

2002.06858v2

Page 1 of 13 Dynamic Spin Injection into Chemical Vapor Deposited Graphene A. K. Patra1,a), S. Singh2,a), B. Barin2, Y. Lee3, J.-H. Ahn3, E. del Barco2,b), E. R. Mucciolo2 and B. Özyilmaz1,4,5,6 ,b) 1Department of Physics, National University of Singapore, 2 Science Dri ve 3, Singapore 117542 2Department of Physics, University of Central Florida, Orlando, Florida USA, 32816 3 School of Advanced Materials Science & Engineering, SKKU Advanced Institute of Nanotechnology (SAINT) , Sungkyunkwan University, Suwon, Republic Kore a 440746 4NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576 5Graphene Research Center, National University of Singapore, Singapore 117542 6NUS Graduate School for Integrative Sciences and Engineering (NGS), National Univer sity of Singapore, Singapore 117456 We demonstrate dynamic spin injection into chemical vapor deposition (CVD) grown graphen e by s pin pumping from permalloy (Py) layer s. Ferromagnetic resonance measurements at room temperature reveal a strong enhancement of the Gilbert damping at the Py/graphene interface , exceeding that observed in even Py/platinum interfaces. Similar results are als o shown on Co/graphene layers . This enhancement in the Gilbert damping is understood as the consequence of spin pumping at t he interface d riven by magnetization dynamics . Our observation s suggest a strong enhancement of spin -orbit coupling in CVD graphene , in agreement with earlier spin valve measurements. a)A. K. Patra and S. Singh contributed equally to this work. b) Autho rs to whom correspondence should be addressed. Electronic mail s: delbarco@physics.ucf.edu and phyob@nus.edu.sg PACS numbers: 72.25. -b, 76.50.+g Page 2 of 13 In spintronic s, where the electron’s spin degree of freedom , rather than its charge , is employed to process in formation , the efficient generation of the large spin current s stands as a key requirement for future spintronic device s and applications. Several approaches to generate pure spin currents have been proposed and are being widely investigated, namely, non-local spin injection [1], spin Hall effect [2-4], and spin pumping [5,6]. Among these , spin pumping offers the advantage of producing spin current s over large ( mesoscopic ) areas [7-13] at ferromagnet ic/non-magnetic (FM/NM) interface s. In addition, dynamical spin pumping is insensitive to a potential impedance mismatch at the FM/NM interface [14], a problem ubiquitous in the non -local spin injection approach. Dynamical s pin pumping consists of generating pure spin current (i.e., with no net charge current) away from a ferromagnet into a non-magnetic material, induced by the coherent precession of the magnetization upon application of microwave stimuli of frequency matching the ferromagnetic resonance (FMR) of the system [15]. Since pure spin currents carry awa y spin angular momentum, in an FMR experiment the transfer of angular momentum from the FM into the NM layer results in an enhance ment of the Gilbert damping in the ferromagnet [5-15]. Most studies of dynamical spin pumping on FM/NM interfaces have made us e of Pt and Pd NM layers, since the large spin -orbit coupling in these systems enables the conversion of the injected spin current into an electric voltage across the N M layer, a phenomenon known as i nverse spin Hall effect (ISHE) . Recently, spin pumping h as been experimentally demonstrated in FM/semiconductor interfaces (e.g. , GaAs [13] and p -type Si [15]). However, there is no experimental report on spin pumping in FM/graphene interface s, though graphene [16] (a two -dimensional layer of carbon atoms ), pos sesses unique electronic properties (e.g. high mobility and gate-tunable charge carrier , among others ), and stands as an excellent material for spin transport due to its large spin coherence length [17]. Page 3 of 13 In this Letter we report experimental FM R studies of Py and Co films and polycrystalline graphene grown by chemical vapor deposition on Cu foils [ 18,19] (henceforth, Co/Gr and Py/Gr , respectively ) performed in a broad -band microwave coplanar waveguide (CPW) spectrometer . The observation of a remarkable broadening of the FMR absorption peaks in the Py/ Gr (88%) and Co/Gr (133%) films demonstrate a strong increase of the Gilbert damping in the FM layer due to spin pumping at the FM/Gr interface and the consequent loss of angular momentum through spin inject ion into the CVD graphene layer . To account for such a remarkable absorption of angular momentum , the spin orbit coupling in CVD graphene must be orders of magnitude large r than what is predicted for pristine, exfoliated graphene . To prepare the FM/Gr sample s, single layer CVD grown graphene [18, 19] was first transferred onto a Si substrate with 300 nm thick SiO 2 layer . The sample was then anneal ed in a H2/Ar environment at 300 °C for 3 hour to remove all organic residue s. For the Py layer we chose Ni80Fe20, a material extensively used for magnetic thin film studies because of its low magneto crystalline anisotropy and its insensitivity to strain. The FM layer (Py/14nm, Co/15nm) was deposited on top of the graphene layer lying over the SiO 2/Si substrate by electron-beam evaporat ion at a base pressure of 310-7 Torr. For the purpose of FMR comparison experiments, a control FM film of the same thickness was deposited simultaneously on the same SiO 2 wafer in an area where graphene was no t present . The schematic of th e FM/Gr samples is shown in Fig. 1-a, together with the Raman spectr um of the CVD graphene before the deposition of Py (Fig. 1-c). The high intens ity of the 2D peak , when compared to the G peak , and the weakness of the D peak , suggests that graphene is sin gle layer and of high quality ( i.e. low degree of inhomogeneity/defects ). Page 4 of 13 Fig. 1 : (Color online) (a) Schematic of the FM/Gr film sample. (b) Schematic of the FMR measurement setup, with the sample placed up -side-down on top of the micro -CPW . (c) Raman spectrum of CVD graphene. FMR measurements were carried out at room temperature with a high -frequency broadband (1 -50 GHz) micro -coplanar -waveguide ( -CPW) [20] using the flip -chip method [ 21- 23], by which the sample is placed up -side-down covering the cen tral part of the CPW (as shown in Fig. 1-b), where the transmission line is constricted to increase the density of the microwave field and enhance sensitivity . The CPW was covered with a 100nm -thick insulating layer of PMMA resist, hardened by electron bea m exposure, to avoid any influence of the CPW, made out of gold, on the sample dynamics. A 1.5 Tesla rotatable electromagnet was employed to vary the applied field direction from the in -plane ( = 0o) to normal -to-the film plane ( = 90o) directions . Fig. 2-a shows t he angular dependence of the FMR field measured at 10 GHz for both Py and Py/Gr films. The rotation plane is chosen to keep the dc magnetic field , H, perpendicular to the microwave field felt by the sample at all times, as shown in Fig. 1-b. The resonance fiel d increases as the magnetic field is directed away from the film plane (i.e. increasing ), as expected for a thin film ferromagnet with in-plane shape magneto -anisotropy. The angular dependence of the FMR field ( HR) can be fitted using the resonance frequency condition given by the Smit and Beljers formula [ 23,24], Page 5 of 13 21HH, (1) where f2 is the angular frequency, /Bg the gyromagnetic ratio, and H1 and H2 are given by H1=Hcos(-)-4Meffsin2 H2=Hcos(-)4Meffcos22K2 Mssin2 , (2) where is the magnetization angle , 2 2 1 cos 4 2 4 4S S S eff MK MK M M is the effective demagnetization field, Ms is the saturation magnetization, and K1 and K2 are the first and second order anisotropy energies, respectively . The best fit s to the data in Fig. 2-a are given by the parameters shown in the third column of Table 1 , together with the corresponding parameters extracted from equivalent measurements on the Co and Co/Gr films (not shown) . Fig. 2 : (color online) (a) Angular dependence of the FMR fi elds measured on both Py and Py/Gr samples at f = 10 GHz with the dc magnetic field, H, applied in a plane perpendicular to the microwave field generated by the CPW at the sample position. (b) In-plane frequency dependence of the FMR fields for both Py and Py/Gr samples. The intercepts with the x-axis give the effective demagnetizing fields of the samples. Page 6 of 13 It is useful to study the resonant behavior by applying the magnetic field at = 0o (parallel configuration) and = 90 (perpendicular configuration), since the frequency behavior of the FMR fields are given respectively by, ,//,2 // 4) 4 ( eff Reff R R M HM HH , (3) where Bg is the gyromagnetic ratio , 1 //,4 4A S eff H M M , 2 1 ,4 4A A S eff H H M M , with S A MK H1 12 and S A MK H2 24 the first and second order anisotropy fields, respectively , which relate to su rface, interface and/or magnetoelastic anisotropy. Note that K 1 > 0 (>> K2) provides out -of-plane anisotropy, competing with the in - plane shape anisotropy . Consequently, a graphical representation of the in - and out -of-plane frequency response of the FMR f ields , conforming to Eq . (3), results in a linear behavior from which the slope and intercept with the magnetic field axis give and the effective demagnetiz ation field s, respectively. The results obtained for the Py and Py/Gr samples are shown in Fig. 2-b and 2-c, and the extracted parameters are listed in the third column of Table 1, together with those extracted from the Co and Co/Gr . Note that the anisotropy fields depend on the selection of the saturation magnetization, w ith theoretical values MS,Py = 9.27 kG (attending to a 20/80 -Ni/Fe ratio and assuming identical densities) , and MS,Co = 17.59 kG. For the Co and Co/Gr films, the effective saturation magnetization ( Meff = 17.7 kG) is similar to the one expected from theo ry, hence there is negligible ou t-of-plane anisotropy (K 1 ~ 0), in agreement with previous studies [ 25]. The situation is different in the case of the Py and Py/Gr, where the small Py anisotropy field HA1 = 1.98 kG grows significantly in the Py/Gr ( HA1 = 3.60 kG), suggesting an increase of the Py surface anisotropy due to the presence of the graphene layer ( i.e. Page 7 of 13 interface effect). Nevertheless, the magnetization remains in the plane of the film for all samples . Theory Sample HR vs. , f Damping Changes Py: Ni80Fe20 geff = 2.10 FeFe S NiNi SFe SNi S effg M g MM Mg 2.0 8.02.0 8.0 Ms = 9.27 kG Fe SNi S M M Ms 2.0 8.0 with 21.2 094.6 NiNi S g kG M 0.2 016.22 FeFe S g kG M Py g = 2.110 = 0.0 113 G = 0.311 GHz K1 increases (interface) Damping increases by ~88% Meff = 7.30 kG H1 = 1.98 kG K1 = 0.73106 erg/cc Py/Gr g = 2.107 = 0.0 213 G = 0.585 GHz Meff,// = 5.70 kG HA1 = 3.60 kG K1 = 1.32106 erg/cc Co g = 2.145 Ms = 17.59 kG Co g = 2. 149 = 0.0 210 G = 1.11 GHz (no K1) Damping increases by ~133% Meff = 17.7 kG Co/Gr g = 2. 149 = 0.0 489 G = 2.59 GHz Meff = 17.5 kG TABLE I: Parameters extracted from the analysis of the data reported in this work. We now focus on the FMR linewidth and its frequency dependence when the magnetic field is applied parallel to the film ( = 0o), from which information about the Gilbert damping (i.e., spin relaxation dynamics) can be directly extracted. The inset to Fig. 3 shows a field derivate of the CPW S 21 transmission parameter obtained when exciting the FMR at 10 GHz in both Py and Py/Gr samples , with HR = 1.28 kG and 1.55 kG, respectively . The peak -to-peak distance repre sents the linewidth , H, of the FMR, whose behavior as a function of frequency is shown for both samples in the main panel of Fig. 3. A remarkable in crease of the FMR linewidth by 88% is observed in the Py/Gr sample , and even higher (133%) in the Co/ Gr fil ms. The change in the linewidth must be attributed to a substantial enhancement of the Gilbert damping in the Page 8 of 13 FM film due to the influence of the graphene directly underneath. The frequency dependence of the FMR linewidth can be written as a contribution f rom two parts : f H H 34 0 , ( 4) where is the parameter of the Gilbert damping SM G . The first term, 0H , accounts for sample -dependent in homogeneous broadening of the linewidth and is independent of frequenc y, while the second term represents the dynamical broadening of the FMR and scales linearly with frequency . Fig. 3: (color online) Frequency dependence of the FRM linewidth for Py , Py/Gr , Co and Co/Gr films obtained with the magnetic field applied at = 0 (in-plane configuration). The inset shows the field derivatives of CPW S 21 transmission parameter (at 10 GHz) of the Py and Py/Gr samples , from which the linewidth, H, is calculated as the peak -to-peak distance. Page 9 of 13 As observed in Fig. 3, the measured l inewidth for both FM and FM/Gr samples increase s linearly with frequency, with negligible inhomogen eous broadening, indicat ing that damping in the FM film can be properly explained by the phenomenological Landau-Lifshitz -Gilbert damp ing model . A similar br oadening of the FMR linewidth is observed in both samples when the field is applied perpendicular to the plane, excluding frequency -dependence inhomogeneous broadening (e.g. two -magnon scattering produced by changes in morphology of the FM surface [26]), as its possible source. By fitting the data in Fig. 3 to Eq. (4) (using 0H = 0), the damping parameter s and G are determined and given in the fourth column of Table 1 for all studied samples. The Gilbert damping increases substantially in the FM/Gr films as a result of the increased linewidth, when compared to the values obtained in the FM sample s (which are comparable with values given in the literature for similar Py and Co films [9,22]). This is our key finding. Remarkably , the change in the damping para meter in the Py/Gr sample ( Py GrPy / = 0.01 ) is even more pronounced than those observed in Py/Pt systems, in which the thick (when compared to graphene) heavy transition metal Pt layer provides the large spin-orbit coupling necessary to absorb (i.e. , relax ) the spin accumulation pumped away from the ferromagnet. The efficiency of spin injection is usually cataloged by means of the interfacial spin-mixing conductance, which is proportional to the additional damping parameter, , as follows: FMSdMg4 , (5) giving g = 5.261019 m-2 for our Py/Gr sample with the thickness of the Py film dFM = 14 nm. The Py/Gr value is substantially larger than those found in other Py/NM systems with a metallic Page 10 of 13 NM layer, e.g. , g = 2.191019 m-2 in Py(Ni 81Fe19:10nm)/Pt(10nm) [9] or g = 2.11019 m-2 in Py(Ni 80Fe20:15nm)/Pt(15nm) [11]. Note that in the cited experiments, the spin -diffusion length of the non -magnetic layer (~10 nm for Pt) is smaller than the layer thickness. This is sign ificant since it explains how the Pt layer is capable of dissipating the spin accumulation generated by the dynamical spin pumping , and account for the loss of angular momentum in the Py . In the case of graphe ne, the enhancement of the damping parameter is more complicated to understand. In a standard FM/NM metallic system , the spin current injected in to the NM layer decays mainly perpendicularly to the interface [ 27], causing the enhancement of the damping parameter to depend on the ratio between the layer thickness and the spin -diffusion length in the NM . However, graphene has effectively zero thickness and, at least theoretically , a very weak intrinsic spin-orbit coupling. Therefore, the spin current must decay in a FM/Gr film parallel and not perpendicul ar to the interface . Furthermore, some sizable spin -orbit coupling must exist in CVD graphene films . The lat ter may also explain the generally observed very short spin relaxation times in lateral CVD graphene spin valves [28]. Recently, small levels of hyd rogen [29] and copper adatoms [30] have been predicted to lead to a strong e nhancement of the spin-orbit coupling, bringing it into meV range. Cu adat oms are certainly likely to be present in the CVD samples utilized in our experiments, pointing at a possi ble explanation for the large spin pumping effect observed in our FM/Gr films . Page 11 of 13 References: [1] F. J. Jadema, A. T. Filip, and B. J. van Wees: Nature (London) 410, 345 (2001) [2] 2.T. Kimura, Y. Ot ani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). [3] S. O. Valenzuela and M. Tinkham, Nature (London ) 442, 176 (2006 ). [4] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. Nitta, and K. Takanashi, Nature Mate r. 7, 125 (2008). [5] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett . 88, 182509 (2006). [6] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010). [7] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002). [8] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003). [9] K. Ando, T. Yoshino, and E. Saitoh, Appl. Phys. Lett. 94, 152509 (2009). [10] Y. Kajiwara, K. H arii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature (London ) 464, 262 (2010) . [11] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann , Phys. Rev. B . 82, 214403 (2011) . [12] K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, Phys. Rev. B 78, 014413 (2008). [13] K. Ando, S. Takahashi, J. leda, H. Kurebayashi , T. Trypiniotis, C.H.W. Barnes, S.Maekawa and E. Sai toh et al. Nature Mater. 10, 655–659 (2011). Page 12 of 13 [14] Arne Brataas , Yaroslav Tserkovnyak , Gerrit E. W. Bauer , Paul J. Kelly , http:// arxiv.org/abs/1108.0385 . [15] Kazuya Ando and Eiji Saitoh , Nature Communications 3, 629 (2012) . [16] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,Y. Zhang, S. V. Dubo nos, I. V. Grigorieva, A. A. Firsov Science 306, 666 (2004). [17] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature 448 (7153), 571 (2007); T. Y. Yang, J. Balakrishnan, F. Volmer, A. Avsar, M. Jaiswal, J. Samm, S. Ali, A. Pachoud, M. Zeng, M. Popinciuc, G. Güntherodt, B. Beschoten, and B. Özyilmaz, Phys. Rev. Lett. 107 (4) (2011); Wei Han and R. Kawakami, Phys. Rev. Lett. 107 (4) (2011); B. Dluba k, M -B. Martin, C. Deranlot, B. Servet, S. Xavier, R. Mattana, M. Sprinkle, C. Berger, W. A. De Heer, F. Petroff, A. Anane, P. Seneor, and A. Fert, Nat . Phys . 8 (7), 557 (2012). [18] X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S. K. Banerjee, L. Colombo, and R. S. Ruoff, Science 324 (5932), 1312 (2009). [19] S. Bae, H. Kim, Y. Lee, X. Xu, J -S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, Y -J. Kim, K. S. Kim, B. Ozyilmaz, J -H Ahn, B. H. Hong, and S. Iijima, Nat. Nano . 5 (8), 574 (2010). [20] W. Barry, I.E.E.E Trans. Micr. Theor. Techn . MTT 34, 80 (1996). [21] G. Counil, J. V. Kim, T. Devolder, C. Chappert, K. Shigeto and Y. Otani, J. Appl. Phys. 95, 5646 (2004). [22] J-M. L. Beaujour, W. Chen, K. Krycka, C. –C. Kao, J. Z. Sun and A. D. Kent, Eur. Phys. J. B 59, 475 -483 (2007). Page 13 of 13 [23] J. J. Gonzalez -Pons, J. J. Henderson, E. del Barco and B. Ozyilmaz , J. Appl. Phys. 78, 012408 (2008). [24] S. V. Vonsovskii, Ferromagnetic Resonance , Pergamon, Oxford, (1996). [25] J-M. L. Beaujour, J. H. Lee, A. D. Kent, K. Krycka and C -C. Kao, Phys. Rev. B 74, 214405 (2006). [26] M. J. Hurben and C. E. Patton, J. Appl. Phys . 83, 4344 -4365 (1998). [27] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [28] A. Avsar, T. Y. Yang, S. Bae, J. Balakrishnan, F. Volmer, M. Jaiswal, Z. Yi, S. R. Ali, G. Gunthe rodt, B. H. Hong, B. Beschoten, and B. Ozyilmaz, Nano letters 11 (6), 2363 (2011). [29] A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103, 026804 (2009). [30] C. Cao, M. Wu, J. Jiang, and H -P. Cheng, Phys. Rev. B . 81, 205424 (2010).

2012-11-02

We demonstrate dynamic spin injection into chemical vapor deposition (CVD) grown graphene by spin pumping from permalloy (Py) layers. Ferromagnetic resonance measurements at room temperature reveal a strong enhancement of the Gilbert damping at the Py/graphene interface, exceeding that observed in even Py/platinum interfaces. Similar results are also shown on Co/graphene layers. This enhancement in the Gilbert damping is understood as the consequence of spin pumping at the interface driven by magnetization dynamics. Our observations suggest a strong enhancement of spin-orbit coupling in CVD graphene, in agreement with earlier spin valve measurements.

Dynamic Spin Injection into Chemical Vapor Deposited Graphene

1211.0492v1

1 Perpendicular magnetic anisotropy in Pt/Co -based full Heusler alloy /MgO thin films structures M.S. Gabor1a), M. Nasui1, A. Timar -Gabor2 1Center for Superconductivity, Spintronics and Surface Science, Physics and Chemistry Department, Technical University of Cluj -Napoca, Str. Memorandumului , 400114 Cluj - Napoca , Romania 2 Interdisciplinary Research Institute on Bio -Nano -Sciences, Babeș -Bolyai University, Str. Treboniu Laurean, 400271, Cluj -Napoca, Romania Abstract Perpendicular magnetic anisotropy (PMA) in ultrathin magnetic structures is a key ingredient for the development of electrically controlled spintronic devices. Due to their relatively large spin -polarization , high Curie temperature and low Gilbert damping t he Co -based full Heusler alloys are of special importance from a scientific and application s point of view. Here, we study the mechanisms responsible for the PMA in Pt/Co -based full Heusler alloy /MgO thin films structures . We show that the ultrathin Heusler films exhibit strong PMA even in the absence of magnetic annealing. By means of ferromagne tic resonance experiments, we demonstrate that the effective magnetization shows a two -regime behavior depending on the thickness of the Heusler layers . Using Auger spectroscopy measurements, we evidence interdiffusion at the underlayer/Heusler interface and the formation of an interfacial CoFe -rich layer which causes the two -regime behavior. In the case of the ultrathin films, th e interfacial CoFe -rich layer promotes the strong PMA through the electronic hybridization of the metal alloy and oxygen orbitals across the ferromagnet /MgO interface . In addition, the interfacial CoFe -rich layer it is also generating an increase of the Gilbert damping for the ultrathin films beyond the spin -pumping effect . Our results illustrate that the strong PMA is not an intrinsic property of the Heusler/MgO interface but it is actively influenced by the interdiffusion, which can be tuned by a proper choice of the underlayer material, as we show for the case of the Pt, Ta and Cr underlayers. a) mihai.gabor@phys.utcluj.ro 2 Introduction Ultrathin films structures showing perpendicular magnetic anisotropy (PMA) are under intensive research for the development of electrically controlled spintronic devices. Particularly, current induced spin–orbit torques (SOTs) in heavy -metal/ferromagnet (FM) heterostructures showing PMA are used to trigger the magnetization switching1,2. Besides , the antisymmetric interfacial Dzyaloshinskii -Moriya3,4 interaction ( iDMI) in similar PMA architectures , if strong enough, can lead to the formation of special chiral structures like skyrmions5 which are drivable by electrical currents6. In the case of the spin transfer torque magnet ic random -access memories (STT -MRAMs), considered a s a poten tial replacement for the semiconductor -based ones, the use of strong PMA materials is required for increased thermal stability7, while high spin polarization and low Gilbert damping is needed to obtain large magnetoresistive ratios and efficient curre nt induced STT switching8,9. Cobalt-based full Heusler alloys are a special class of ferromagnetic materials that attract an increased scientific interest , since the ir theoretical prediction of half-metallicity10. These compounds are described by the formula Co2YZ, where Y is a transition metal, or a mixture of two trans ition metals , and Z is a main group , or a mixture of two main group sp element s. Large magnetoresistive ratios are experimentally demonstrated in certain Co2YZ based in -plane11-18 and out -of-plane19,20 magnetized magnetic tunnel junctions (MTJs) and r elative ly low Gilbert dampin g parameters were determined for some compounds21- 28. Furthermore , PMA was evidenced for Co2FeAl/ MgO19,29-33, Co 2FeAl 0.5Si0.5/MgO34-37, Co2FeSi/MgO38,39 or Co2FexMn 1-xSi/MgO40,41 structures with different non-magnetic underla yers. In some of the cases, an annealing stage was necessary to induce PMA, while for the other the perpendicular magnetiz ation was achieved even in the as -deposited state. The origin of the strong PMA in th is type of structures is still under debate . It could be related to both the oxidation at the Heusler/MgO interface42,43 and to the spin -orbit interaction effects at the heavy -metal underlayer/Heusler interface44,45. Moreover, it was recently pointed out that in t he case of Co 2FeAl/MgO the diffusion of Al towards the MgO layer during annealing plays an important role for the stabiliz ation of the PMA46,47. The precise knowledge and control of the mechanisms responsible for PMA is essential in order to be able to develop viable spintronic application s. Therefore, i n this paper, we study the underlying physics governing the PMA for Co2FeAl, Co2FeAl 0.5Si0.5, Co2FeSi and Co2Fe0.5Mn 0.5Si Heusler alloy thin films sandwiched between Pt and MgO layers. We show that b elow a certain critical thickness all the Heusler films show strong PMA even in the 3 absence of magnetic annealing . Additionally, using ferromagnetic resonance experiments , we demonstrate that, depending on the thickness of the Heusler layers, the effective perpendicular magnetic anisotropy shows a two-regime behavior. After excluding other possible mechanism s, we evidence using Auger spectroscopy measu rements, that the diffusion of the lighter elements towards the Pt underlayer and the formation of an interfacial CoFe -rich layer causes the two-regime behavior . In the case of the ultrathin films , this interfacial CoFe -rich layer promotes the strong PMA through the hybridization of the [Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbital s at the interface and is also responsible for the increased Gilbert damping . Our study reveals that the strong PMA is not intrinsic to the Heusler/MgO interface . It is strongly influenced by the interdiffusion and can be adjusted by a proper choice of the underlayer material , as we show for the case of the Pt, Ta and Cr underlayers. Experimental All the samples studied here were grown at room temperature on thermally oxidized silicon substrates in a magnetron sputtering system having a base pressure lower than 2×10-8 Torr. The main samples have the following structure: Si/SiO 2//Ta (3 nm )/Pt (4 nm)/ FM (0.8-10 nm)/MgO (1 nm)/ Ta (3 nm), where FM stands for Co2FeAl (CFA), Co 2FeAl 0.5Si0.5 (CFAS), Co 2FeSi (CFS), Co 2Fe0.5Mn 0.5Si (CFMS) or CoFeB (CFB) , depending on the sample. Additional samples were grown, and their structure will be discussed later in the text. The metallic layers were deposited by dc sputtering under an argon pressure of 1 mTorr, while the MgO layer was grown by rf sputtering under an argon pressure of 10 mTorr. The Heusler alloy s thin films were sputtered from stoichiometric targets. The 3 nm thick Ta buffer layer was grown directly on the substrate to minimize the roughness and to facilitate the (111) texturing of the upper Pt layer. The 1 nm thick MgO layer was deposited to induce perpendicular m agnetic anisotropy on the Heusler thin film7. An additional 3 nm th ick Ta capping layer was sputtered to protect the samples from oxidation due to air exposure. The structure of the samples was characterized by x -ray diffraction (XRD) using a four - circle diffractometer. The static magnetic properties have been investigated using a Vibrating Sample Magnetometer (VSM) , while the dynamic magnetic properties by using a TE(011) cavity Ferromagnetic Resonance (FMR) setup working in X -band (9.79 GHz ). Auger spectra have been recorded in derivative mode, using a cylindrical mirror analyzer spectrometer working at an electron beam energy of 3keV. Depth profile analysis have been performed by successive recording of the Auger spectra and Ar ion sputter -etching of the surface of the samples by using a relatively low ion energy of 600 eV. 4 Results and discussions Figure 1 (a) shows 2θ/ω x-ray diffraction patterns recorded for four representative Pt (4 nm)/ Co2YZ (10 nm)/ MgO (1 nm) samples . Irrespective of the Heusler composition, the patterns show the (111) and (222) peaks belonging to the Pt layer , the (022) peak arising from the Heusler films and the (001) peak of the Si substrate . This indicates that the Pt layer has a (111) out -of-plane texture, while the Heusler films are (011) out -of-plane textured. Laue oscillations are observable around the (111) Pt reflection which confirms the good crystalline quality for the Pt films48. Moreover, ϕ -scan measurements (not shown here) indicate that both the Pt underlayer and the Heusler fil ms have no in -plane texturing but show an in-plane isotropic distribution of the crystallites . No peak belon ging to the Ta capping layer was observed, indicating that the film is in an amorphous or nanocrystalline state. The static magnetic properties of our films were characterized by VSM measurements . Figure 2 show s hysteresis loops measured with the magnetic field applied perpendicular to the plane of the samples, for representative Heusler films thickness es. In order to remove th e substrate diamagnetic contribution, we fitted the large field data with a linear function and extract ed the linear slope from the raw data. Regardless of the ir composition , all the Heusler films show a similar behavior. Above a critical spin-reorientation transition thickness the samples show in-plane magnetic anisotropy . This is indicated by the shape of the hysteresis loop s in Fig. 2 (a)–(d), which is typical for a hard axis of magnetization, showing a continuous rotation of the magnetization up to saturation. Below th is critical thickness, the samples show PMA , which is attested by the square shaped hysteresis loops in Fig. 2 (e) –(h). We also determined the saturation magnetization (𝑀𝑆) and the effective thickness es of the ferr omagnetic layers using hysteresis loop measurements and the procedure described in 31. The effective thicknesses of the ferromagnetic layers are used throughout the paper and the 𝑀𝑆 is found to be 790 ± 70 emu/cm3, 660 ± 50 emu/cm3, 935 ± 75 emu/cm3 and 895 ± 75 emu/cm3 for CFS, CFMS, CFAS and CFA samples, respectively. In order to get more insights on the magnetic anisotropy properties of our films, we have performed FMR measurements with the magnetic field applied at diffe rent θH angles (defined in the inset of Fig. 4) with respect to the normal direction of the layers . Figure 3 shows typical FMR spectra for various fie ld angle s recorded for a 2.4 nm thick Pt/CFAS sample . We define the resonance field HR as the intersection of the spectr um with the base line, and the linewidth HPP as the distance between the positive and negative peaks of the spectrum. Figure 4 shows the θH dependence of the HR and of the linewidth HPP for the 2.4 5 nm thick Pt/CFAS sample. In order to extract the relevant FMR parameters, we analyzed the θH dependence of the FMR spectrum using a model in which the total energy per unit volume is given by 𝐸=−𝑀𝑆𝐻cos(𝜃𝐻−𝜃𝑀)+2𝜋𝑀𝑆2cos2𝜃𝑀−𝐾⊥cos2𝜃𝑀, (1) where the first term is the Zeeman energy, the second term is the demagnetizing energy, and the last term is the magnetic anisotropy energy. The 𝑀𝑆 is the saturation magnetization, 𝜃𝐻 and 𝜃𝑀 are the field and magnetization angles defined in the inset of Fig. 4, and the 𝐾⊥ is the effective perpendicular magnetic anisotropy constant. From eq. 1 and the Landau -Lifshitz -Gilbert equation, one can der ive the resonance condition as49 (𝜔 𝛾)2 =𝐻1×𝐻2, (2) where 𝜔 is the angular frequency of the microwave , 𝛾 is the gyromagnetic ratio , given by 𝛾=𝑔𝜇𝐵ℏ where 𝑔 is the Landé g-factor, 𝜇𝐵 is the Bohr magneton and ℏ is the reduced Plan ck constant , and with 𝐻1 and 𝐻2 given by 𝐻1=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀, (3) 𝐻2=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀, (4) where 4𝜋𝑀eff is the effective magnetization defined as 4𝜋𝑀eff=4𝜋𝑀𝑆−2𝐾⊥𝑀𝑆⁄ and 𝐻𝑅 is the resonance field. For each value of 𝜃𝐻, the 𝜃𝑀 at resonance is calculated from the energy minimum condition 𝜕𝐸 𝜕𝜃𝑀=0 ⁄ . Hence , the 𝐻𝑅 dependence on 𝜃𝐻 can be fitted by Eq. (2)-(4) using 4𝜋𝑀eff and 𝑔 as adjustable parameters . A typical fit curve is s hown in Fig. 4(a). Figure 5 shows the 𝑔 factor depen dence on the thickness of the Heusler layers for samples with different Heusler layer composition s. Depending on the thickness , two regimes are discernable . For relatively large thickness es, above 2.5-3 nm, the 𝑔 factor shows rather constant value s between 2.07 and 2.11, depending on the type of the Heusler layer . For lower thickness es, 𝑔 shows a monotonous decrease, regardless of the Heusler layer composition. This is an interface effect and it is usually attributed to the fact that at the interfaces , due to the symmetry breaking, the orbital motion is n o longer entirely quenched and will contribute to the gyromagnetic ratio50,51. Another possibility, which cannot be exclude d in our case, is the reduction of the 𝑔 factor due to intermixing between the ferromagnetic Heusler layer and non - magnetic materials at the interfaces50. Figure 6 shows the effective magnetization 4𝜋𝑀eff dependence on the inverse thickness of the ferromagnetic layer for samples with different composition s. It is to be mentioned that the 4𝜋𝑀eff was determined from FMR experiments only for samples with in-plane magnetic anisotropy (positive 4𝜋𝑀eff). In the case of ultrathin samples showing perpendicular magnetic anisotropy (negative 4𝜋𝑀eff), due to the 6 strong linewidth enhancement , it was not possible to obtain reliable resonance curves. Therefore, in this case the 4𝜋𝑀eff was estimated from VSM measurements. Generally, it is considered that the effective perpendicular magnetic anisotropy co nstant 𝐾⊥ can be written as the sum of a volume (𝐾𝑉), which includes magneto -crystalline and strain related anisotropies, and a surface (𝐾𝑆) contribution : 𝐾⊥=𝐾𝑉+𝐾𝑆𝑡⁄, where 𝑡 is the thickness of the ferromagnetic layer . Thus, the effective magnetization can be written as 4𝜋𝑀eff=(4𝜋𝑀𝑆−2𝐾𝑉 𝑀𝑆)−2𝐾𝑆 𝑀𝑆1 𝑡. (5) The above relation implies a linear dependence of the effective magnetization on the inverse thickness of the ferromagnetic layer. However, as shown in Fig.6 the Heusler samples do not show a single linear dependence for the entire thickness range, but two regimes above and below a certain critical thickness. Using the 𝑀𝑆 values determined from VSM measurements and b y fitting the experimental data in the large thickness regime to eq. (5) , we extract a surface anisotropy constant 𝐾𝑆 for the CFA and CFAS of 0.24 ± 0.03 erg/ cm2 and 0.22 ± 0.02 erg/cm2 and a volume contribution 𝐾𝑉 of (1.27 ± 0. 69)×106 erg/cm3 and (1.51 ± 0.7)×106 erg/cm3, respectively. In the case of the CFMS and CFS, the 𝐾𝑆 was negligible small within the error bars and the 𝐾𝑉 was found to be (0.44 ± 0.38)×106 erg/cm3 and ( 0.51 ± 0. 4)×106 erg/cm3, respectively. Using the as extracted values of th e anisotropy constants , we can calculate , for example, in the case of the CFAS samples a spin-reorientation tra nsition thickness of around 0.55 nm. This is clearly not in agreement with the experimental data , as seen from Fig. 2 and 6 , already a 1 nm thick CFAS film shows strong PMA and it is spontaneous perpendicular ly magnetized . This is a consequence of the fact that the 1 nm thick CFAS film falls within the second anisotropy regime below the critical thickness . The occurrence of this second anisotropy regime with larger effective perpendicular magnetic anisotropy can have several explanations. For such thin films one must always consider the possible influences of the surface roughness. If the roughness is relatively large , an in -plane demagnetization field will develop at the edges of the terraces which will reduce the shape anisotropy and fav or perpendicular magneti zation . This is equivalent to the emergence of an additional dipolar surface anisotropy contribution52. The roughness is a parameter which is not easily quantifiable experimentally in such thin multilayer structures. However, it is reasonable to expect to be comparable for similar heterostructures in which the Heusler alloy film is replaced with a CFB layer . Atomic force microscopy topography images (not shown here) recorded for heterostructure w ith CFB and CFAS layers are featureless and show a similar RMS roughness. As such, if the low thickness anisotropy regime is due to the roughness it must be observable also in the case of CFB samples. However, this is not the case , as shown in Fig. 6, the CFB samples show 7 a single linear behavior for the whole range of thickness . Fitting the data to eq. (5), allowed us to extract for CFB samples a surface anisotropy contribution 𝐾𝑆 of 0.79 ± 0.04 erg/cm2 and a negligible small 𝐾𝑉 volume contribution , in line with previous reports7,53. These findings suggest that the roughness is not responsible for the two regimes behavior observed in the case of the Heusler samples. Another possible physical mechanism which can explain the presence of the two regimes is the strain variation due to coherent –incoherent growth transition54,55. Within this model, below the critical thickness , the ferromagnetic layer grows uniformly strained in order to account for the lattice misfit with the adjacent layer s. Above the critical thickness, the strains are partially relaxed through the formation of misfit dislocations. The changes in the magnetoelastic anisotropy contributions corresponding to this structural transition can be responsible for t he presence of the two regimes54,55. This scenario is likely in the case of the Heusler samples , since both the bottom Pt layer the upper Heusler film grow out-of-plane textured. In order to test this hypothesis, we have deposited two additional sets of samples. The first set consisted of Si/SiO 2//Ta ( 6 nm)/ CFAS (tCFAS)/MgO (1 nm)/Ta (3 nm) samples . The motivation to grow this type of samples was to obtain Heusler films with no out -of-plane texturing. Indeed, x -ray diffra ction measurement [Fig. 1(b)], performed on a Ta/CFAS sample with a Heusler layer thickness of 10 nm , did not indicate the presence of any diffraction peak s, except for the one belonging to the Si substrate. This suggest that both the Ta and the CFAS films are either nanocrystalline or amorphous . Thus , in this type of structure we do not expect the presence of the coherent –incoherent growth transition. The second set of samples consisted of epitaxial MgO (001)//Cr ( 4 nm)/CFAS (tCFAS)/MgO (1 nm)/Ta (3 n m) structures . The x -ray diffraction measurement [Fig. 1 (b)], performed on a Cr/CFAS sample with a Heusler layer thickness of 10 nm, indicate s the exclusive presence of the (001) type reflections from the MgO substrate and the Cr and CFAS layers . This confirms the epitaxial growth of the stacks, except for the Ta capping layer, which is amorphous. Having in view the epitaxial growth we might expect for these samples a possible coherent – incoherent growth transition, eventually at higher CFAS thick nesses having in view the relative low mismatch between the CFAS lattice and the 45° in-plane rotated Cr lattice (0.7%). The effective magnetization dependence on the inverse thickness of the ferromagnetic layer for amorphous Ta/CFAS and epitaxial Cr/CFAS samples alongside with the Pt/CFAS samples is shown in Fig. 7 . Is to be mentioned that in the for the Ta/CFAS samples the PMA was obtained for thicknesses below 1.6 nm , while for the Cr/CFAS the PMA was not achieved even for thicknesses down to 1 nm. Interestingly, in the case of the epitaxial Cr/CFAS samples , for which one might expect possible coherent –incoherent growth transition, a single linear behavior for the whole thickness range is observ ed. In the case amorphous Ta/CFAS samples, for which the coherent –incoherent growth transition is not expected, a two-regimes behavior can 8 be distinguished . Although we cannot rule a possible coherent -incoherent growth transition at larger thicknesses, t he results indicate this mechanism is not responsible for the two regimes behavior that we observe at relatively low thicknesses and other mechanism s must be at play. By fitting the high thickness regime data from Fig.7 to eq. (5) we extracted for Ta/CFAS samples a surface anisotropy contribution 𝐾𝑆 of 0.27 ± 0.08 erg/cm2. The volume contribution, 𝐾𝑉, was determine d to the be negligible small, as expected for untextured films. Remarkably, the 𝐾𝑆 for the Ta/CFAS samples is similar within the error bar s to one obtained for the Pt/CFAS samples. Moreover, even in the low thickness regime the 𝐾𝑆 might be assumed similar for the two sets of samples. However, we must consider the large uncertainty having in view the sparse data points available for fitting in the low thickness regime. Even so, the clear difference between the two sets of samples is that the Ta/CFAS one shows a larger critical thickness (around 2.4 nm ) that separates the two anisotropy regimes , as compared t o the Pt/CFAS one (around 1.5 nm) . This suggests that the possible mechanism responsible for the two -regime behavior might be related to the atomic diffusion at the Pt/CFAS and Ta/CFAS interface s. It is well known that Ta is prone to diffusion of light elements56. Therefore , a larger critical thickness for the Ta/CFAS samples will imply a larger atomic diffusion at the Ta/CFAS interface compared to the Pt/CFAS one . To test th e hypothesis of the interdiffusion , we performed Auger electron spectroscopy (AES) analyses on th e three set of samples : Pt/CFAS, Cr/CFAS and Ta/CFAS. AES is a surface sensitive technique which can give information about the chemical composition of the surface with a depth detection limit of 1 -2 nm. We started from 10 nm thick CFAS layer samples and first Ar ion etched the CFAS films down to 4 nm thickness and recorded the AES spectra. Subsequently, t he Ar ion etching and AES spectra recording was repeated in steps of 1 nm until reaching the underlaye r/CFAS interface. The etching rate of CFAS was previously calibrated using ex-situ x-ray reflectometry measurements. Figure 8 (a) shows two spectra recorded for the Pt/CFAS sample, one after etching the CFAS layer down to 4 nm (Pt/CFAS 4 nm) and the other one after etching the CFAS layer down to 1 nm of thickness (Pt/CFAS 1 nm). In the case of the Pt/CFAS 4 nm spectr um the peaks of Co and Fe are visible alongside with the peaks fro m Al and Si. The inset of Fig. 8(a) depicts a n enlargement of the Pt/CF AS 4 nm spectrum around the peaks of Al and Si. The amplitude of the Co and Fe peaks is m uch larger than the amplitude of the of Al and Si ones. This is due to the higher concentration and higher Auger relative sensitivity of the Co and Fe compared to the Al and Si. In the case of the Pt/CFAS 1 nm the spectrum shows the peaks from Co and Fe , with a lower amplitude, and the peaks from the Pt underlayer. The presence of the Co and Fe peaks together with the Pt peaks is not surprising . It is owed to the possible interdiffusion layer at the interface and to the finite depth resolution of the AES which probes both the CFAS layer and the Pt underlayer. The Al and Si peaks 9 are not observable , which can be associated to the relatively low amplitude of the Al and Si falling below the detection limit of the measurement. To test this possibility, we acquired Auger spectra in a narrow energy window around the Al peak , using a longer acquisition time and averaging 10 spectra for e ach recoded spectrum. We selected the Al peak and not the Si one because of its larger ampli tude. These spectra recorded for the Pt/CFAS, Ta/CFAS and Cr/CFAS samples after etching the CFAS layer down to 4, 3, 2 and 1 nm are shown in Fig.8 (b)-(d). In the case of the Pt/CFAS sample the Al peak is observable for CFAS thickness es down to 1 nm , while in the case of Ta /CFAS for thicknesses down to 2 nm. Interestingly, in the case of the Cr/CFAS sample the Al peak is visible even for a CFAS thickness of 1 nm, although with lower amplitude. These findings suggest th at at the underlayer/CFAS interface there is a diffusion of the light er elements (Al and most likely also Si) towards the underlayer, with different degree, depe nding on the nature of the underla yer. As shown schematically in Fig. 8, due to th is lighter elements diffusion a CoFe -rich layer form s at the underlayer/CFAS interface. The extent of the CoFe rich layer depends on the nature of the underlayer . It has the largest thickness for the Ta underlayer (between 2 and 3 nm) , it is decreasing for the Pt underlayer (between 1 and 2 nm) and it is most likely non -existing or extremely thin (below 1 nm) in the case of the Cr underlayer. The presence of th e CoFe r ich layer agrees with our findings concerning the occurrence of the high and the low effective PMA regime s depend ing on the thickness of the Heusler layer. In the case of the Pt/CFAS /MgO samples , the low effective PMA regime occurs for a CFAS layer thickness above 1.6 nm. In this case, the bottom interface consists of Pt/CoFe -rich layer, while the top one of CFAS/MgO . In principle, both interfaces could contribute to PMA through Co-O hybridization in the case of the Co- terminated CFA S/MgO interface43 or through the d –d hybridization between the spin-split Co 3d bands and the Pt layer 5d bands with large spin-orbit coupling44,45. However, their contribution to PMA is small and, as we previously mentioned, would not stabilize perpendicular magnetization except for extremely thin CFAS layer s. In the case of the high effective PMA regime (below 1.6 nm) , the bottom interface is similar consisting of Pt/CoFe -rich layer and will contribute negligibl y to PMA . However, the top interfa ce is now constituted of CoFe -rich layer/MgO and will induce strong PMA through the hybridization of the [Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. The premise that the strong PMA is induced by the CoFe -rich layer/MgO interface is also consistent with our observat ions regarding the dependence of the magnetic anisotropy on the nature of the underlayer. As seen in Fig. 7, i n the case of the Cr/CFAS samples, where no CoFe -rich layer was evidenced, there is only one anisotropy regime with a relatively low effective PMA . In the case of the Ta/CFAS sa mples, the high effective PMA regime is present starting from a larger CFAS thickness , as compared to de case of Pt/CFAS samples , which is in agreement with the thicker 10 CoFe -rich layer observed for the Ta/CFAS relative to the Pt/CFAS ones. It is to be mentioned that in the case of Ru/CFA/MgO and Cr/CFA/Mg O annealed samples Al diffusion towards the MgO but not towards the underlayer was previously observed46,47. The lack of Al diffusion towards the Cr underlayer is in agreement with our findings. In the case of the aforementioned studies, the thermal annealing of the samples was necessary to facilitate the Al diffusion and to achieve strong PMA. In our case, for the Pt and Ta underlayer , we attain strong PMA in the low thickness regime without the need of thermal annealing. This indicates that for the Pt and Ta underlayer s the [Al,Si] diffusion takes place during the growth of the CFAS film, which results in the formation of the interfacial CoFe -rich layer directly during deposition . A further deposition of MgO on this CoFe -rich layer will generate the strong PMA through the hybridizati on of the [Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. Having in view the si milar behavior of the magnetic anisotropy for the CFA, CFAS, CFMS and CFS Heusler alloys thin films that we study here , it is reasonable to assume that in all the cases there is a diffusion of the lighter elements (Al, Si) towards the Pt underlayer and the formation of the CoFe -rich interfacial layer , which , when MgO is deposited on top, will give rise to the strong PMA in the low thickness regime. We now discuss the thickness dependence of the Gilbert damping parameter extracted from the θH dependence of the linewidth HPP. It is known that generally the linewidth is given by a sum o f extrinsic and intrinsic contr ibution as49,57-59: 𝐻PP=𝐻PPint+𝐻PPext, (5) 𝐻PPint=𝛼(𝐻1+𝐻2)|d𝐻𝑅 d(𝜔𝛾⁄)|, (6) 𝐻PPext=|d𝐻𝑅 d(4𝜋𝑀eff)|Δ(4𝜋𝑀eff)+|d𝐻𝑅 d𝜃𝐻|Δ𝜃𝐻+Δ𝐻TMS, (7) where, 𝛼 is the intrinsic Gilbert dampi ng parameter and the three terms in equation (7) are the linewidth enhancement due to the anisotropy distribution , due to deviation from planarity of the films and due to the two-magnon scattering. In the case of our films, the θH dependence of the linewidth HPP is well fitted using only the intrinsic contribution and the extrinsic enhancement due to the anisotropy distribution. For this, |d𝐻𝑅d(𝜔𝛾⁄) ⁄ | and |d𝐻𝑅d(4𝜋𝑀eff) ⁄ | are numerically calculated using Eqs. (1)-(4) and the HPP vs. θH experimental dependence is fi tted to Eq. (5) using 𝛼 and Δ(4𝜋𝑀eff) as adjustable parameters49. An example of a fit curve is depicted in Fig. 4(b) for the case of the 2.4 nm thick Pt/CFAS sample . Figure 9 shows the 𝛼 dependence on the inverse ferromagnetic layer (1/t) thickness for the Pt/CFA, Pt/CFS, Pt/CFMS , Pt/CF AS and Pt/CFB samples. We will first discuss the case of CFB, where a linear dependence is observed . The linear increase of the Gilbert damping parameter with 1/t is expected and it is due to the angular momentum loss due to the spin pumping effect in the Pt layer. In this type of structures it was 11 shown60 that the total damping is given by 𝛼=𝛼0+𝛼SP𝑡⁄, where 𝛼0 is the Gilbert damping of the ferromagnetic film and 𝛼SP is due to the spin pu mping effect. By linear fitting the data in Fig. 9 we obtain a Gilbert damping parameter for the CFB of 0.0028 ± 0.0003 , in agreement with other reports61,62. In the case of the CFAS films, the linear dependence is observed only for the large thickness region and by fitting this data we obtain a Gilbert da mping parameter of 0.00 53 ± 0.00 12, consistent wit h previously reported value s for relative ly thick er films25. The low thickness data deviates from the linear dependence . This behavior is similar for all the other studied Heusler films, with the low thickness deviation being even more pronounced. The strong increase of the damping can be related to the [Al,Si] diffusion and the formation of the interfacial CoFe -rich layer. Since the [Al,Si] diffusion is more important for thinner films , it will have a stronger impact on the chemical composition relative to the thicker ones. The relatively small damping of the Co based full Heusl er alloys is a consequence of the ir specific electronic structure21. Consequently , deviations from the correct stoichiometry , which is expected to have a n important effect on the electronic structure , will lead to a strong increase of the damping, as shown, for example, by ab- initio calculation in the case of Al deficient CFA films47. Therefore, the increase of the damping beyond the spin pumping effect for the thinner Heusler films is explained by the interfacial CoFe -rich layer format ion. Conclusions We have studied the mechanism s responsible for PMA in the case of Co2FeAl, Co 2FeAl 0.5Si0.5, Co 2FeSi and Co 2Fe0.5Mn 0.5Si Heusler alloy thin films sandwiched between Pt and MgO layers. We showed that the ultrathin Heusler films exhibit strong PMA irrespective of their composition. The effective magnetization displays a two -regime behavior depending on the thickness of the Heusler layers. The two - regime behavior is generated by the formation of a n CoFe -rich layer at the underlayer/Heusler interface due to the interdiffusion. The strong PMA observed in the case of the ultrathin films can be explained by the electronic hybridization of the CoFe -rich metal lic layer and oxygen orbitals across the ferromagne t/MgO interface . The formation of the interfacial CoFe -rich layer causes the increase of the Gilbert damping coefficient beyond the spin pumping for the ultrathin Heusler films. Our results illustrate that the strong PMA is not an intrinsic property of the Heusler/MgO interface, but it is actively influenced by the interdiffusion, which can be tuned by a proper choice of the underlayer material. 12 FIG. 1. (a) 2θ/ω x-ray diffraction patterns recorded for four representative Pt/Co 2YZ/MgO samples having a thickness of the Heusler layer of 10 nm. The patterns show the (111) and (222) peaks belonging to the Pt layer , the (022) peak from the Heusler films and the (001) peak of the Si substrate. (b) 2θ/ω x-ray diffraction patterns for the Ta/CFAS (10 nm)/MgO and Cr/CFAS (10 nm)/MgO samples indicating the amorphous or epitaxial growth of the CFAS layer , respectively. 13 FIG. 2. Hysteresis loops measured with the magnetic field applied perpendicular to the plane of the samples . Depending on the thickness of the Heusler layers, the samples show in-plane magnetic anisotropy (a)-(d) or perpendicular magnetic anisotropy (e) -(h). 14 FIG. 3. Typical FMR spectra measured at 9.79 GHz for different θH field angles for a 2.4 nm thick Pt/CFAS sample. 15 FIG. 4. (a) Resonance field HR and (b) linewidth HPP dependence on the θH field angle for a 2.4 nm thick Pt/CFAS sample . The inset s hows a schematic of the measurement geometry. The points stand for experimental data while the lines represent the result of the theoretical fits, as described in text. 16 FIG. 5. g factor dependence on the thickness of the Heusler layers for samples with different Heusler layer composition. 17 FIG. 6. The effective magnetization 4effM dependence on the inverse thickness of the ferromagnetic layer for samples with different composition s. The points are experimental data while the lines are linear fits. In the case of the Heusler samples two linear fits correspond to the two anisotropy regimes. 18 FIG. 7. The effective magnetization 4effM dependence on the inverse thickness of the ferromagnetic layer for amorphous Ta/CFAS and epitaxial Cr/CFAS samples. The data for Pt/CFAS is also shown for comparison. The points are experimental data while the lines are linear fits. 19 FIG. 8. (a) AES spectra recoded for the Pt/CFAS sample after etching the CFAS layer down to 4 and 1 nm, respectively . The inset shows a zoom around de Al and Si peaks. AES spectra recorded around the Al peak after etching the CFAS layer down to 4, 3, 2 and 1 nm for the (b) Pt/CFAS, (c) Ta/CFAS and (d) Cr/CFAS samples. Schematic representation of the [Al,Si] diffusion to wards the underlayer and the interfacial CoFe -rich layer formation . 20 FIG. 9. Gilbert damping parameter ( 𝛼) dependence on the inverse ferromagnetic layer (1/t) thickness for the Pt/CFA, Pt/CFAS , Pt/CFMS, Pt/CFS and Pt/CFB samples. 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2019-10-29

Perpendicular magnetic anisotropy (PMA) in ultrathin magnetic structures is a key ingredient for the development of electrically controlled spintronic devices. Due to their relatively large spin-polarization, high Curie temperature and low Gilbert damping the Co-based full Heusler alloys are of special importance from a scientific and applications point of view. Here, we study the mechanisms responsible for the PMA in Pt/Co-based full Heusler alloy/MgO thin films structures. We show that the ultrathin Heusler films exhibit strong PMA even in the absence of magnetic annealing. By means of ferromagnetic resonance experiments, we demonstrate that the effective magnetization shows a two-regime behavior depending on the thickness of the Heusler layers. Using Auger spectroscopy measurements, we evidence interdiffusion at the underlayer/Heusler interface and the formation of an interfacial CoFe-rich layer which causes the two-regime behavior. In the case of the ultrathin films, the interfacial CoFe-rich layer promotes the strong PMA through the electronic hybridization of the metal alloy and oxygen orbitals across the ferromagnet/MgO interface. In addition, the interfacial CoFe-rich layer it is also generating an increase of the Gilbert damping for the ultrathin films beyond the spin-pumping effect. Our results illustrate that the strong PMA is not an intrinsic property of the Heusler/MgO interface but it is actively influenced by the interdiffusion, which can be tuned by a proper choice of the underlayer material, as we show for the case of the Pt, Ta and Cr underlayers.

Perpendicular magnetic anisotropy in Pt/Co-based full Heusler alloy/MgO thin films structures

1910.13107v1

Nonlinear network dynamics for interconnected micro-grids Dario Bauso June 7, 2021 Abstract This paper deals with transient stability in interconnected micro-grids. The main contribution involves i) robust classication of transient dynamics for dierent intervals of the micro-grid parameters (synchronization, inertia, and damping); ii) exploration of the analogies with consensus dynamics and bounds on the damping coecient separating underdamped and overdamped dynamics iii) the extension to the case of disturbed measurements due to hackering or parameter uncertainties. Keywords: Synchronization; consensus; Nonlinear control; Transient stability. 1 Introduction This paper investigates transient stability of interconnected micro-grids. First we develop a model for a single micro-grid combining swing dynamics and synchronization, inertia and damping parameters. We focus on the main characteristics of the transient dynamics especially the insurgence of oscillations in underdamped transients. The analysis of the transient dynamics is then extended to multiple interconnected micro-grids. By doing this we relate the transient characteristics to the connectivity of the graph. We also investigate the impact of the disturbed measurements (due to hackering or parameter uncertainties) on the transient. 1.1 Main theoretical ndings The contribution of this paper is three-fold. First, for the single micro-grid we identify intervals for the parameters within which the behavior of the transient stability has similar characteristics. This shows robustness of the results and extends the analysis to cases where the inertia, damping and synchronization parameters are uncertain. In particular we prove that underdamped dynamics and oscillations arise when the damping coecient Dario Bauso is with the Department of Automatic Control and Systems Engineering, The University of Sheeld, Mappin Street Sheeld, S1 3JD, United Kingdom, and with the Dipartimento di Ingegneria Chimica, Gestionale, Informatica, Meccanica, Universit a di Palermo, V.le delle Scienze, 90128 Palermo, Italy.fd.bauso@sheffield.ac.uk garXiv:1708.07296v1 [math.OC] 24 Aug 2017is below a certain threshold which we calculate explicitly. The threshold is obtained as function of the product between the inertia coecient and the synchronization parameter. Second, for interconnected micro-grids, under the hypothesis of homogeneity, we prove that the transient stability mimics a consensus dynamics and provide bounds on the damping coecient for the consensus value to be overdamped or underdamped. This result is meaningful as it sheds light on the insurgence of topology-induced oscillations. These bounds depend on the topology of the grid and in particular on its maximum connectivity, namely, the maximum number of links over all the nodes of the network. We also observe that the consensus value changes dramatically with increasing damping coecient. This implies that the micro-grid, if working in islanding mode, can synchronize to a frequency which deviates from the nominal one of 50 Hz. This nding extends to smart-grids with dierent inertia but same ratio between damping and inertia coecient. Third, we extend the analysis to the case where both frequency and power ow mea- surements are subject to disturbances. Using a traditional technique in nonlinear analysis and control we isolate the nonlinearities in the feedback loop, and analyze stability under some mild assumptions on the nonlinear parameters. The obtained result extends also to the case where the model parameters like synchronization coecient, inertia and damping coecients are uncertain. This adds robustness to our ndings and proves validity of the results even under modeling errors. To corroborate our theoretical ndings a case study from the Nigerian distribution network is discussed. 1.2 Related literature This study leverages on previous contributions of the authors in [2] and [3]. In [2] the author studies exible demand in terms of a population of smart thermostatically con- trolled loads and shows that the transient dynamics can be accommodated within the mean-eld game theory. In [3] the author extends the analysis to uncertain models in- volving both stochastic and deterministic (worst-case) analysis approaches. The analysis of interconnected micro-grids builds on previous studies provided in [5]. Here the authors link transient stability in multiple electrical generators to synchronization in a set of cou- pled Kuramoto oscillators. The connection between Kuramoto oscillators and consensus dynamics is addressed in [7]. A game perspective on Kuramoto oscillators is in [8], where it is shown that the synchronization dynamics admits an interpretation as game dynam- ics with equilibrium points corresponding to Nash equilibria. The observed deviation of the consensus value from the nominal mains frequency in the case of highly overdamped dynamics can be linked to ineciency of equilibria as discussed in [9]. This study has beneted from some graph theory tools and analysis eciently and concisely exposed in [4]. The model used in this paper, which combines swing dynamics with synchronization, inertia and damping parameters has been inspired by [6]. The numerical analysis has been conducted using data provided in [1]. This paper is organized as follows. In Section 2, we model a single micro-grid. In Section 3, we turn to multiple interconnected micro-grids. In Section 4, we analyze the impact of measurement disturbances. In Section 5, we provide numerical studies on the Nigerian grid. Finally, in Section 6, we provide conclusions.2 Model of a single micro-grid Consider a single micro-grid connected to the network, refer to it as the ith micro-grid. Let us denote by Pithe power ow into the ith micro-grid. Also let fibe the frequency deviation of micro-grid iandfja virtual signal representing the frequency of the mains. By applying dc approximation, the power Pievolves according to _Pi=Tij(fjfi) =Tijeij; (1) whereTijis the synchronizing coecient. This coecient is obtained as the inverse of the transmission reactance between micro-grid iandj. In other words, the power Pidepends on the frequency error eij=fjfi. The physical intuition of this is that in response to a positive error we have power injected into the ith micro-grid from the jth micro-grid. Vice versa, a negative error induces power from micro-grid itoj. The dynamics for fifollows a traditional swing equation _fi=Di Mifi+Pi Mi; (2) whereMiandDiare the inertia and damping constants of the ith micro-grid, respectively. By denoting fi=x(i) 1,Pi=x(i) 2,fj=x(j) 1, and by considering fjas an exogenous input to theith micro-grid, the dynamics of the ith micro-grid reduces to the following second- order system" _x(i) 1 _x(i) 2# =Di Mi1 Mi Tij0" x(i) 1 x(i) 2# +0 Tij x(j) i: (3) Figure 1 shows the block representation and corresponding transfer function of the dynamical system (3). fj + eijTij s1 Mis+Difi Pi Figure 1: Block representation of the ith micro-grid. Theorem 1 Dynamics (3) is asymptotically stable. Furthermore, let Di>2p TijMithen the origin is an asymptotically stable node. Vice versa, if Di<2p TijMithen the origin is an asymptotically stable spiral. Proof. For the rst part, stability derives from Tr(A) =Di Mi, whereTr(A) is the trace of matrixAand from
( A) =Tij Mi>0, where
( A) is the determinant of matrix A. Let us recall that stability depends on the eigenvalues of Aand that the expression of the eigenvalues is given by 1;2=Tr(A)p Tr(A)24
(A) 2 =1 2 Di Miq (Di Mi)24Tij Mi :(4)Tr(A)2<4
(A) saddle pointsTr(A)2>4
(A) unstable nodes a.s. nodesa.s. spirals unst. spirals Tr(A)
(A) Figure 2: Classication of equilibrium points. As the trace Tr(A) is strictly negative and the determinant
( A) is strictly positive, then this corresponds to any point in the fourth quadrant in Fig 2, which characterizes stable systems. As for the rest of the proof, we know that if Di>2p TijMithenTr(A)2>4
(A) and the origin is an asymptotically stable node. This corresponds to any point in the fourth quadrant in Fig 2 outside the parabolic curve, whereby the system is stable and no oscillations occur. The parabolic curve identies the set of points for which Tr(A)2= 4
(A). The last case is when Di<2p TijMiwhich implies Tr(A)2<4
(A) and therefore the origin is an asymptotically stable spiral. This corresponds to any point in the fourth quadrant in Fig 2, inside the parabolic curve whereby the system is stable but oscillations may occur due to imaginary parts in the eigenvalues. The above theorem sheds light on the role of the dierent parameters in the transient stability of the micro-grid. Example 1 In particular, let the synchronization coecient be Tij= 1 and the inertia coecient be M= 1 and investigate the role of the damping coecient D. From (4) the transient dynamics is determined by the eigenvalues 1;2=Dip D2 i4 2:We can conclude that ifD > 2all eigenvalues are real and negative and no oscillations arise. The slowest eigenmode is determined by the smallest (in modulus) eigenvalue, which isDi+p D2 i4 2. Dierently, if D2we have complex eigenvalues given by 1;2=Di 2ip D2 i4 2 and we observe damped oscillations. The damping factor depends on the real part Re(1;2) =Di 2while oscillation frequencies are related to the imaginary part Im(1;2) =p D2 i4 2. Example 2 In this example we set the damping coecient D= 1 and the inertia coe- cientM= 1 and investigate the role of the synchronization coecient Tij. Again, from(4), the eigenvalues governing the transient dynamics are 1;2=1p 14Tij 2:Then we have the following cases: ifTij<1 4the eigenvalues are all real and negative and we observe no oscillations. The transient is dominated by the slowest eigenmode, which in turn is determined by the smallest (in modulus) eigenvalue, i.e.1+p 14Tij 2. Unlikewise, if Tij>1 4the eigenvalues are complex and given by 1;2=1 2ip 14Tij 2 in correspondence to which the transient dynamics shows damped oscillations. The damping factor is determined by the real part Re(1;2) =1 2and the oscillation frequencies are determined by the imaginary part Im(1;2) =p 14Tij 2. The above theorem and examples identify intervals for the parameters within which the behavior of the transient stability is unchanged. This provides robustness to our results and extend the analysis to cases where the inertia, damping and synchronization parameters are uncertain. 3 Multiple interconnected micro-grids Let us now consider a network G= (V;E) of interconnected smart-grids, where Vis the set of nodes, and Eis the set of arcs. Figure 3 displays an example of interconnection topology. Nodes represent smart-grids units and arcs represent power lines interconnections. We use shades of gray to emphasize dierent levels of connectivity of the smart-grids. The connectivity of a grid is indicated by the degree of the node. We recall that for undirected graphs the degree of a node is number of links with an extreme in node i. We denote by dithe degree of node i. Figure 3: Graph topology indicating smart-grids and interconnections. Building on model (3) developed for the single grid, we derive the following macroscopicdynamics for the whole grid: 2 666666664_x(1) 1... _x(n) 1 _x(1) 2... _x(n) 23 777777775=2 666666664D1 M1::: 01 M1::: 0 0... 0 0...0 0:::Dn Mn0:::1 Mn T11::: T 1n 0...0 ...... Tn1:::Tnn 0::: 03 7777777752 666666664x(1) 1... x(n) 1 x(i) 2... x(n) 23 777777775: In the above set of equations, the block matrix L:=2 64T11:::T1n ... Tn1::: Tnn3 75 is the graph-Laplacian matrix. Given a weighted graph its components L= [lij]i;j2f1;:::;ng are given by lij=Tij ifi6=j;P h=1;h6=iTihifi=j:(5) Note that given a Laplacian matrix, its row-sums are zero, its diagonal entries are non- negative, and its non-diagonal entries are nonpositive. The above set of equations can be rewritten in compact form as follows _X1 _X2 =" Diag Di Mi Diag 1 Mi L 0# |{z } AX1 X2 : (6) We also recall that L= [lij]i;j2f1;:::;ngwhere for an unweighted and undirected graph we have lij=8 < :1 if (i;j) is an edge and not self-loop ; d(i) ifi=j; 0 otherwise.(7) We are ready to establish the next result. Let us denote by spanf1g=f2Rn: 92Rs:t: =1g. Furthermore, let the following consensus set be dened as C=f2Rn:2spanf1g;min jxj(0)max jxj(0)g: Theorem 2 Let a network of homogeneous micro-grids be given, and set Di=Dfor all i. LetMi= 1 for alli, andTij= 1 for any (i;j)2E. Then dynamics (6) describes a consensus dynamics, i.e., lim t!1Xi(t) =x i2C; i = 1;2: Furthermore, let D >p4ithen the consensus value vector (x 1;x 2)Tis an asymptot- ically stable node. Vice versa, if D <p4ithen (x 1;x 2)Tis an asymptotically stable spiral.Proof. Let us start by nding the roots of det(IA). To this purpose, we recall that for any generic block matrix it holds det(A B C D ) =det(DABC);ifBD=DB. (8) Then, from the above we have det(IA) =det" I+Diag Di Mi Diag 1 Mi L  I# =det(2I+I
Diag (Di Mi) +Diag (1 Mi)
L):(9) Under the homogeneity assumption Di=Dfor alli, we have det 2I+I
Diag Di Mi +Diag 1 Mi
L =det (2+D)I+L =Qn i=1 (2+D)i ;(10) whereiis theith eigenvalue ofL. The roots of (10) can be obtained by solving 2+Di= 0 from which we have + i=D+p D2+4i 2;  i=Dp D2+4i 2: (11) From the above, after noting that the real part of the eigenvalues is negative, we can conclude that system (6) is asymptotically stable. Remark 1 The result stated in the above theorem applies also to the case where the micro- grids have dierent inertia but the same ratio D=Di Mifor alli2V. In this case we need to consider the Laplacian matrix of the corresponding weighted graph ~L=Diag (1 Mi)Land the associated eigenvalues. We next recall some properties of the Laplacian spectrum and use such properties to investigate the insurgence of topology-induced oscillations. The maximal eigenvalue ~ n of a symmetric Laplacian matrix L=LTinRnnsatises the following lower and upper bounds which are degree-dependent: dmax~n2dmax; (12) where the maximum degree is dmax = maxi21;:::;ndi[4, Chapter 6]. We also observe that the eigenvalues appearing in (11) refer to the negative Laplacian, and therefore we have i=~ifor every eigenvalue iof the negative Laplacian Land ~iof the Laplacian L. Corollary 1 The following properties hold:All eivenvalues + i; ifori= 1;:::;n are real and negative if Dp8dmax; There exists at least one complex eigenvalue/eigenmode if Dp4dmax: Corollary 2 Given a chain topology of n3nodes, for which dmax = 2 the following properties hold: All eivenvalues + i; ifori= 1;:::;n are real and negative if D4; There exists at least one complex eigenvalue/eigenmode if D2p 2: 3.1 Example of two interconnected micro-grids In this section, we specialize the above results to the case of two interconnected micro- grids. The interconnection topology is a chain one with two nodes, and the maximal degree isdmax = 1. A graph representation is displayed in Fig. 4. + eji Tij s1 Mjs+Djfj Pj fj+ eij Tij s1 Mis+Difi Pi Figure 4: Block representation of two interconnected micro-grids. Dynamics (6) can be rewritten as 2 6664_x(i) 1 _x(j) 1 _x(i) 2 _x(j) 23 7775=2 664D1 M101 M10 0D2 M201 M2 T11T12 0 0 T12T110 03 7752 6664x(i) 1 x(j) 1 x(i) 2 x(j) 23 7775: (13) The Laplacian of the weighted graph is given by ~L=Diag (1 Mi)L=1 M10 01 M211 1 1 : (14) AssumingD=D1 M1=D2 M2, from Corollary 1 we infer that All eivenvalues + i; ifori= 1;:::;n are real and negative if Dp 8; There exists at least one complex eigenvalue/eigenmode if D2: In other words, if the ratio between the damping coecient and the inertia of each micro- grid is greater thanp 8 then we certainly have an overdamped dynamics, and observe no overshoots and no oscillations. Dierently, if the ratio between the damping coecient and the inertia of each micro-grid is less than 2 then we certainly have an underdamped dynamics, and observe overshoots and oscillations.4 Absolute stability In this section we extend the analysis to the case where both frequency and power ow measurements are subject to disturbances. Using a traditional technique in nonlinear analysis and control we isolate the nonlinearities in the feedback loop, and analyze stability under some mild assumptions on the nonlinear parameters. Likewise in the previous section we consider two interconnected micro-grids, and as- sume that each micro-grid can be described in terms of power ow Piand frequency fi. Assuming disturbed measurements on fi, the evolution of the power ow is given by _Pi=Tij(fj (fi)) =Tij~eij; (15) whereTijis the synchronizing coecient as in the previous section and where the new term (:) is a sector nonlinearity satisfying the following assumption. Assumption 1 Function (fi;t)is time-varying and satises the sector condition 0 (fi)~kfi: Now, the power Pidepends on a disturbed measure of the frequency error ~ eij:=fj (fi). The dynamics for fistill follows a traditional swing equation, which now involves disturbed measurements of the frequency (fi) and of the power ow (Pi): _fi=Di Mi (fi) + (Pi) Mi+!: (16) In the above model, MiandDiare the inertia and damping constants of the ith micro-grid, respectively. Similarly to the previous section, we denote fi=x(i) 1,Pi=x(i) 2,fj=x(j) 1, and by considering fjas an exogenous input to micro-grid i, the dynamics of micro-grid ireduces to the following second-order system _x=" _x(i) 1 _x(i) 2# =Di Mi1 Mi Tij0 |{z} A" (x(i) 1) (x(i) 2)# +1 0 0Tij |{z} B! x(j) i :(17) The block system of the ith micro-grid, which admits the state space representation (17), is displayed in Fig. 5. Building on the Kalman-Yakubovich-Popov lemma, absolute stability is linked to strictly positive realness of Z(s) =I+KG(s) whereK=~kIandG(s) is the transfer function of linear part of system (17) which is obtained as G(s) =CT[sIA]1Bwhere we set A=Di Mi1 Mi Tij0 ; B =1 0 0Tij ; C =1 0 0 1 : We recall from Theorem 1 that matrix Ais Hurwitz.fj+ eTij s1 Mis+Difi (fi;t)Pi! Figure 5: Block system representing micro-grid i. The idea now is to isolate the nonlinearities in the feedback loop and introduce a new variable for them, say . Let us rst obtain the transfer function associated to the dynamical system (17): G(s) =CT[sIA]1B =1 s(s+Di Mi)+Tij Mis1 Mi Tijs+Di Mi
1 0 0Tij =1 s(s+Di Mi)+Tij Mi" sTij Mi Tij(s+Di Mi)Tij# =1
(sIA)" sTij Mi Tij(s+Di Mi)Tij# ;(18)where
(sIA) =s(s+Di Mi) +Tij Mi. Then, for Z(s) we obtain Z(s) =I+KG(s) =1 0 0 1 +k s(s+Di Mi)+Tij Mi" sTij Mi Tij(s+Di Mi)Tij# =1 0 0 1 +k
(sIA)" sTij Mi Tij(s+Di Mi)Tij# =1
(sIA)" ks+
(sIA) kTij Mi kTijk(s+Di Mi)Tij+
(sIA)# =1 s(s+Di Mi)+Tij Mi
 ks+s(s+Di Mi) +Tij MikTij Mi kTij k(s+Di Mi)Tij+s(s+Di Mi) +Tij Mi =1 s(s+Di Mi)+Tij Mi
 s2+ (Di Mi+k)s+Tij MikTij Mi kTij s2+ (Di Mi+kTij)s+Tij Mi+kTijDi Mi : Note that matrix Ais Hurwitz. This implies that also Z(s) is Hurwitz as the poles of Z(s) coincide with the eigenvalues of A. We use this in the proof of absolute stability of the dynamical system (17) established next. Theorem 3 Let the dynamical system (17) be given where Ais Hurwitz. Furthermore, let us consider the sector nonlinearities as in Assumption 1. Then, Z(s)is strictly positive real and system (17) is absolutely stable. Proof. We rst prove that Z(s) is strictly positive real. For this to be true, the following conditions must hold true: Z(s) is Hurwitz, namely the poles of all entries of the matrix Z(s) have negative real parts; Z(j!) +Z(j!)>0;8!2R; Z(1) +ZT(1)>0. For the rst condition note that Z(s) is Hurwitz as its poles are the roots of s(s+ Di Mi) +Tij Mi= 0, which coincide with the values obtained in (4) and which we rewrite here for convenience: 1;2=1 2 Di Miq (Di Mi)24Tij Mi . As for the second condition,Z(j!) +Z(j!)>0;8!2R, let us obtain for Z(j!) andZ(j!) the following expressions: Z(j!) =1 Tij Mi!2+Di Mij!
Tij Mi!2+ (Di Mi+k)j! kTij Mi kTijTij Mi+kTijDi Mi!2+ (Di Mi+kTij)j! : Z(j!) =1 Tij Mi!2Di Mij!
Tij Mi!2(Di Mi+k)j! kTij Mi kTijTij Mi+kTijDi Mi!2(Di Mi+kTij)j! ; By combining the expressions above for Z(j!) andZ(j!) we then obtain Z(j!) +Z(j!) =1
(j!IA)
(j!IA)
(j!IA)
Tij Mi!2+ (Di Mi+k)j! kTij Mi kTijTij Mi+kTijDi Mi!2+ (Di Mi+kTij)j! +
(j!IA)
Tij Mi!2(Di Mi+k)j! kTij Mi kTijTij Mi+kTijDi Mi!2(Di Mi+kTij)j! =1 Tij Mi!22  Di Mij!2z11z12 z21z22 ;(19) where we set z11andz22as follows: z11= 2[!42!2Tij Mi+!2Di Mi(Di Mi+k) + (Tij Mi)2] = 2[(!2Tij Mi)2+!2Di Mi(Di Mi+k)]; z22= 2[!4!2(Tij Mi+kTijDi Mi) +!2Di Mi(Di Mi+kTij) !2Tij Mi+Tij Mi(Tij Mi+kTijDi Mi) +!2Di Mi(Di Mi+kTij) = 2[(!2Tij Mi)2!2kTijDi Mi +Tij MikTijDi Mi+!2Di Mi(Di Mi+kTij)] = 2[(!2Tij Mi)2+!2(Di Mi)2+Tij MikTijDi Mi]:From the above equation we then have Z(j!) +Z(j!) =1 Tij Mi!22  Di Mij!2
 2[(!2Tij Mi)2+!2Di Mi(Di Mi+k)] z21 z12 2[(!2Tij Mi)2+!2(Di Mi)2+Tij MikTijDi Mi] >0;for all!: The last inequality follows from the trace of the above matrix being positive. To see this note that z11+z22 = 2[(!2Tij Mi)2+!2Di Mi(Di Mi+k)] +2[(!2Tij Mi)2+!2(Di Mi)2+Tij MikTijDi Mi]>0:(20) As for the third condition, namely Z(1) +ZT(1)>0, we have that lim!!1z12= lim!!1z21= 0; lim!!1z11= lim!!1z22= 2:(21) Then we obtain that Z(1) +ZT(1) = 2I>0. We can conclude that also the third condition is veried. Now we wish to show that there exists a Lyapunov function V(x) =xTx, where  = [ij]2R22is symmetric. After dierentiation with respect to time and using (17) we obtain_V(t;x) = _xTx+xTx =xTATx+xTAx TBTxxTB = [x1x2]Di Mi1 Mi Tij0T1112 2122x1 x2 +[x1x2]1112 2122Di Mi1 Mi Tij0x1 x2 [ 1 2]1 0 0Tij1112 2122x1 x2 [x1x2]1112 21221 0 0Tij B 1 2 ; where we denote (t;y) = [ 1 2]T. From Assumption 1 and the property of rst and third sector nonlinearities we have 2 T( Ky)0. Furthermore, from symmetry ofmatricesPandK=~kI, the time derivative of the candidate Lyapunov function can be rewritten as _V(t;x)xT(AT +PA)x2xTB 2 T( Ky) =xT(AT + A)x2xTB + 2 TKCx2 T =xT(AT + A)x+ 2xT(CTKB) 2 T = [x1x2] Di Mi1 Mi Tij0T1112 2122 +1112 2122Di Mi1 Mi Tij0x1 x2 +2[x1x2]k0 0k 1112 21221 0 0Tij 1 2 2[ 1 2] 1 2 : The right-hand side of the above inequality is negative if there exist matrices  2R22 and a positive scalar such that AT + A=T; B=CTKp 2T;(22) or in explicit form Di Mi1 Mi Tij0T1112 2122 +1112 2122Di Mi1 Mi Tij0 =1112 2122T1112 2122 1112 2122 ; 1112 21221 0 0Tij =k0 0k p 21121 1222 ; By introducing the solutions of the above in terms of ,  and , the time derivativeof the candidate Lyapunov function can be rewritten as _V(t;x)xTxxTTx+ 2p 2xTT 2 T =xTx[xp 2 ]T[xp 2 ] xTx [x1x2]1121 1222x1 x2 : It is well known that from the Kalman-Yakubovich-Popov lemma, there exist solutions in terms of , , and satisfying the above set of matrix equalities, as the transfer function Z(s) is positive real and this concludes our proof. Remark 2 The above theorem has been obtained under the hypothesis that both frequency and power ow measurements are subject to disturbances. The same result extend straight- forwardly also to the case where the model parameters Tij,MiandDiare uncertain. 5 Simulations This section provides simulation studies to corroborate the theoretical results developed in the previous sections. The analysis is based on open source data relating to a part of the Nigerian grid obtained from [1]. The data set shows the one-line diagram of part of the distribution network including the geographical location of generators and load buses. Figure 6 displays the one-line diagram with the geographical names. Yobe Borno Adamawa TarabaGombeBauchiPlateauKadunaKanoJigawa Katsina Figure 6: One-line diagram of part of Nigerian grid [1]. From the one-line diagram we obtain the graph representation showing the intercon- nection between bus loads as in Fig. 7. The graph is characterized by 11 nodes and 10 arcs. Most nodes have degree 1 or 2 except for Gombe, and Kano which have degree 4, and 3, respectively. The graph is undirected, i.e. the in uence of smart-grid ionjis bidirectional.YobeBorno Adamawa TarabaGombeBauchiPlateauKadunaKanoJigawa Katsina Figure 7: Graph representation of part of Nigerian grid [1]. The numerical studies involve two sets of simulations. The rst set of simulations has been conducted considering the following normalized parameters: number of smart-grids n= 11, damping constant D= 1;3;6 for three consecutive runs of simulations; Inertial constantM= 1; Synchronizing coecient T= 1; Horizon window involving N= 500 iterations; Step size dt=:01. The parameters and the dynamics is normalized in an interval [0;1]. For instance the initial state of each grid is a randomized bidimensional vector in the interval [0 ;1]. To simulate periodic disturbances, the initial state is reini- tialized every 10 sec. To obtain realistic plots we rescale the state variable around 50 Hz for the frequency and 30 MWh for the power ow. Figure 8 displays the evolution of the frequency of each smart-grid. Frequencies are measured in Hz and are centered around 50 Hz which is the nominal value. Oscillations remain within 1% of the nominal value, i.e., in the interval [49 :95;50:05]. From top to bottom we consider an increasing damping constantD= 1;3;6 which re ects in damped oscillations and smaller time constants. Figure 9 displays the evolution of the power ows in each smart-grid. Power ows are measured in MWh and are centered around the nominal value of 30 MWh. From the plots we observe that oscillations remain within 3 :3% of the nominal value, i.e., in the interval [29:00;31:00] MWh. From top to bottom the damping constant is increasing and equal toD= 1;3;6. Note that the maximal degree of the network is dmax = 4 and therefore forD= 1;3 we have D <p4dmax = 4 and oscillations emerge as we have complex eigenvalues for + i; i;forA. Unlikewise for D= 6 it holds D >p8dmax =p 32 and therefore no oscillations and no complex eigenvalues emerge. The maximal eigenvalue of the Laplacian has been obtained as ~ n= 5:1748. In a second set of simulations, we isolate one smart-grid from the rest of the power network and investigate the transient response under disturbances in the measurement of frequency and power. Such disturbances are modeled using the paradigm developed in Section 4. In particular we consider a rst and third quadrant nonlinearity in the feedback loop. The function is periodic and we take for it the expression (t) = 1 + sin(ft) in [0;2], wherefis the frequency, tis time, and is a factor increasing the periodicity of the oscillation. For the second set of simulations we consider the following normalized parameters: number of smart-grids n= 1, damping constant D= 1; Inertial constantM= 1; Synchronizing coecient T= 1; periodicity factor = 1;5;10 for three consecutive runs of simulations; Horizon window involving N= 1000 iterations; Step sizeFigure 8: Time series of smart-grids frequencies in Hz. dt=:01; The initial state of each grid is a randomized bidimensional vector in the interval [0;1]. Both variables are rescaled around 50 Hz for the frequency and 30 MWh for the power ow. Figure 10 displays the time evolution of the frequency of each smart-grid (left) and power ow (right). As in the previous simulation example, frequencies are measured in Hz and are centered around 50 Hz which is the nominal value. We observe that oscillations remain within 1% of the nominal value, i.e., in the interval [49 :95;50:05]. From top to bottom the damping constant is D= 1;3;5 and this implies a higher damping, smaller time constants, and faster convergence. Power ows are measured in MWh and are centered around the nominal value of 30 MWh. The plots show that oscillations remain within 3 :3% of the nominal value, i.e., in the interval [29 :00;31:00] MWh. From top to bottom the damping constant is increasing and equal to D= 1;3;5. 6 Discussion and conclusions For single and multiple interconnected micro-grids, we have studied transient stability, namely the capability of the micro-grids to remain in synchronism even under cyber- attacks or model uncertainties. First we have showed that transient dynamics can be robustly classied depending on specic intervals for the micro-grid parameters, such asFigure 9: Time series of smart-grids power ows in MWh. synchronization, inertia, and damping parameters. We have then turned to study the analogies with consensus dynamics. We have obtained bounds on the damping coecient which determine wether the network dynamics is underdamped or overdamped. Such a result is meaningful as in the case of underdamped dynamics we observe oscillation around the consensus value, whereas in the case of underdamped dynamics we observe a deviation of the consensus value from the nominal mains frequency. The bounds are linked to the connectivity of the network. We have also extended the stability analysis to the case of disturbed measurements due to hackering or parameter uncertainties. Using traditional nonlinear analysis and the Kalman-Yakubovich-Popov lemma we have rst isolated the nonlinear terms in the feedback loop and have showed that nonlinearities do not compromise the stability of the system. There are three key directions for future work. First we wish to relax constraints on the nature of the disturbances. Indeed here we have assumed that such disturbances can be modeled using rst and third quadrant nonlinearities. Such nonlinearities tends to vanish around the equilibrium points. In reality, disturbances due to hackering can impact the systems even at the equilibrium, thus leading to synchronization deciency. A second direction involves the analysis of the impact of stochastic disturbances on the transient stability. Concepts like stochastic stability, stability of moments, and almost sure stability will be used to classify the resulting stochastic transient dynamics. Finally,Figure 10: Time series of smart-grids power ows in MWh. a third direction involves the extension to the case of a single or multiple heterogeneous populations of micro-grids. In this context we will try to gain a better insights on scal- ability properties and emergent behaviors. The latter is a terminology used in complex network theory to address macroscopic phenomena arising from microscopic behavioral patterns. References [1] F. K. Ariyo, M. O. Omoigui. Investigation of Nigerian 330 kV Electrical Network with Distributed Generation Penetration { Part I: Basic Analyses. Electrical and Electronic Engineering , Scientic & Academic Publishing, 3(2), 49{71, 2013. [2] F. Bagagiolo, D. Bauso. Mean-eld games and dynamic demand management in power grids. Dynamic Games and Applications , 4(2), 155{176, 2014. [3] D. Bauso. Dynamic demand and mean-eld games, IEEE Transactions on Automatic Controls , in press 10.1109/TAC.2017.2705911[4] F. Bullo. Lectures on Network Systems , Version 0.95, 2017, http://motion.me.ucsb.edu/book-lns. [5] F. D or er, F. Bullo. Synchronization and Transient Stability in Power Networks and Nonuniform Kuramoto Oscillators. SIAM Journal on Control Optimization , 50(3), 1616{1642, 2012. [6] T. Namerikawa, N. Okubo, R. Sato, Y. Okawa, M. Ono. Real-Time Pricing Mecha- nism for Electricity Market With Built-In Incentive for Participation. IEEE Trans- actions on Smart Grid , 6(6), 2714{2724, 2015. [7] R. Olfati-Saber, J. A. Fax, R. M. Murray. Consensus and Cooperation in Networked Multi-Agent Systems. Proceedings of the IEEE , vol. 95, no. 1, pp. 215{233, 2007. [8] H. Yin, P. G. Mehta, S. P. Meyn, U. V. Shanbhag, Synchronization of Coupled Oscillators is a Game. IEEE Transactions on Automatic Control, 57(4) (2012) 920{ 935. [9] H. Yin, P. G. Mehta, S. P. Meyn, U. V. Shanbhag. On the Eciency of Equilibria in Mean-Field Oscillator Games, Dynamic Games and Applications, 4(2) (2014) 177{ 207.

2017-08-24

This paper deals with transient stability in interconnected micro-grids. The main contribution involves i) robust classification of transient dynamics for different intervals of the micro-grid parameters (synchronization, inertia, and damping); ii) exploration of the analogies with consensus dynamics and bounds on the damping coefficient separating underdamped and overdamped dynamics iii) the extension to the case of disturbed measurements due to hackering or parameter uncertainties.

Nonlinear network dynamics for interconnected micro-grids

1708.07296v1

Draft version March 21, 2023 Typeset using L ATEX default style in AASTeX631 Nonlinear Damping and Field-Aligned Flows of Propagating Shear Alfv en Waves with Braginskii Viscosity Alexander J. B. Russell1 1School of Science and Engineering, University of Dundee, Dundee, DD1 4HN, Scotland, UK ABSTRACT Braginskii MHD provides a more accurate description of many plasma environments than classical MHD since it actively treats the stress tensor using a closure derived from physical principles. Stress tensor eects nonetheless remain relatively unexplored for solar MHD phenomena, especially in nonlin- ear regimes. This paper analytically examines nonlinear damping and longitudinal ows of propagating shear Alfv en waves. Most previous studies of MHD waves in Braginskii MHD considered the strict linear limit of vanishing wave perturbations. We show that those former linear results only apply to Alfv en wave amplitudes in the corona that are so small as to be of little interest, typically a wave energy less than 1011times the energy of the background magnetic eld. For observed wave ampli- tudes, the Braginskii viscous dissipation of coronal Alfv en waves is nonlinear and a factor around 109 stronger than predicted by the linear theory. Furthermore, the dominant damping occurs through the parallel viscosity coecient 0, rather than the perpendicular viscosity coecient 2in the linearized solution. This paper develops the nonlinear theory, showing that the wave energy density decays with an envelope (1+ z=Ld)1. The damping length Ldexhibits an optimal damping solution, beyond which greater viscosity leads to lower dissipation as the viscous forces self-organise the longitudinal ow to suppress damping. Although the nonlinear damping greatly exceeds the linear damping, it remains negligible for many coronal applications. Keywords: Alfv en waves (23), Solar corona (1483), Solar coronal heating (1989), Solar coronal holes (1484), Solar wind (1534), Magnetohydrodynamics (1964), Space plasmas (1544), Plasma astrophysics (1261), Plasma physics (2089) 1.INTRODUCTION Alfv enic waves are a ubiquitous feature of natural plasmas, including the solar corona (Tomczyk et al. 2007; De Pontieu et al. 2007; Lin et al. 2007; Okamoto et al. 2007) and solar wind (Coleman 1967; Belcher & Davis 1971). In solar physics, these waves contain sucient energy to heat the open corona and accelerate the fast solar wind (McIntosh et al. 2011), and they damp signicantly within a solar radius above the surface (Bemporad & Abbo 2012; Hahn et al. 2012; Hahn & Savin 2013; Hahn et al. 2022). How these Alfv enic waves damp in astrophysical and space plasmas is an important question that has remained open for almost a century (see early papers by Alfv en 1947 and Osterbrock 1961; modern reviews by De Moortel & Browning 2015, Arregui 2015 and Van Doorsselaere et al. 2020; and historical perspectives by Russell 2018 and De Moortel et al. 2020). Most theoretical knowledge about solar Alfv enic waves is based on \classical" magnetohydrodynamics (MHD), a mathematical framework that originated from intuitive coupling of Maxwell's equations and Euler equations of inviscid hydrodynamics (Hartmann 1937; Alfv en 1942, 1943, 1950; Batchelor 1950) and became widely adopted in large part due to its success providing insight into diverse natural phenomena (see e.g. Priest 2014). However, classical MHD is Corresponding author: Alexander J. B. Russell a.u.russell@dundee.ac.ukarXiv:2303.11128v1 [astro-ph.SR] 20 Mar 20232 Russell Start: General MHD with Braginskii πSet geometryLinearise in , includes setting in .b/B0h=̂zπ removed. Dissipation by only.η0η2Full calculation of QRatio of heating terms: Q0/Q2Linearise in , setting . (ωiτi)−2η1,η2≪η0Dissipation by . Nonlinear in .η0b/B0 Vanishing waves (bB0)2≪(ωiτi)−2Coronal regime (bB0)2≫(ωiτi)−2 Figure 1. Schematic paths of reasoning. The vertical branch gives priority to smallness of the wave amplitude and concludes that damping is a linear process governed by the perpendicular viscosity coecient 2, e.g. §8 of Braginskii (1965). The horizontal branch gives priority to the smallness of 2=0, leading to nonlinear damping via 0. This paper follows the diagonal branch, which includes deriving the validity condition for the two outcomes. Nonlinear damping via 0is appropriate for most coronal applications. one member of a larger family of plasma descriptions, some of which oer a more complete description of the plasma. This paper analytically examines Alfv en wave damping in the more general framework of Braginskii MHD, which unlike classical MHD, retains the anisotropic viscous stress tensor. A number of authors, including §8 of Braginskii (1965), have previously investigated viscous damping of Alfv en waves in the linear limit of vanishingly small wave amplitude. When priority is given to smallness of the wave amplitude, the problem becomes framed as a matter of how anisotropic viscosity aects velocities that are perpendicular to the magnetic eld (the direction of which is treated as unchanging). With this approximation, damping is determined by the \perpendicular" viscosity coecient 2, which is extremely small in the corona. It was thus originally concluded that viscous damping is very weak for coronal Alfv en waves unless they have very short wavelengths. This path of reasoning is shown as the vertical branch in Fig. 1. There is, however, another way to view the problem. Viscous damping of Alfv en waves can alternatively be considered with priority given to the largeness of parallel viscosity coecient 0. Given that 2=0&1011is typical in the corona (Hollweg 1985), even a very small component of vparallel to the total magnetic eld Bwould be expected to produce major departures from linear theory. This path of reasoning is shown as the horizontal branch in Fig. 1. The second viewpoint of the problem takes impetus from the observation that (unless wave amplitudes vanish entirely) Alfv en waves do have a non-zero velocity component parallel to the total magnetic eld. Two eects contribute to this, which are separated if one expands V
B=V
(b+B0), where B0is the equilibrium magnetic eld and bis the magnetic perturbation. First, V
bis non-zero for an Alfv en wave, since the velocity perturbation perpendicular toB0is aligned with the magnetic eld perturbation b. In other words, de ection of the magnetic eld from its equilibrium direction implies there is a non-zero velocity component parallel to the total magnetic eld. Second, in compressible plasma, the magnetic pressure of the magnetic eld perturbation drives a nonlinear ponderomotive ow parallel to the equilibrium magnetic eld (e.g. Hollweg 1971). The ponderomotive ow makes V
B0nonzero as well.Nonlinear Alfv en Waves with Viscosity Tensor 3 Both of these eects allow for the possibility of nonlinear viscous damping via the large parallel viscosity coecient 0. With the benet of modern observations (e.g. McIntosh et al. 2011; Morton et al. 2015), it is known that normalised wave amplitudes b=B 0V=vA0:1 are typical for the base of an open coronal eld region, for example. The \smallness" of the square of this ratio is very modest in comparison to the extreme largeness of 0=2. Thus, it is likely from the outset that viscous damping of Alfv en waves will be a nonlinear process governed by 0and the wave amplitude. This paper provides mathematical evidence that this heuristic analysis holds true, along with detailed examination of the consequences. Various previous studies have explored eects of Braginskii viscosity on MHD waves since Braginskii (1965). In solar physics, the eect of linearized Braginskii viscosity was revisited from the 1980s to the mid-1990s through the lens of phase mixing and resonant absorption, with the aim of determining how including the viscosity tensor modies these scale-shortening processes and their heating properties. At the time, it was common practice in solar MHD wave theory to work with linearized equations. Thus, due to linearization, Steinolfson et al. (1986), Hollweg (1987), Ruderman (1991), Ofman et al. (1994) and Erdelyi & Goossens (1995) obtained analytical and numerical results that strictly apply to Alfv en waves of vanishing amplitude. In adjacent elds, the eect of anisotropic viscosity on MHD waves has also been investigated with an eye on MHD turbulence and the solar wind. Of particular note, Montgomery (1992) advocated that Braginskii viscosity is important in hot tenuous plasmas, that in many circumstances it should be treated using parallel ion viscosity, and that plasma motions may self-organise to suppress damping. He further applied these ideas to anisotropy in MHD turbulence, on the basis that a quasi-steady turbulence is composed of the undamped modes. Quantitative elaboration in Montgomery (1992) was based on a linear normal mode analysis, which captures linear damping of magnetoacoustic waves by parallel viscosity, but excludes nonlinear viscous damping of Alfv en waves. The conclusion that a linearized stress tensor damps Alfv en waves only negligibly, while damping magnetoacoustic waves signicantly, was further reinforced by related work by Oughton (1996, 1997). Similar ideas to ours regarding the importance of nonlinearity were advocated by Nocera et al. (1986), who modelled Alfv en waves subject to the 0part of the Braginskii viscous stress tensor, retaining the leading-order nonlinear terms in the wave perturbations. Consistent with the argument above, their calculations found that coronal Alfv en waves damp nonlinearly by parallel viscosity. The current paper complements and extends the previous analysis by Nocera et al. (1986), with the goal of producing a comprehensive understanding of the nonlinear damping and eld-aligned ows of propagating shear Alfv en waves with Braginskii viscosity. A limitation of the mathematical techniques used in this paper is that they exclude certain other nonlinear eects that may be important in plasmas, such as nonlinear interactions between waves. Numerical investigations will be required in future to verify the analytical theory presented here, compare the relative importance of viscous damping and other nonlinear eects such as parametric decay instability, and consider interactions between nonlinear processes in Braginskii MHD. This paper is organised as follows. §2 provides scientic background on single- uid Braginskii MHD and its relationship to other single- uid plasma models. §3 quantitatively examines Alfv en wave heating by the full Braginskii viscous stress tensor, demonstrating the importance of nonlinear 0terms and compressibility, and obtaining the wave decay properties for the weakly viscous limit using energy principles. In §4, we argue that in highly viscous limit, viscous heating is suppressed by self-organisation of the ponderomotive ow, which implies that viscosity strongly alters the eld-aligned ow associated with Alfv en waves in this regime. §5 further strengthens the analysis, using multiple scale analysis to obtain the decay properties without restrictions on the Alfv enic Reynolds number, assuming the framework of Braginskii MHD. The paper nishes with discussion in §6 and summary of main conclusions in §7. 2.BRAGINSKII MHD Braginskii MHD is an important plasma description that treats anisotropic viscosity and thermal conduction using rigorous closure from physical principles. This section provides a short primer on single- uid Braginksii MHD, its connection with pressure (or temperature) anisotropy, and its relation to classical MHD and the CGL double-adiabatic equations. As is described in various plasma textbooks, uid variables can be rigorously and robustly dened as velocity moments of the underlying particle distribution functions. Transport equations for each particle species are then4 Russell derived by taking moments of the kinetic Boltzmann equation, and combined to obtain the single uid equations. Recommended presentations can be found in Schunk & Nagy (2009) Chapter 7 and the Appendix of Spitzer (1962). Assuming quasi-neutrality, and conservation of mass, momentum and energy, this process yields the mass continuity equation, @ @t+r
(V) = 0; (1) momentum equation DV Dt=r
P+G+jB; (2) energy equation, D Dt3 2p +5 2pr
V=:rVr
q+j
(E+VB); (3) higher-order transport equations if required, and the generalized Ohm's law. The pressure tensor Pthat appears in Eq. (2) is the most fundamental representation of the internal forces associated with thermal motions of particles. It is symmetric, so it represents six degree of freedom. The momentum equation can also be reformulated by introducing the scalar pressure and stress tensor as p=1 3Trace ( P) =1 3P;  =Pp; (4) whereis the Kronecker delta. So dened, the stress tensor is symmetric and traceless. These denitions gives the replacementr
P=rpr
. Deriving transport equations by moment taking meets with a fundamental closure problem: the transport equation for each uid variable depends on a higher-order variable, producing an innite regress unless the system can be closed by other considerations. The method of closure is therefore a major distinguishing feature between dierent uid models for plasmas. It is also a major source of validity caveats. Various dierent methods of closure produce governing equations that conserve mass, momentum and energy, since these properties are already built into Eqs. (1){ (3). However, the dierent models discussed below disagree on the internal forces and heating, and can therefore produce dierent behaviors. Classical MHD (Hartmann 1937; Alfv en 1942, 1943; Batchelor 1950) corresponds to a closure treatment in which the stress tensor and the heat ow vector are dropped from Eqs. (2) and (3). Dropping the stress tensor can be justied when particle collisions or other forms of particle scattering such as wave-particle interactions are frequent enough that the pressure tensor remains very close to isotropic. The resulting MHD equations are valid for many situations, for instance modelling static equilibria, or dynamic situations in which the divergence of the stress tensor remains small compared to the Lorentz force. It is nonetheless a truncation since higher order variables are set to zero rather than approximated. Furthermore, collisionality in environments such as the solar corona is low enough that the stress tensor can become signicant for various dynamic phenomena, including MHD waves. Braginskii MHD uses a less restrictive method of closure. As is detailed by Braginskii (1965), when the collisional mean free path is signicantly shorter than length scales over which uid quantities vary, the heat ow vector takes the form of an anisotropic thermal conduction, and the stress tensor takes the form of an anisotropic viscosity. Closure can therefore be achieved by expressing qandin terms of lower-order uid variables, which are traditionally derived using methods similar to Chapman & Cowling (1939) or Grad (1949). The anisotropy inherent in qandcan be appreciated heuristically, by considering the helical motion of charged particles in magnetized plasmas. The mean free path parallel to the magnetic eld is the same as for unmagnetized plasmas, implying that transport parallel to the magnetic eld is the same as for unmagnetized plasmas. Meanwhile, the mean free path perpendicular to the magnetic eld is the gyroradius, which is typically much less than the mean free path parallel to the magnetic eld, which supresses perpendicular transport. Hence both thermal conduction and viscous stresses are anisotropic with respect to the magnetic eld direction, often extremely so. The full Braginskii stress tensor, used in §3, involves ve viscosity coecients. A useful simplication, used in §5, is that for strong magnetizations, ii1, the parallel 0coecient greatly exceeds the other viscosity coecients. Hence, one can often simplify by neglecting the smaller coecients (although, as shown in §3 it can be necessary to retain other viscosity coecients if length scales are highly anisotropic). In this simplication, one has the followingNonlinear Alfv en Waves with Viscosity Tensor 5 covariant expressions for parallel viscosity (Lifshitz & Pitaevskii 1981; Hollweg 1986): =30 hh 3 hh 3 @V; Qvisc= 30 hh 3 @V2 ;(5) where h=B=jBjis the unit vector in the direction of the magnetic eld. These expressions are dierent to the isotropic viscosity that appears in the Navier-Stokes equations, owing to the anisotropy introduced by the magnetic eld. Parallel viscosity is closely related to pressure anisotropy. As pointed out by Chew et al. (1956), when ii1 the particle Lorentz force makes the pressure tensor gyrotropic, giving it the form P=p?+ (pjjp?)hh: (6) This is a signicant simplication, since the six degrees of freedom of a general pressure tensor have been replaced with two variables, pjjandp?. The denitions in Eqs. (4) then yield p= (pjj+ 2p?)=3 and = (pjjp?) hh 3 : (7) Equation (7) shows that pressure anisotropy has an equivalent stress tensor, which is proportional to pjjp?. Furthermore, Eqs. (5) and (7) both have the form (hh=3), so equivalence of the stress tensors reduces to equivalence of the scalar factors in the two equations. An illuminating analysis of the conditions under which they converge has been written by Hollweg (1985, 1986), the most important condition being that collisions (or other processes such wave-particle interactions) relax the pressure anisotropy driven by velocity gradients to an extent that the pressure is only weakly anisotropic. Classical MHD, for comparison, assumes that pressure anisotropy can be neglected altogether. For low collisionality, the quasistatic approximation in Braginskii MHD ceases to be valid and strong pressure anisotropy may develop. Under these conditions, separate evolution equations can be derived for pjjandp?(Chew et al. 1956; Hollweg 1986). However, the closure problem rears its head again, because those equations depend on the heat ow vector. A simple approach to obtaining a closed system is to ignore the heat ow vector, thus obtaining the CGL double adiabatic equations (Chew et al. 1956), which are commonly used for collisionless plasma. More sophisticated approaches also exist that solve for the evolution of the pressure anisotropy or the evolution of the stress tensor, retaining the heat ow vector and closing by other means. The works by Balescu (1988); Schunk & Nagy (2009); Zank (2014); Hunana et al. (2019a,b, 2022) provide further reading on this topic. Summarising, there exists a family of adjacent (sometimes overlapping) single- uid models for plasmas. The most appropriate choice for a particular problem and/or context depends on the collisionality. When MHD timescales are greater than the ion collision time, Braginskii MHD provides rigorous closure and treats the internal forces and heat ow more accurately than classical MHD. 3.ALFV EN WAVE HEATING BY BRAGINSKII VISCOSITY 3.1. Model We quantitatively examine the viscous dissipation for an Alfv en wave, which is a transverse wave polarized so that the magnetic perturbation is perpendicular to the equilibrium magnetic eld and the wavevector. Setting the equilibrium magnetic eld in the z-direction, the magnetic perturbation in the x-direction and the wavevector in the yz-plane, we consider a total magnetic eld of the form B=b(y;z;t )ex+B0ez: (8) This ansatz automatically satises r
B= 0. For the velocity eld we assume the form V=Vx(y;z;t )ex+Vz(y;z;t )ez: (9)6 Russell TheVxis the dominant velocity component. In linearized theory it would be the only component of V. Additionally, we have explicitly included a higher-order Vzterm that represents the nonlinear ponderomotive ow parallel to the equilibrium magnetic eld, which is driven by gradients of the magnetic pressure perturbation b2=20associated with a nite-amplitude Alfv en wave (e.g. Hollweg 1971). The Vzterm can be dropped when the plasma is incompressible (see §3.3), however it is required for a nonlinear treatment of compressible plasma and aects the wave heating via the parallel viscosity coecient 0(as remarked in §1). The expression for Vzin classical MHD is given later in Eq. (29). In a full solution, derivatives of b2=20with respect to ygive rise to an additional nonlinear y-component of V, which in turn produces a nonlinear y-component of B. These terms are not shown explicitly in Eqs. (8) and (9). Such terms were included by Nocera et al. (1986) and appear not to aect our main conclusions, provided the perpendicular wavelength of the Alfv en wave is suciently large. The viscous force is determined from the viscous stress tensor by Fvisc; =@ @x; (10) and the viscous heating rate is determined using Qvisc=@V @x; (11) where2fx;y;zg,2fx;y;zg, thexare components of the position vector, Vare components of Vand repeated indices imply summation in the Einstein convention. A vital point is that the viscous stress tensor depends on the direction of the magnetic eld given by the unit vector h=B=jBj, which for our Alfv en wave model in Eq. (8) has hx=bp B2 0+b2; hy= 0; hz=B0p B2 0+b2; (12) withh2 x+h2 z= 1. Our analysis diers from many past works by considering hx6= 0 and identifying the dominant heating contribution at the end, as opposed to setting hx= 0 before evaluating the damping eect on Alfv en waves. Applying formulas from §4 of Braginskii (1965) (equivalent matrix expressions are given by Hogan 1984), the stress tensor is related to ve viscosity coecients by =2X i=0iWi+4X i=3iWi: (13) The gyroviscous 3and4terms do not contribute to heating, so evaluating the heating rate Qviscrequires W0=3 2 hh1 3 hh1 3 W; W1= ? ? +1 2? hh W; W2= ? hh+? hh
W;(14) whereis the Kronecker delta, ? =hh; (15) and the rate of strain tensor is W=@V @x+@V @x2 3r
V: (16)Nonlinear Alfv en Waves with Viscosity Tensor 7 For the shear Alfv en wave geometry described by Eq. (9), the Witensors become W0= hxhz@zVx+2 3h2 x
@zVz
0 B@3h2 x1 0 3hxhz 01 0 3hxhz0 3h2 z11 CA; (17) W1=hx(hz@zVxhx@zVz)0 B@h2 z0hxhz 0 1 0 hxhz0h2 x1 CA+ (hx@yVxhz@yVz)0 B@0hz0 hz0hx 0hx01 CA; (18) W2= 12h2 x
@zVx2hxhz@zVz
0 B@2hxhz0 12h2 x 0 0 0 12h2 x02hxhz1 CA+ (hx@yVx+hz@yVz)0 B@0hx0 hx0hz 0hz01 CA; (19) The viscous heating rate with hx6= 0 retained is thus Qvisc=0 3 3hxhz@zVx+ (23h2 x)@zVz
2+1h2 x(hz@zVxhx@zVz)2+1(hz@yVxhx@yVz)2 +2 12h2 x
@zVx2hxhz@zVz
2+2(hx@yVx+hz@yVz)2:(20) 3.2. Two small parameters As anticipated in §1 (e.g. Fig. 1), two parameters determine the relative importance of individual terms in Eq. (20). The rst small parameter is ( b=B 0)2, the ratio of the wave's magnetic energy density to the energy density of the background magnetic eld, which enters through hxandhz. In the modern era, extensive observations of coronal MHD waves (Nakariakov & Verwichte 2005; De Moortel & Nakariakov 2012) allow ( b=B 0)2to be quantied with good certainty, directly from resolved wave observations or indirectly from spectral line widths. For example, Morton et al. (2015) studied waves at the base of a coronal open eld region using both approaches and reported a wave speed vA= 400 km s1and wave motions at v= 35 km s1. Both measurements are consistent with earlier ndings for coronal holes and the quiet Sun (e.g. McIntosh et al. 2011). From observations like these, h2 x(b=B 0)2(v=vA)2102. The second small parameter is ( ii)2, which sets the viscosity coecients 1and2relative to0. The value of iican vary signicantly in the corona, but if magnetic null points are excluded one obtains values similar to the estimates made by Hollweg (1985), who found 3 :4105for a solar active region and 7 :2105near the base of a coronal hole. We therefore expect ( ii)2.1011under common conditions, and 1and2simplify to 2=6408 5125( ii)20;  1=1 42: (21) The numerical coecients in Eq. (21) are obtained in the limit ( ii)2!0, e.g. from Eq. (73) of Hunana et al. (2022). They are approximate for nite ( ii)2but have a high degree of accuracy because the corrections to the coecients are of the order of ( ii)2.1011. Inspecting Eq. (21) and considering ( ii)2.1011, the2and1 coecients are both vastly smaller than 0. The smallness of ( b=B 0)2and the smallness of ( ii)2compete to make dierent terms dominate the viscous heating. If one tries to simplify Eq. (20) by setting ( b=B 0)2to zero, then hx= 0 andVz= 0 givesQvisc=2(@zVx)2+1(@yVx)2, as obtained by Braginskii (1965). On the other hand, if one tries to simplify by rst taking ( ii)2to zero then only 0terms remain, suggesting a dierent conclusion. Thus, the quantitative results recover the two branches shown in Fig. 1. To correctly determine the damping under coronal conditions, one must carefully compare terms in the full Eq. (20), bearing in mind that there are two small parameters, which we do now (diagonal branch in Fig. 1). 3.3. Heating rate for incompressible plasma The analysis for incompressible plasma is relatively straightforward, which makes it a natural starting point for discussion. The assumption of incompressibility is appropriate for liquid metals or high-beta plasmas, but not, we note, for the corona. The use of coronal wave amplitudes and magnetizations in this section is therefore intended to be instructive only, with the compressible nite-beta treatment that follows later in this paper being required to treat the corona.8 Russell In the incompressible case, r
V= 0 applied to our Alfv en wave geometry implies Vz= 0. Thus Eq. (20) with 1=1 42simplies to Qvisc= 30h2 xh2 z+2 12h2 x
2+1 4h2 xh2 z (@zVx)2+21 4h2 z+h2 x (@yVx)2: (22) The terms involving @zVxset the viscous dissipation due to wavelengths parallel to the equilibrium magnetic eld, and we rst ask whether dissipation due to parallel wavelengths is dominated by the linear 2contribution that has been widely recognised since Braginskii (1965), or the nonlinear 0contribution. The ratio of the two terms inside the curly brackets in Eq. (22) is 3h2 x0 21h2 x (12h2x)2+1 4h2x(1h2x) 3h2 x0 22:4h2 x( ii)2; (23) where the rst step simplies using h2 x1 (for the observed value of h2 x102, retaining the terms in the square bracket increases the ratio by 2.8%, so this approximation is both accurate and conservative) and the substitution for 0=2is by Eq. (21). For the coronal wave amplitudes and iivalues noted in §3.2, this ratio exceeds 109, with the nonlinear damping via 0dominating the heating rate by that factor. In other words, the viscous dissipation of Alfv en waves via derivatives aligned with the equilibrium magnetic eld is a factor 109stronger than predicted by linear theory. We now evaluate the role of derivatives perpendicular to the equilibrium magnetic eld by comparing the nonlinear 0term in Eq. (22) to the term involving @yVx. The ratio of these heating rate terms is 12h2 x0 2? jj21h2 x 1 + 3h2x 12h2 x0 2? jj2 9:6h2 x( ii)2? jj2 ; (24) where?andjjare the wavelengths perpendicular and parallel to the equilibrium magnetic eld (for the observed value ofh2 x102, the approximation of the terms in h2 xis accurate to 3.9%). For the coronal parameters noted above, if?jjthen the nonlinear 0term again dominates by a factor that exceeds 109. For smaller transverse wavelengths, the nonlinear 0term dominates whenever ?&105jj. If one considers a wave speed of 400 km s1 and a frequency of 3 mHz, consistent with the observations by Morton et al. (2015), the condition that the nonlinear 0 term dominates becomes ?&800 m. Given that CoMP has imaged Alfv enic waves using 3 Mm pixels, this condition appears to be met by a very large margin, making the nonlinear 0dissipation dominant over the 2linear dissipation. The purpose of deriving Eq. (22) and the ratios on the left hand sides of Eq. (23) and (24) such that they include all appearances of hxandhzis that they can be evaluated exactly for a given value of hx. This makes it explicit that our conclusions are insensitive to the precise value of hx, only that the value of hxis broadly consistent with coronal observations. While that approach is most comprehensive, the same conclusions can also be reached by separately simplifying each term in Eq. (22) using h2 x1 andh2 z= 1h2 x1 to obtain the less cumbersome formula Qvisc= 30h2 x+2 (@zVx)2+1(@yVx)2; (25) and comparing terms to reach the same conclusions. 3.4. Compressible plasma with large Re Under typical coronal conditions, the thermal pressure is too small to prevent compression of the plasma by nonlinear magnetic pressure forces, thus a nonlinear Vzdevelops that is known as the ponderomotive ow (Hollweg 1971). This ow component aects the viscous heating rate via the parallel viscosity coecient 0, hence compressible theory is required for nonlinear viscous damping of Alfv en waves in plasma. We dene the Alfv enic Reynolds number as Re =vA kjj0: (26) This dimensionless parameter diers from the traditional Reynolds number since it refers to the Alfv en speed vA= B=p0instead of a typical uid velocity. This distinction mirrors that between the Lundquist number and magnetic Reynolds number in resistive MHD. Justication for dening Re according to Eq. (26) will be found in the detailedNonlinear Alfv en Waves with Viscosity Tensor 9 mathematical solutions in §5, in which it is found to be a natural parameter of the system (also see Nocera et al. 1986). In this section, the ponderomotive Vzwill be related to Vxusing expansions in the amplitude of the primary wave elds. Several assumptions are used to accomplish this. First, we make use of the result that a travelling wave solution propagating in the positive z-direction has @ @tvA@ @z; (27) wherevAis the wave speed. For simplicity it is assumed that derivatives of background quantities are suciently weak to play a higher-order role on the dynamics. We also simplify here by replacing full treatment of thermal conduction with two thermodynamic cases: adiabatic and isothermal. Finally, it is assumed that Re is large enough that viscous forces can be neglected at leading order when evaluating Vz, which makes it possible to obtain an algebraic relationship betweenVzandb2. This assumption will be removed for §5, in which the eect of viscous forces on VxandVzis included. Thex-components of the momentum and induction equations are unaected by the ponderomotive ow at leading order in the wave amplitude. From them one recovers the Wal en relation for propagating Alfv en waves, b=B 0=Vx=vA. At leading order, the z-component of the momentum equation is 0@Vz @t+@ @z p+b2 20 = 0; (28) where the viscous force has been neglected since we currently consider the limit of large Re. Using Eq (27) and integrating yields an algebraic relationship between Vz,pandb2. In an adiabatic treatment, the energy equation yieldsp= p0Vz=vA, hence we obtain Vz vA=1 2(1)b B02 ; (29) where =cs vA2 = 2p B2 0=20 : (30) In an isothermal treatment, the ideal gas law p=RT yieldsp=p 0== 0=Vz=vA. This does not change the form of Eq. (29); instead, the isothermal case is recovered simply by setting = 1 in the denition of . Deningas the square of the ratio of the sound speed ( cs=p p0=0) to the Alfv en speed diers slightly from the convention of dening as the ratio of thermal pressure to magnetic pressure, due to the factor =2, which is 5/6 for an adiabatic monoatomic gas, and 1 =2 for an isothermal model. Dening as the speed ratio squared leads to cleaner mathematics for many MHD wave problems, including this one, and it has therefore become established practice in MHD wave theory. The= 1 singularity in Eq. (29) arises because cs=vAimplies resonance between the Alfv en wave and an acoustic wave, which resonantly transfers energy between the waves. In this specic case, Eq. (27) does not apply because it does not account for evolution due to resonance. Similarly, Eq. (28) assumes that VzvAto simplify the convective derivative, and the solution in Eq. (29) does not satisfy this condition in the immediate vicinity of = 1. The= 1 resonance and the singularity in Eq. (29) are not of concern for most coronal applications, which typically have  <0:2, but there are special cases in which it is of interest, such as waves propagating towards coronal magnetic nulls or across the= 1 layer in the lower solar atmosphere. Russell et al. (2016) have previously applied such nonlinear resonant coupling to the problem of sunquake generation by magnetic eld changes during solar ares. Dierentiating Eq. (29) and employing the relation b=B 0=Vx=vAyields @Vz=1 (1)hx hz@Vx; (31) which can be used to eliminate Vzfrom Eq. (20). To leading order in h2 xin each viscosity coecient, we nd Qvisc= C0h2 x+2 (@zVx)2+1(@yVx)2; (32) where C=1 313 12 : (33)10 Russell Some special cases are noteworthy. The incompressible results of §3.3 are recovered for !1 , which gives Vz!0 andC!3. Similarly, the cold plasma solution is recovered by setting = 0, which gives Vz= (b=B 0)2=2 and C= 1=3. The nonlinear heating rate for cold plasma ( = 0) is a factor nine smaller than for incompressible plasma ( !1 ), which demonstrates the importance of compressibility for this problem. Furthermore, C() is monotonically decreasing between= 0 and= 1=3. Since 0<  < 1=3 for most coronal applications, the dissipation rate due to nonlinear Braginskii viscosity in these environments is reduced compared to the cold plasma solution. For example, given = 0:1, the heating rate is approximately 60% of the value for cold plasma. It is therefore evident that compressibility and nite-beta eects must be treated when assessing viscous dissipation of Alfv en waves. Another important feature is that Chas a zero for = 1=3. This is one circumstance in which Vz vA=3 4Vx vA2 =3 4b B02 ; (34) which causes cancellation within the 0contribution to Qvisc. That a particular organisation of Vz=vAcan suppress nonlinear viscous dissipation is an important novel nding that §4 explores further in the context of low Re. The nal feature of Cis the singularity at = 1. As noted earlier in this section, cs=vAimplies that the Alfv en wave is in resonance with a sound wave, which transfers energy between the Alfv en wave and the sound wave. Caution is needed around the resonance, since resonant energy transfer cannot be described using Eq. (27), which was used to derive Eq. (33). Evaluation of ratios of heating terms from Eq. (32) proceeds as for the comparison in Sec. 3.3, but with 0multiplied byC=3. The top-level conclusions remain intact: heating by the Braginskii viscous stress tensor is dominated by an 0term that is nonlinear in the wave amplitude, and for coronal values, heating due to the nonlinear 0term is many orders of magnitude larger than the heating due to the linear 1and2terms. 3.5. What wave amplitude is linear? An important implication of the preceding analysis is that nonlinear eects become signicant for anisotropic viscosity at far lower wave amplitudes than they do for other terms in the MHD equations. Linearizing the Braginskii viscous stress tensor is only appropriate when h2 x(b=B 0)2( ii)2, which in the corona corresponds to a requirement that the wave energy density is less than 1011times the energy density of the background magnetic eld, far too small to be relevant to coronal energetics. Waves that have small enough amplitudes to be governed by linear viscous damping theory would be unobservable and have no eect on the coronal energy balance. Thus, for coronal Alfv en waves, viscosity must be treated nonlinearly in the wave amplitude, as well as anisotropically due to the magnetic eld. Interestingly, this linearization condition is far more stringent than the linearization condition for other terms in the MHD equations, whereby h2 xis normally compared to unity. The extreme dierence in these linearization conditions is due to the large 0=2ratio produced by the strong magnetization. 3.6. Damping scales for large Re (energy derivation) It is of major interest to know the time and length scales over which waves damp. This section provides a relatively simple derivation of the decay scales for nonlinear viscous damping of propagating shear Alfv en waves for large Re, using energy principles. Dropping the 1and2terms from Eq. (32), the heating rate due to the 0parallel viscosity coecient for large Re is Qvisc=0 313 12b B02 (@zVx)2: (35) A wave energy decay time can be dened according to d=hEwi=hQvisci, whereh:idenotes the time average over a wave period, and Ewis the wave energy density. The corresponding decay length is Ld=vAd. For forward propagating Alfv en waves, Ew0V2 x, andVx=acos() where=kjj(zvAt). Hence, Ew=0a2(1 + cos(2)) 2: (36) Similarly, using Eq. (35) with ( b=B 0)2(Vx=VA)2, Qvisc=0k2 jj 3v2 A13 12 a4(1cos(4)) 8; (37)Nonlinear Alfv en Waves with Viscosity Tensor 11 which give the fast time averages hEwi=0a2 2; (38) hQi=0k2 jja4 24v2 A13 12 : (39) The wave energy decay scales are therefore d=120 0k2 jj(a=vA)21 132 ; (40) Ld=120vA 0k2 jj(a=vA)21 132 : (41) Equations (40) and (41) show that waves with larger kjj(equivalently, higher frequencies) are damped on shorter scales. We also remark that since Lddepends on the amplitude of Vx(the constant a), the decay envelope is non- exponential. The damping properties are elaborated on more fully in §5, in which the assumption of large Re is removed and the functional form of the wave envelope is determined. 4.SELF-ORGANISED VISCOUS FLOW In the limit Re!0, the viscous force in the z-component of the momentum equation risk becoming extremely large, unless the ow self-organises to prevent this. Correspondingly, in the limit Re !0, strong dissipation will prevent waves from propagating, unless Vzis determined by viscosity. One can therefore expect self-organisation of the ow pattern for Alfv en waves in highly viscous plasma (small values of Re), which is a concept previously advanced by Montgomery (1992). To investigate quantitatively, we analyze the highly magnetised regime ii1, simplifying the stress tensor and heating rate by retaining only the 0parallel viscosity coecient. Inspecting Eq. (5), components of are proportional to ( hh=3)@V, andQviscis proportional to the square of this expression. Applying the shear Alfv en wave geometry of Eqs. (8) and (9) and simplifying by h2 x1,  hh 3@V @xhx@Vx @z+2 3@Vz @z: (42) Viscous forces and heating can be suppressed, allowing Alfv en wave propagation, if the ow self-organises to keep this expression close to zero. Using the Alfv en wave relation to substitute hxb=B 0Vx=vAand integrating, we nd that for small Re Vz vA=3 4Vx vA2 =3 4b B02 : (43) The relation specied by Eq. (43) appeared previously in the dierent context of §3.4, where it was seen that viscous dissipation of Alfv en waves in high Re plasma is suppressed for the special case of = 1=3. The ow pattern required to produce cancellation within the 0part ofQviscis independent of Re and , but it occurs for dierent reasons in the two cases: in §3.4 it arose as a special case of ponderomotive ow with nite ; when Re is small, it occurs because of self-organisation through viscous forces. This novel result demonstrates that decay scales and other properties derived in §3 should not be extrapolated to small Re. Instead, we expect that as Re !0, the viscous force organises the ow such that Vzobeys Eq. (43), for which dissipation is suppressed by cancellation within the 0part ofQvisc. 5.MULTIPLE SCALE ANALYSIS Section 3 used methods of analysis based on heating rates and energy principles. Section 5 now takes a complementary approach of solving the full set of governing equations using multiple scale analysis, to reinforce the results of §3, extend to general Re by including the eect of the viscous force on V, and obtain additional results including the functional form of the nonlinear decay.12 Russell 5.1. Comparison to Nocera et al. (1986) We preface the multiple scale analysis part of this paper with some remarks about related calculations by Nocera et al. (1986). Their work and ours both concentrate on 0viscosity as the main source of wave damping, treating this nonlinearly in the wave amplitude (the horizontal branch of Figure 1). Also in common, both treat ponderomotive and niteeects. The previous work of Nocera et al. (1986) derived a version of the viscous stress tensor that includes the leading order eect of hx6= 0 in the 0term. Terms in the viscosity tensor were then compared, concluding like our §3 (but by dierent arguments) that the nonlinear 0term exceeds contributions from other viscosity coecients when (b=B 0)2( ii)2(their Eq. (3.13)). The two studies thus agree on the dominance of nonlinear 0viscosity. Nocera et al. (1986) then found a decay length using the following strategy. A self-consistent perturbation ordering was introduced, then the x-components of the momentum and induction equations were combined to obtain a single equation for Vx, which at linear order is a wave equation. Next, all variables apart from Vxwere eliminated from the leading-order nonlinear term. Finally, they concluded from a stability analysis that waves with k?= 0 are damped nonlinearly, with a decay time that has the same form as our Eq. (40) (their Eq. (5.7), given in terms of normalised variables). The detailed derivation that follows in §5.2 draws inspiration from the framework developed by Nocera et al. (1986). We have also taken the opportunity to make several changes that we regard as improvements, most importantly: 1. Nocera et al. (1986) assumed that the fast time average of Vzis zero, which necessitated adding a non-zero constant of integration to Vz. By contrast, we will set the constant of integration to zero, which is the only choice for which an Alfv en wave driver switching on at one boundary does not unphysically send an instantaneous signal to innity. Additional support for our choice comes from simulations of nonlinear longitudinal ows produced by Alfv en waves (e.g. McLaughlin et al. 2011), which are consistent with the constraint used in our work. 2. The stability analysis in §5 Nocera et al. (1986) is replaced with a multiple scale analysis of the type covered in Chapter 11 of Bender & Orszag (1978). 3. Nocera et al. (1986) made their wave envelope a function of z+vAt. We treat the envelope as time-independent and thus explicitly investigate damping of a propagating wave with respect to distance. 4. Our derivation provides the envelope of Vxas well as the decay length. 5. Our solution is valid for general Re, whereas Nocera et al. (1986) solved for the decay scales in the low-viscosity limit of high Re only. Equally, Nocera et al. (1986) treated cases that we do not, including the possibility of k?large enough for coupling between the Alfv en and fast modes to alter the wave properties (referred to in their paper as the case of phase mixed waves). 5.2. Detailed solution 5.2.1. Geometry and perturbations We assume the Alfv en wave geometry of Eqs. (8) and (9), set @=@y0 to concentrate on waves without short perpendicular scales, and introduce density and pressure perturbations andptogether with a self-consistent perturbation ordering that has Vx=vAb=B 01=2andVz=vA= 0p=p 0. The viscosity 0and background quantities B0,0andp0are treated as locally homogeneous for simplicity. 5.2.2. Nonlinear wave equation Starting from the ideal induction equation, @B @t=r(VB); (44) we have@b @tB0@Vx @z=@ @z(bVz) (exact); (45) where the linear terms have been grouped on the left hand side and the nonlinear term on the right hand side.Nonlinear Alfv en Waves with Viscosity Tensor 13 The momentum equation is @V @t+ (V
r)V =@ @x p+B2 20 +1 0(B
r)B@ @x: (46) When0contributions dominate the viscous force, Eqs. (13) and (17) give xz= 30b B0b B0@Vx @z+2 3@Vz @z +O(5=2) (47) so thex-component of Eq. (46) becomes @Vx @tB0 00@b @z= 0@Vx @tVz@Vx @z+30 0@ @zb B0b B0@Vx @z+2 3@Vz @z +O(5=2); (48) where linear terms and nonlinear terms have again been placed on opposite sides of the equation. Taking the time derivative of Eq. (48) and using Eq. (45) to eliminate bfrom the linear terms, @2 @t2v2 A@2 @z2 Vx=v2 A@2 @z2b B0Vz @ @t 0@Vx @t+Vz@Vx @z +30 0@2 @t@zb B0b B0@Vx @z+2 3@Vz @z +O(5=2): (49) Interpreting Eq. (49), the linear terms (on the left hand side) correspond to a wave equation with wave speed vA. The leading nonlinear terms (those shown explicitly on the right hand side) include the leading-order eect of the anisotropic viscosity, which enters at the same order as the leading nonlinear terms that appear in perturbative nonlinear theory of ideal Alfv en waves. Next, we eliminate bandfrom theO(3=2) nonlinear terms in Eq. (49). Equations (45) and (48) are solved at linear order by the Alfv en wave relation b B0=Vx vA+O(3=2): (50) We choose the negative sign so waves travel in the positive zdirection, giving b B0=Vx vA+O(3=2): (51) The travelling wave behaviour of the linear solution together with assumption that the wave envelope changes over a distance controlled by the leading order nonlinear terms in Eq. (49) allows replacement @ @t=vA@ @z+O(): (52) The density perturbation is governed by the mass continuity equation @ @t+r
(V) = 0; (53) which gives for our shear Alfv en wave @ @t=0@Vz @z+O(2): (54) Then, using Eq. (52) and integrating,  0=Vz vA+O(2): (55) The constant of integration has been set to zero, for reasons discussed in §5.1. Using these results, Eq. (49) becomes @2 @t2v2 A@2 @z2 Vx=vA@2 @z2(VxVz) +30 0@2 @t@zVx vAVx vA@Vx @z2 3@Vz @z +O(5=2): (56)14 Russell Now that the problem has been reduced to the two variables VxandVz, it is convenient to make the dependence explicit by introducing dimensionless variables vandwdened by Vx(z;t) =1=2vAv(z;t); Vz(z;t) =vAw(z;t): (57) Expressing Eq. (56) in the dimensionless variables vandw, and dropping the non-explicit higher order terms from the right hand side, we seek solutions to @2 @t2v2 A@2 @z2 v= v2 A@2 @z2(vw) +0 0@2 @t@z@v3 @z2v@w @z : (58) 5.2.3. Multiple scale analysis Equation (58) is now solved using multiple scale analysis (e.g. Bender & Orszag 1978). Applying this technique, one introduces a new variable Z=zthat denes a long length scale, and the perturbation expansions v(z;t) =v0(z;Z;t ) +v1(z;Z;t ) +::: (59) w(z;t) =w0(z;Z;t ) +w1(z;Z;t ) +:::: (60) Derivatives are treated using the chain rule as though zandZare independent variables and setting d Z=dz=. Thus, @v @z=@v0 @z+@v0 @Z+@v1 @z +O(2); (61) @2v @z2=@2v0 @z2+ 2@2v0 @Z@z+@2v1 @z2 +O(2); (62) with equivalent expressions for derivatives of w. Substituting into Eq. (58), collecting 0terms, and thus solving the homogeneous wave equation @2 @t2v2 A@2 @z2 v0= 0; (63) obtains d'Alembert's solution v0(z;Z;t ) =f(zvAt;Z) +g(z+vAt;Z): (64) For forward propagating waves, the function gis zero. The corresponding w0is obtained by integrating the z-component of the momentum equation, Eq. (46), which gives Vz vA=1 0v2 A p+b2 20+zz +O(2): (65) From Eqs. (13) and (17), we have zz= 20b B0@Vx @z+2 3@Vz @z +O(2): (66) A substitution for the pressure perturbation pis obtained by integrating the energy equation, @p @t+V
rp+ pr
V= ( 1)Qvisc: (67) The viscous heating Qviscis of orderO(2), so integration gives the adiabatic relation p p0= Vz vA+O(2): (68) Alternatively, one can consider isothermal conditions using p=p 0=Vz=vAfrom the ideal gas law, which is recovered from Eq. (68) by setting = 1.Nonlinear Alfv en Waves with Viscosity Tensor 15 Using Eqs. (66) and (68), and eliminating bterms using Eq. (51), Eq. (65) can be expressed as  1+4 30 0vA@ @z w=1 2+0 0vA@ @z v2; (69) where= (cs=vA)2and dropped terms are O(). Whenvandware expanded according to Eqs. (59) and (60), an equation identical to (69) connects w0andv0. Inspecting Eq. (69), it is evident that obtaining wfor a known vin general requires solving a rst order linear partial dierential equation. In the limit where the viscous terms can be neglected, the problem simplies to the algebraic w=v2=2(1) relation used in Sec. 3.4. Similarly, when the viscous terms dominate, one obtains the w= (3=4)v2 relation for viscously self-organised parallel ow discussed in Sec. 3.4. For the detailed solution in this section, we retain the complete set of forces that determine Vz, solving the full Eq. (69). Solution is facilitated by considering the special case where v0oscillates sinusoidally in time. For the rest of this derivation we therefore set v0=A(Z)ei+A(Z)ei;  =kjj(zvAt); (70) whereA(Z)2Canddenotes the complex conjugate. Representing Ain polar form, A(Z) =R(Z)ei(Z); (71) Eq. (70) is equivalent to v0(z;Z;t ) = 2R(Z) cos(kjj(zvAt) +(Z)): (72) From inspection, 2 R(Z) is the local amplitude and (Z) is a phase shift. We have found the complex form in Eq. (70) more convenient to work with in the following. We now solve for the corresponding w0. Noting that v2 0=A2e2i+ 2jAj2+A2e2i; (73) wherejAj2=AA, we seek a solution of the form w0=De2i+De2i+ ^w0: (74) Substituting into Eq. (69), terms in e0give ^w0=jAj2 1; (75) while terms in e2iand e2iindependently give D=A2;  =1 2(1 + 4ikjj0=0vA) (1+ (8=3)ikjj0=0vA): (76) The solution for w0can also be expressed without complex numbers. Making explicit the real and imaginary parts of=r+ii, we have the real constants r=1 2(1+ (32=3)(Re)2) ((1)2+ (64=9)(Re)2); (77) i=2 3(13)(Re)1 ((1)2+ (64=9)(Re)2); (78) where Re = 0vA=kjj0consistent with Eq. (26). It is then easily shown that w0 2R(Z)2=rcos(2(kjj(zvAt) +(Z)))isin(2(kjj(zvAt) +(Z))) +1 2(1): (79) To deduceR(Z) and(Z), we return to analysing Eq. (58). The 1terms in Eq. (58) give the inhomogeneous partial dierential equation @2 @t2v2 A@2 @z2 v1= 2v2 A@2v0 @Z@z+v2 A@2 @z2(v0w0) +0 0@2 @t@z@v3 0 @z2v0@w0 @z : (80)16 Russell Thev0andw0terms drive v1, and the solution for v1will have a secular contribution (i.e. one or more terms that grow relative to corresponding solutions of the homogeneous equation) if terms on the right hand side resonate with the solution to the undriven wave equation. In the specic case where v0is given by Eq. (70), secular terms in the solution forv1will restrict the domain for which v0is a valid approximation if the right hand side of Eq. (80) contains eior eiterms. The central idea in multiple scale analysis is to solve for the A(Z) that makes the resonance disappear, makingv0a durable approximation for v. Using Eqs. (70) and (74){(76), eiterms vanish from the right hand side of Eq. (80) if and only if 1 AdA dZ=kjjjAj2 2 i +1 1 +kjj0 0vA(34) : (81) The same condition also removes the eiterms. Changing to polar form, Eq. (71) implies 1 AdA dZ=1 RdR dZ+id dZ: (82) Hence, the real and imaginary parts of Eq. (81) yield the real ordinary dierential equations dR dZ=1 2R3; (83) d dZ=2R2; (84) where 1=kjjkjj0 0vA(34r)i ; (85) 2=kjj 2 r+1 1kjj0 0vA4i : (86) Eqs. (83) and (84) govern the local amplitude and phase drift of the Alfv en wave respectively ( c.f.Eq. (72)). Our main interest is in R(Z), which determines how the waves decay. Equation (83) is a separable rst order dierential equation. The solution is R(Z) =R(0)p 1 +1R(0)2Z: (87) For1>0 the wave envelope decays non-exponentially, over a damping length that is inversely proportional to the square of the initial wave amplitude. Having obtained R(Z), the solution for (Z) is obtained by directly integrating Eq. (84). Using Eq. (87), (Z) =(0) +2 1ln1 +1R(0)2Z: (88) 5.2.4. Solution in original variables Having ensured corrections to vv0remain of order (b=B 0)21, the multiple scale analysis is concluded by usingv0as the approximation for v. Returning to the original variables, Vx(z;t) =a0p 1 +z=Ldcos kjj(zvAt) + (2=1) lnj1 +z=Ldj+0
; (89) wherea0is the amplitude of Vx(0;t),0sets the initial phase of the wave (at z= 0,t= 0) and Ld=4 1(a0=vA)2(90) is the decay length. Using Eqs. (77), (78) and (85), 1=kjj 3Re(13)2 (1)2+ (64=9)(Re)2; (91)Nonlinear Alfv en Waves with Viscosity Tensor 17 where Re is the Reynolds number for the wave, as dened by Eq. (26). Thus, Ld=12Re kjj(a0=vA)2(1)2+ (64=9)(Re)2 (13)2: (92) If one neglects the Re2term in the numerator of Eq. (92), then Ldagrees exactly with the formula in Eq. (41) that we derived from energy principles. The formula for Ldin Eq. (92) is more general since it was derived without direct assumptions about the value of Re = kjj0=0vA, although the multiple scale analysis requires that the combination of parameters kjj,a2 0and Re are such that waves damp over a signicantly longer scale than the wavelength. 5.3. Non-exponential decay and interpretation of damping length As a general principle, the Alfv en wave energy density Ew=V2 xdecays more rapidly than the perturbation Vx, due to the quadratic power. For exponential decay this is re ected in a factor two dierence in the respective e1decay lengths. For the non-exponential decay produced by nonlinear viscous damping, the situation is handled dierently. The sameLddescribesVxandEw, however they have dierent functional forms. The velocity amplitude decays as (1 +z=Ld)1=2(see Eq. (89)), while the wave energy density decays as (1 + z=Ld)1. Therefore, over a distance Ld, the velocity amplitude reduces by a factorp 2 and the energy density halves. 5.4. Inclusion of thermal conduction The multiple scale analysis can also be modied to include explicit thermal conduction. Since thermal conduction is highly anisotropic, we include the parallel thermal conduction, setting the heat ow vector to q=Kjj(h
rT)h; (93) whereKjjis the coecient of parallel thermal conduction, and temperature T=p=RwhereRis the gas constant. The energy equation with heat ow is @p @t+V
rp+ pr
V= ( 1)(Qviscr
q); (94) which replaces Eq. (67). It follows that p=p 0is related to vz=vAby the partial dierential equation  1 + @ @zp p0= + @ @zVz vA+O(2): (95) where  =( 1)Kjj 0RvA(96) is a conductive length scale. In the limit of weak thermal conduction,  !0 givesp=p 0= Vz=vA, recovering the adiabatic case treated above. Similarly, for strong thermal conduction,  !1 givesp=p 0=Vz=vA, recovering the isothermal case. Introducing a dimensionless pressure variable cdened byp=p0c(z;t), expanding c(z;t) =c0(z;Z;t )+c1(z;Z;t )+ :::and setting c0=Ce2i+Ce2i+ ^c0, terms in e0in Eq. (95) yield ^ c0= ^w0, and terms in e2iyieldC= D, where = + 2ikjj 1 + 2ikjj: (97) Solving further, an equation D=A2analogous to Eq. (76) is obtained but with replaced by the complex-valued p0=0v2 Ain the formula for . Meanwhile, (75) and (81) are unchanged, retaining the real-valued = p0=0v2 A. The wave amplitude is therefore governed by results identical to Eqs. (83) and (85), with the aforementioned change in the denition of .18 Russell 6.DISCUSSION 6.1. Optimum damping Inspecting Eq. (92), the formula for Ldkjjhas a minimum with respect to the Alfv enic Reynolds number at Re = 8=(3j1j). Thus, shear Alfv en waves with kjj= 30vAj1j=30are damped in the fewest number of wavelengths, which we refer to as optimum damping. The optimally damped waves have Ld jj=32 j1j (a0=vA)2(13)2: (98) When1, the right hand side of Eq. (98) is approximately ten divided by the square of the normalised wave amplitude. Hence, while nonlinear viscous damping can in principle damp Alfv en waves in a small number of wave- lengths, this requires large amplitudes a=vA1, or1. For a more typically encountered amplitudes a=vA101 and low, one ndsLd=jj&1000, making nonlinear viscous damping negligible for many coronal situations. 6.2. Viscous self-organisation The suppression of nonlinear viscous damping for small Re (highly viscous plasma) does not mean that viscous eects are unimportant in this regime. On the contrary, nonlinear damping is suppressed for small R ebecause viscous forces organise the parallel ow associated with the Alfv en wave to approach the relationship Vz=vA= (3=4)(Vx=vA)2. This modication of the parallel ow plays a crucial role in avoiding signicant nonlinear damping in highly viscous plasma, when modelled using Braginskii MHD. 6.3. Validity constraints Throughout this paper, we have assumed that 6= 1 to avoid resonant wave coupling. This condition holds throughout most of the corona, so it is appropriate for our primary applications. Additionally, the multiple scale analysis in §5 uses(Vx=vA)2(b=B 0)2as a small parameter, one consequence of which is that the nonlinearly damping occurs over a distance considerably greater than the parallel wavelength. As noted in §3.2, transverse coronal waves are observed in open-eld regions with 102, making weakly nonlinear theory appropriate for such situations. Obtaining nonlinear viscous solutions in the resonant and strongly nonlinear regimes nonetheless remain interesting future challenges for plasma theory. Applicability of this paper's results to physical problems is also constrained to conditions under which Braginskii MHD can be rigorously applied. As discussed in §2, the traditional derivation of Braginskii MHD assumes that the collisional mean free path is less than the macroscopic scales. Comparing the mean free path parallel to the magnetic eld to the parallel wavelength, this condition can be given as kjjmfp<1, wheremfp=vTii,vTi=p kBT=miand iis the ion collision time. Using the formula (Braginskii 1958, 1965; Hollweg 1985) 0= 0:96nkBTi; (99) and the denition of Re in Eq. (26), one can show that kjjmfp<1,1=2Re =cs kjj0>1: (100) In other words, Braginskii MHD requires that the Reynolds number based on the sound speed is greater than unity. One should therefore be cautious about applying small Alfv enic Reynolds number results such as viscous self-organisation to real low- plasmas. 6.4. Formulas for applications For applications to real plasmas, the following formulas are convenient. In cases where the parallel viscosity coecient is determined by Coulomb collisions, 0= 0:96nkBTi=22 C1017T5=2; (101) where this formula is stated in S.I. units with Tin kelvin, and Cis the Coulomb logarithm (e.g. Hollweg 1985). The Reynolds number dened in Eq. (26) can then be expressed as Re = 5:81020CB2f1T5=2; (102)Nonlinear Alfv en Waves with Viscosity Tensor 19 also in S.I. units, where f=vAkjj=2is the wave frequency. This formula makes explicit the dependences on frequency, magnetic eld strength and temperature. The Alfv enic Reynolds number is smallest when the plasma has high temperature and low magnetic eld strength, and for higher frequency waves. Finally, we express the damping length in Eq. (92) as a function of frequency and the mean square velocity V2 x =a2 0=2, which gives Ld=3 v3 ARe fhV2xi(1)2+ (64=9)(Re)2 (13)2: (103) 6.5. Waves in a coronal open-eld region Outgoing transverse waves in the magnetically open solar corona contain sucient energy to heat the open corona and accelerate the fast solar wind (McIntosh et al. 2011; Morton et al. 2015), and they are observed to damp signicantly within a solar radius above the Sun's surface (Bemporad & Abbo 2012; Hahn et al. 2012; Hahn & Savin 2013; Hahn et al. 2022). Heating at these altitudes is also thought to be important for producing the observed rapid acceleration of the fast solar wind (Habbal et al. 1995; McKenzie et al. 1995). The problem of how the outgoing waves damp has not been conclusively solved, although one leading hypothesis is turbulent cascade driven by interactions with downgoing Alfv en waves (Hollweg 1986; Heyvaerts & Priest 1992; Matthaeus et al. 1999; Cranmer et al. 2007; Verdini et al. 2010; Miki c et al. 2018) produced either by re ection from density inhomogeneities (van Ballegooijen & Asgari-Targhi 2016; Pascoe et al. 2022) or by parametric decay instability (Galeev & Oraevskii 1963; Derby 1978; Goldstein 1978; Shoda et al. 2019; Hahn et al. 2022). Here, we demonstrate that Braginskii viscosity does not cause signicant damping of Alfv en waves at the altitudes at which the traditional derivation of Braginskii MHD holds. For concreteness, we consider the Sun's northern polar open-eld region on 27 March 2012, using observational values reported by Morton et al. (2015). Enhanced wave power was present around f= 5 mHz, which suggests Alfv enic waves produced by p-modes (Morton et al. 2019). We will calculate damping lengths for this frequency, noting that Re and Lddepend on f, withLdf2in the limit of high Re. Morton et al. (2015) inferred that the Alfv en speed was nearly constant with vA= 400 km s1on their domain ofr= 1:05 to 1:20R. For temperature, we set T= 1:6106K, the formation temperature of the Fe XIII lines used by the CoMP instrument, which implies the proton thermal speed VTi=p kBT=miis 115 km s1. Hence, in for an isothermal equation of state = 0:083 and1=2= 0:29. For the wave velocity amplitude, Morton et al. (2015) recommended that the reported non-thermal line width should be used, which varies with altitude. Starting with lowest altitude observed by Morton et al. (2015), r= 1:05R, we setn= 1014m3,B= 2104T and take the rms value of Vxas 35 km s1. We therefore nd C= 19 and Re = 28. Since 1=2Re = 8>1, Braginksii MHD applies and we evaluate Ld= 4:2108km600R. Atr= 1:20, we set n= 1013m3,B= 6105T and take the rms value of Vxas 50 km s1. The observed parameters therefore give C= 21 and Re = 2 :7. Since1=2Re = 0:81, this altitude is close to the maximum at which the assumptions by which Braginskii MHD is traditionally derived remains valid (for this particular open eld region, and assuming Eq. (101)). Evaluating the damping length here returns Ld= 4:2107km61R. We conclude that Braginskii viscosity does not cause signicant wave damping below r= 1:2R, which is consistent with observational results that Alfv enic wave amplitudes in coronal holes follow ideal WKB scaling out to around this altitude (Cranmer & van Ballegooijen 2005; Hahn & Savin 2013). Between the altitudes we have examined, Ldreduces by an order of magnitude. If one were to extrapolate using high Re or incompressible results, it would appear that viscous damping becomes important near the altitudes at which the waves are observed to damp. We are cautious about making such an assertion for two reasons. First, as discussed in §6.1, our results show that for Re <8=(3j1)) the damping length in a Braginskii MHD model increases again as the eld-aligned ow self organises to supress viscous damping. Secondly, as the plasma becomes increasingly collisionless (1=2Re<1) the traditional derivation of Bragniskii MHD falters. Intriguingly, it may be signicant that the onset of wave damping broadly coincides with the altitude at which Braginskii MHD can no longer be condently applied if one invokes the 0expression for Coulomb collisions given in Eq. (101). This correspondence is suggestive that the wave damping observed in coronal holes may involve collisionless and heat ow eects not found in the most common uid models. 6.6. Future work The present types of analyses should be extended in future to other types of propagating transverse MHD waves. The nonlinear longitudinal ow that accompanies propagating torsional Alfv en waves diers from its counterpart for20 Russell propagating shear Alfv en waves (Vasheghani Farahani et al. 2011) and it will be of interest to investigate how this dierence aects the nonlinear viscous damping. It is similarly desirable to determine how nonlinear viscosity aects propagating kink waves (Edwin & Roberts 1983). For propagating shear Alfv en waves, viscous damping appears most promising near the cs=vAsingularity, which must be treated using dierent methods to those used in this paper. The solar wind frequently has 1, while= 1 occurs in the lower solar atmosphere and in the vicinity of coronal nulls points. Hence, this case is of considerable physical interest. One challenge for application to magnetic nulls is that the magnetic eld unit vector h=B=jBj is not dened at the null itself, so one must be careful to evaluate the Braginskii stress tensor using appropriate calculations, e.g. see recent discussion by MacTaggart et al. (2017). A further challenge is to develop a theory of nonlinear viscous damping applicable to strongly nonlinear waves with amplitudes bB0and greater. The results of the multiple-scale analysis in §5 are rigorous only for the weakly nonlinear case, in which (b=B 0)2can be treated as a small parameter and it is assumed that the damping length is signicantly longer than the wavelength. Strongly nonlinear Alfv en waves with bB0are a feature of the solar wind, and while the low collisionality of the solar wind means that Braginskii MHD may not be an appropriate framework for that application, extending the current work to strongly nonlinear waves remains an interesting problem. There is a diverse collection of MHD wave problems beyond wave damping for which viscous eects are likely to be signicant. Prime among these are nonlinear phenomena involving Alfv en waves, for which the nonlinear viscosity tensor enters the equations at the same order as the eect of interest. For example, standing Alfv en waves drive signicantly stronger eld-aligned ows than occur for propagating waves because standing Alfv en waves create inhomogeneous time-averaged magnetic pressure. There could also be signicant value in investigating how viscosity modies wave interactions, including Alfv en wave collisions and parametric decay instability (Galeev & Oraevskii 1963; Derby 1978; Goldstein 1978), which are central to leading hypotheses of wave heating in the magnetically open solar corona. Finally, we point to the continuing need for basic plasma physics research to provide increasingly rigorous derivation and validation of the appropriate uid equations for weakly collisional and collisionless plasma, in the face of the closure problem summarised in §2. As discussed in §2 and 6.3, Braginskii MHD breaks down at higher altitudes in the corona as the plasma becomes increasingly collisionless (see Eqs. (100) and (102)). The CGL double-adiabatic equations and other models that evolve the stress tensor may provide a more suitable framework in these conditions. Hunana et al. (2019a,b, 2022) provide recent discussions of such models and their limitations. Alternatively, it may be necessary for the solar waves community to more widely adopt non- uid plasma models. However, tractability of kinetic models remains a limiting factor, especially in light of the large separations between kinetic and macroscopic scales that are characteristic of the Sun's corona. Eloquent comments on these matters can be found in Montgomery (1996). 7.CONCLUSIONS This paper has investigated the properties of propagating shear Alfv en waves subject to the nonlinear eects of the Braginskii viscous stress tensor. The main points are as follows: 1. For many plasma environments, including the low-altitude solar corona, Braginskii MHD provides a more accurate description of plasma than classical MHD does, by rigorously treating the stress tensor and thermal conduction. Stress tensor eects nonetheless remain relatively unexplored for many solar MHD phenomena. 2. The dominant viscous eects for propagating shear Alfv en waves are nonlinear in the wave amplitude and occur through the \parallel" viscosity coecient, 0. Theoretical results based on linearizing the stress tensor with respect to the wave amplitude are only valid for amplitudes satisfying ( b=B 0)2( ii)2. Such waves would be energetically insignicant under normal coronal conditions, hence nonlinear treatment is required. 3. Compressibility and pressure aect the nonlinear eld-aligned ow associated with shear Alfv en waves, hence they impact nonlinear wave damping. Both must be included to produce accurate coronal results. 4. Braginskii viscosity damps propagating shear Alfv en waves nonlinearly, such that the primary wave elds band Vxdecay as (1 + z=Ld)1=2, where the decay length Ld=12Re kjj(a0=vA)2(1)2+ (64=9)(Re)2 (13)2:Nonlinear Alfv en Waves with Viscosity Tensor 21 Here,a0is the initial velocity amplitude of the wave, = (cs=vA)2and Re =vA=kjj0is the Alfv enic Reynolds number of the wave. The energy density decays as (1 + z=Ld)1. 5. Optimal damping (the minimum normalised damping length kjjLd) is obtained when Re = 8 =(3j1j). For low plasma and ( a0=vA).101, one nds Ld=jj&1000, indicating that nonlinear viscous damping is negligible for many coronal situations. 6. The asymptotic behaviour that Ld!1 in the highly viscous regime Re !0 is attributed to self-organisation of the parallel ow by viscous forces such that Vz=vA(3=4)(Vx=vA)2, which suppresses dissipation. 7. Applicability of the Braginskii MHD solutions to real plasmas is constrained by the traditional derivation of Braginskii MHD assuming that kjjmfp<1 which is equivalent to 1=2Re =cs=kjj0>1. In other words, Braginskii MHD requires that the Reynolds number based on the sound speed is greater than unity. We therefore recommend that only the damping results for large Alfv enic Reynolds number should be applied to real coronal plasma, using the simplied formula Ld= 12Re(1)2=(kjj(a0=vA)2(13)2)) that has been derived in this paper by two dierent techniques: energy principles and multiple scale analysis. 8. Application to transverse waves observed in a polar open-eld region concludes that nonlinear Braginskii viscosity does not cause signicant damping of the waves at the altitudes at which the assumptions by which Braginskii MHD is traditionally derived remain valid ( r.1:2Rfor the considered region and wave properties). Intrigu- ingly, the observed onset of wave damping broadly coincides with the altitude at which Braginskii MHD can no longer be condently applied if one invokes the 0expression for Coulomb collisions given in Eq. (101). This work was prompted by and beneted from conversations with Paola Testa, Bart De Pontieu, Vanessa Polito, Graham Kerr, Mark Cheung, Wei Liu, David Graham, Joel Allred, Mats Carlsson, Iain Hannah and Fabio Reale during a research visit to LMSAL funded by ESA's support for the IRIS mission (August 2018) and meetings of International Space Science Institute (Bern) International Team 355 (November 2018). 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P. 2014, Transport Processes in Space Physics and Astrophysics, Vol. 877 (Springer), doi: 10.1007/978-1-4614-8480-6

2023-03-20

Braginskii MHD provides a more accurate description of many plasma environments than classical MHD since it actively treats the stress tensor using a closure derived from physical principles. Stress tensor effects nonetheless remain relatively unexplored for solar MHD phenomena, especially in nonlinear regimes. This paper analytically examines nonlinear damping and longitudinal flows of propagating shear Alfv\'en waves. Most previous studies of MHD waves in Braginskii MHD considered the strict linear limit of vanishing wave perturbations. We show that those former linear results only apply to Alfv\'en wave amplitudes in the corona that are so small as to be of little interest, typically a wave energy less than $10^{-11}$ times the energy of the background magnetic field. For observed wave amplitudes, the Braginskii viscous dissipation of coronal Alfv\'en waves is nonlinear and a factor around $10^9$ stronger than predicted by the linear theory. Furthermore, the dominant damping occurs through the parallel viscosity coefficient $\eta_0$, rather than the perpendicular viscosity coefficient $\eta_2$ in the linearized solution. This paper develops the nonlinear theory, showing that the wave energy density decays with an envelope $(1+z/L_d)^{-1}$. The damping length $L_d$ exhibits an optimal damping solution, beyond which greater viscosity leads to lower dissipation as the viscous forces self-organise the longitudinal flow to suppress damping. Although the nonlinear damping greatly exceeds the linear damping, it remains negligible for many coronal applications.

Nonlinear Damping and Field-aligned Flows of Propagating Shear Alfvén Waves with Braginskii Viscosity

2303.11128v1

arXiv:1207.3864v1 [math.DS] 17 Jul 2012Asymptotic Dynamics of a Class of Coupled Oscillators Driven by White Noises Wenxian Shena1, Zhongwei Shena, Shengfan Zhoub2 aDepartment of Mathematics and Statistics, Auburn Universi ty, Auburn 36849, USA bDepartment of Applied Mathematics, Shanghai Normal Univer sity, Shanghai 200234, PR China Abstract : This paper is devoted to the study of the asymptotic dynamic s of a class of coupled second order oscillators driven by white n oises. It is shown that anysystemofsuchcoupledoscillators withpositiveda mpingandcoupling coefficients possesses a global random attractor. Moreover, when the damping and the coupling coefficients are sufficiently large, the globa l random attractor is a one-dimensional random horizontal curve regardless of the strength of the noises, and the system has a rotation number, which impli es that the oscillators in the system tend to oscillate with the same fre quency eventually andthereforethesocalled frequencylockingissuccessful . Theresultsobtained in this paper generalize many existing results on the asympt otic dynamics for a single second order noisy oscillator to systems of coupled second order noisy oscillators. They show that coupled damped second order osc illators with large damping have similar asymptotic dynamics as the limit ing coupled first order oscillators as the damping goes to infinite and also tha t coupled damped secondorderoscillators havesimilarasymptoticdynamics astheirproperspace continuous counterparts, which are of great practical impo rtance. Keywords : Coupled second order oscillators; white noises; random at tractor; random horizontal curve; rotation number; frequency locki ng AMS Subject Classification : 60H10, 34F05, 37H10. 1 Introduction This paper is devoted to the study of the asymptotic dynamics of the following system of second order oscillators driven by additive noises: d˙uj+αduj+K(Au)jdt+βg(uj)dt=fjdt+ǫjdWj, (1.1) wherej∈Zd N:={j= (j1,...,jd)∈Zd: 1≤j1,...,jd≤N},ujis a scalar unknown function of tandu= (u1,u2,···,uNd)⊤,αandKare positive constants, Ais anNd×Ndmatrix and ( Au)j stands for the jth component of the vector Au,β∈R,gis a periodic function, fjandǫjare 1The first author is partially supported by NSF grant DMS-0907 752 2The third author is supported by National Natural Science Fo undation of China under Grant 10771139, and the Innovation Program of Shanghai Municipal Education Com mission under Grant 08ZZ70 1constants, and {Wj(t)}j∈Zd Nare independent two-sided real-valued Wiener processes. M oreover, Aandgsatisfy (HA)Ais anNd×Ndnonnegative definite symmetric matrix with eigenvalues den oted byλi, i= 0,1,...,Nd−1satisfying that 0 =λ0< λ1≤ ··· ≤λNd−1, λ0is algebraically simple, and (1,...,1)⊤∈RNdis an eigenvector corresponding to λ0= 0. (HG)g∈C1(R,R)has the following properties g(x+κ) =g(x),|g(x)| ≤c1,|g′(x)| ≤c2,∀x∈R, wherec1>0,c2>0andκ >0is the smallest positive period of g. System (1.1) appears in many applied problems including Jos ephson junction arrays and coupled pendula (see [11], [13], [22], [27], etc.). Physica lly,αin (1.1) represents the damping of the system and Kis the coupling coefficient of the system. (1.1) then represen ts a system of Ndcoupled damped oscillators independently driven by white n oises. System (1.1) also arises from various spatial discretizati ons of certain damped hyperbolic partial differential equations. For example, the Nd×NdmatrixAin (1.1) includes the dis- cretization of negative Laplace operator −∆ with Neumann or periodic boundary conditions defined as follows: (Au)j= (Au)(j1,j2,...,jd)=1 h2/bracketleftbig 2duj−u(j1+1,j2,...,jd)−u(j1,j2+1,...,jd)−···−u(j1,j2,...,jd+1) −u(j1−1,j2,...,jd)−u(j1,j2−1,...,jd)−···−u(j1,j2,...,jd−1)/bracketrightbig , with Neumann boundary condition u(j1,...,ji−1,0,ji+1,...,jd)=u(j1,...,ji−1,1,ji+1,...,jd), u(j1,...,ji−1,N+1,ji+1,...,jd)=u(j1,...,ji−1,N,ji+1,...,jd) or periodic boundary condition u(j1,...,ji−1,0,ji+1,...,jd)=u(j1,...,ji−1,N,ji+1,...,jd), u(j1,...,ji−1,N+1,ji+1,...,jd)=u(j1,...,ji−1,1,ji+1,...,jd) forj= (j1,...,jd)∈Zd Nandi= 1,...,d. Thus, (1.1) with uj=u(j1h,···,jih,···,jdh) (h=L/N),Abeing as above, fj=f,ǫj=ǫ, andWj=Wis a spatial discretization of the following problem d˙u+αdu−K∆udt+βg(u)dt=fdt+ǫdW,inU×R+(1.2) with Neumann boundary condition or periodic boundary condi tion, i.e., ∂u ∂n= 0 on ∂U×R+ or u|Γj=u|Γj+d,∂u ∂xj/vextendsingle/vextendsingle/vextendsingle Γj=∂u ∂xj/vextendsingle/vextendsingle/vextendsingle Γj+d, j= 1,...,d, 2where Γ j=∂U∩{xj= 0}, Γj+d=∂U∩{xj=L},j= 1,...,dandU=/producttextd i=1(0,L). Note that ifg(u) = sinu, (1.2) is the so called damped sine-Gordon equation, which i s used to model, for instance, the dynamics of a continuous family of junctions ( see [25]). Two of the main dynamical aspects about coupled oscillators and damped wave equations considered in the literature are the existence and structur e of global attractors and the phe- nomenon of frequency locking. A large amount of research has been carried out toward these two aspects for a variety of systems related to (1.1). See for example, [17, 18, 20, 22, 28] for the study of coupled oscillators with constant or periodic e xternal forces; [12, 19, 21, 25, 26] for the study of the deterministic damped sine-Gordon equat ion; [8, 16, 23] for the study of coupled oscillators driven by white noises; and [10, 24, 29] for the study of stochastic damped sine-Gordon equation. Many of the existing works focus on th eexistence of global attractors and the estimate of the dimension of the global attractors. In [8 , 23, 24], the existence and structure of random attractors of stochastic oscillators and stochas tic damped wave equations are studied. In particular, the asymptotic dynamics of a single second or der noisy oscillator, i.e., (1.1) with N= 1, is studied in [23]. The author of [23] proved the existenc e of a random attractor which is a family of horizontal curves and the existence of a rotation number which implies the frequency locking. In [8], the authors considered a class of coupled fir st order oscillators driven by white noises. Among those, the existence of a one-dimensional ran dom attractor and the existence of a rotation number are proved in [8]. The system of coupled firs t order oscillators considered in [8] is of the form duj+K(Au)jdt+βg(uj)dt=fjdt+ǫdWj, j∈Zd N. (1.3) Note that, by resealing the time variable by t→t α, (1.1) becomes 1 αd˙uj+duj+K(Au)jdt+βg(uj)dt=fjdt+ǫdWj, j∈Zd N. (1.4) Hence, (1.3) can be formally viewed as the limiting system of (1.1) as the damping coefficient α goes toinfinite. In[24], theauthorsinvestigated theexist ence andstructureof randomattractors of damped sine-Gordon equations of the form (1.2) with Neuma nn boundary condition, which is a space continuous counterpart of (1.1) as mentioned abov e. However, many important dynamical aspects including the ex istence of global attractor and the occurrence of frequency locking have been hardly studie d for coupled second order oscillators of the form (1.1) driven by white noises. It is of great intere st to investigate the extent to which the existing results on asymptotic dynamics of a singl e second order noisy oscillator may be generalized to systems of coupled second order noisy osci llators. Thanks to the relations between (1.1) and (1.3) and between (1.1) and (1.2), it is als o of great interest to explore the similarity and difference between the dynamics of coupled dam ped second order oscillators and its limiting coupled first order oscillators as the damping c oefficient goes to infinite and between the dynamics of coupled damped second order oscillators and their proper space continuous counterparts. The objective of this paper is to carry out a st udy along this line. In particular, we study the asymptotic or global dynamics of (1.1), includi ng the existence and structure of global attractor in proper phase space and the success of fre quency locking. In order to do so, as usual, we first change (1.1) to some system of coupled first order random equations. Assume N≥2 andd≥1 (N= 1reduces tothesingle noisy oscillator case considered in [23]). Let u= (uj)j∈Zd N,g(u) = (g(uj))j∈Zd N,f= (fj)j∈Zd N,W(t) = (ǫjWj(t))j∈Zd N. Then, 3(1.1) can be written as the following matrix form, d˙u+αdu+KAudt+βg(u)dt=fdt+dW(t). (1.5) Let Ωj= Ω0={ω0∈C(R,R) :ω0(0) = 0} equipped with the compact open topology, Fj=B(Ω0) be the Borel σ-algebra of Ω 0andPjbe the corresponding Wiener measure for j∈Zd N. Let Ω =/producttext j∈Zd NΩj,Fbe the product σ-algebra on Ω and Pbe the induced product Wiener measure. Define ( θt)t∈Ron Ω via θtω(·) =ω(·+t)−ω(t), t∈R. Then, (Ω ,F,P,(θt)t∈R) is an ergodic metric dynamical system (see [1]). Consider t he Ornstein- Uhlenbeck equation, dz+zdt=dW(t), z∈RNd. (1.6) Letz(θtω) = (zj(θtω))j∈Zd Nbe the unique stationary solution of (1.6) (see [1, 2, 9] for t he existence and various properties of z(·)). Letv= ˙u−z(θtω). We obtain the following equivalent system of (1.5),/braceleftBigg ˙u=v+z(θtω), ˙v=−KAu−αv+f−βg(u)+(1−α)z(θtω).(1.7) To study the global dynamics of (1.1), it is therefore equiva lent to study the global dynamics of (1.7). Observe that the natural phase space for (1.7) is E:=RNd×RNdwith the standard Euclidean norm. Thanks to the presence of the damping, it is e xpected that (1.7) possesses a global attractor in certain sense. However, due to the uncon trolled component of the solutions along the direction of the eigenvectors of the linear operat or in the right of (1.7) corresponding to the zero eigenvalue, there is no bounded attracting sets i nEwith the standard Euclidean norm, which will lead to nontrivial dynamics. There is also s ome additional difficulty if one studies (1.7) in Ewith the standard Euclidean norm due to the zero limit of some eigenvalues of the linear operator in the right of (1.7) as α→ ∞. The later difficulty does not appear for coupled first order oscillators studied in [8] and for a singl e noisy oscillator considered in [23]. We will overcome the difficulty by using some equivalent norm o nEand considering (1.7) in some proper quotient space of Eand prove the existence of a global random attractor as well as the existence of a rotation number of (1.7). To be more precise, let C=/parenleftbigg0I −KA−αI/parenrightbigg . (1.8) By simple matrix analysis, the eigenvalues of Care given by (see [14, 20] for example) µ± i=−α±√ α2−4Kλi 2, i= 0,1,...,Nd−1. (1.9) Note that µ+ 0= 0, which requires some special consideration for the solut ions along the direction of the eigenvector η0= (1,...,1,0,...,0)⊤corresponding to µ+ 0. We overcome this difficulty by considering (1.7) in the cylindrical space E1/κη0Z×E2, whereE1= span{η0},E2is the space 4spanned by all the eigenvectors corresponding to non-zero e igenvalues of C(see section 4 for details). We then prove (1)For any α >0andK >0, system (1.1)possesses a global random attractor (which is unbounded along the one-dimensional space E1and bounded along the one-codimensional space E2) (see Theorem 4.2, Corollary 4.3 and Remark 4.4). It is expected physically that when the damping coefficient α→ ∞, the dynamics of (1.7) becomes simpler or the structure of the global attractor of ( 1.7) becomes simpler. However, µ+ i→0 asα→ ∞fori= 1,2,···,Nd−1, which gives rise to some difficulty for studying the structure of the global attractor in Ewith the standard Euclidean norm. We introduce an equivalent norm on Eto overcome this difficulty (see section 3 for the introductio n of the equivalent norm, the choice of such equivalent norm was first discovered in [15]) and prove (2)WhenαandKare sufficiently large, the global random attractor of (1.1)is a one-dimensional random horizontal curve (see Theorem 5.3 and Corollary 5.4), a nd the rotation number (see Def- inition 6.1) of (1.1)exists (see Theorem 6.3 and Corollary 6.4). Note that roughly a real number ρ∈Ris called the rotation number of (1.1) or (1.7) if for any solution {uj(t)}j∈Zd Nof (1.1), the limit lim t→∞uj(t) texists almost surely for any j∈Zd Nand lim t→∞uj(t) t=ρfora.e. ω∈Ω and j= 1,2,···,Nd (see Definition 6.1 and the remark after Definition 6.1). Henc e if (1.1) has a rotation number, then the oscillators in the system tend to oscillate with the same frequency eventually and therefore the so called frequency locking is successful. (1) and (2) above are the main results of the paper. They make a n important contribution to the understanding of coupled second order oscillators dr iven by noises. Property (1) shows that system (1.1) is dissipative along the one-codimension al space E2. By property (2), the asymptotic dynamics of (1.1) with sufficiently large αandKis one dimensional regardless of the strength of noise. Property (2) also shows that all the so lutions of (1.1) tend to oscillate with the same frequency eventually almost surely and hence f requency locking is successful in (1.1) provided that αandKare sufficiently large. The results obtained in this paper generalize many existing results on the asymptotic dynam- ics for a single damped noisy oscillator to systems of couple d damped noisy oscillators. They show that coupled damped second order oscillators with larg e damping have similar asymptotic dynamics as the limiting coupled first order oscillators as t he damping goes to infinite and hence one may use coupled first order oscillators to analyze qualit ative properties of coupled second order oscillators with large damping, which is of great prac tical importance. They also show that coupled damped second order oscillators have similar a symptotic dynamics as their proper space continuous counterparts and hence one may use finitely many coupled oscillators to study qualitative properties of damped wave equations, which is o f great practical importance too. The rest of the paper is organized as follows. In section 2, we present some basic concepts and properties for general random dynamical systems. In sectio n 3, we provide some basic settings about (1.1) and show that it generates a random dynamical sys tem. We prove in section 4 the existence of a global random attractor of the random dynamic al system φgenerated by (1.1) for anyα >0 andK >0. We show in section 5 that the global random attractor of φis a random horizontal curve and show in section 6 that (1.1) has a rotati on number, respectively, provided thatαandKare sufficiently large. 52 Random Dynamical Systems In this section, we collect some basic knowledge about gener al random dynamical system (see [1, 4] for details). Let ( X,d) be a complete and separable metric space with Borel σ-algebra B(X). Definition 2.1. Acontinuous random dynamical system over (Ω ,F,P,(θt)t∈R)is a(B(R+)× F ×B(X),B(X))-measurable mapping ϕ:R+×Ω×X→X,(t,ω,x)/ma√sto→ϕ(t,ω,x) such that the following properties hold: (1)ϕ(0,ω,x) =xfor allω∈Ω; (2)ϕ(t+s,ω,·) =ϕ(t,θsω,ϕ(s,ω,·))for alls,t≥0andω∈Ω; (3)ϕ(t,ω,x)is continuous in xfor every t≥0andω∈Ω. For given x∈XandE,F⊂X, we define d(x,F) = inf y∈Fd(x,y) and dH(E,F) = sup x∈Ed(x,F). dH(E,F) is called the Hausdorff semi-distance fromEtoF. Definition 2.2. (1) A set-valued mapping ω/ma√sto→D(ω) : Ω→2Xis said to be a random set if the mapping ω/ma√sto→d(x,D(ω))is measurable for any x∈X. Ifω/ma√sto→d(x,D(ω))is measurable for any x∈XandD(ω)is closed (compact) for each ω∈Ω, thenω/ma√sto→D(ω) is called a random closed (compact) set . A random set ω/ma√sto→D(ω)is said to be bounded if there exist x0∈Xand a random variable R(ω)>0such that D(ω)⊂ {x∈X:d(x,x0)≤R(ω)}for allω∈Ω. (2) A random set ω/ma√sto→D(ω)is called tempered provided that for some x0∈XandP-a.e. ω∈Ω, lim t→∞e−βtsup{d(b,x0) :b∈D(θ−tω)}= 0for allβ >0. (3) A random set ω/ma√sto→B(ω)is said to be a random absorbing set if for any tempered random setω/ma√sto→D(ω), there exists t0(ω)such that ϕ(t,θ−tω,D(θ−tω))⊂B(ω)for allt≥t0(ω), ω∈Ω. (4) A random set ω/ma√sto→B1(ω)is said to be a random attracting set if for any tempered random setω/ma√sto→D(ω), we have lim t→∞dH(ϕ(t,θ−tω,D(θ−tω),B1(ω)) = 0for allω∈Ω. (5) A random compact set ω/ma√sto→A(ω)is said to be a global random attractor if it is a random attracting set and ϕ(t,ω,A(ω)) =A(θtω)for allω∈Ωandt≥0. 6Theorem 2.3. Letϕbe a continuous random dynamical system over (Ω,F,P,(θt)t∈R). If there is a random compact attracting set ω/ma√sto→B(ω)ofϕ, thenω/ma√sto→A(ω)is a global random attractor ofϕ, where A(ω) =/intersectiondisplay t>0/uniondisplay τ≥tϕ(τ,θ−τω,B(θ−τω)), ω∈Ω. Proof.See [1, 4]. 3 Basic Settings In this section, we give some basic settings about (1.1) and s how that it generates a random dynamical system. First, let Y= (u,v)⊤andF(θtω,Y) = (z(θtω),f−βg(u) +(1−α)z(θtω))⊤. System (1.7) can then be written as ˙Y=CY+F(θtω,Y), (3.1) whereCis as in (1.8). Recall that z(θtω) = (zj(θtω))j∈Zd Nis the unique stationary solution of (1.6). Note that the random variable |zj(ω)|is tempered and the mapping t/ma√sto→zj(θtω) isP-a.s. continuous (see [1, 2]). More precisely, there is a θt-invariant ˜Ω⊂Ω withP(˜Ω) = 1 such that t/ma√sto→zj(θtω) is continuous for ω∈˜Ω andj∈Zd N. We will consider (1.7) or (3.1) for ω∈˜Ω and write ˜Ω as Ω from now on. LetE=RNd×RNdandFω(t,Y) :=F(θtω,Y), thenFω(·,·) :R×E→Eis continuous intand globally Lipschitz continuous in Yfor each ω∈Ω. By classical theory of ordinary differential equations concerning existence and uniqueness of solutions, for each ω∈Ω and any Y0∈E, (3.1) has a uniqueness solution Y(t,ω,Y0),t≥0, satisfying Y(t,ω,Y0) =eCtY0+/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0))ds, t≥0. (3.2) Moreover, it follows from [1] that Y(t,ω,Y0) is measurable in ( t,ω,Y0). Hence (3.1) generates a continuous random dynamical system on E, Y:R+×Ω×E→E,(t,ω,Y0)/ma√sto→Y(t,ω,Y0). (3.3) Define a mapping φ:R+×Ω×E→Eby φ(t,ω,φ0) =Y(t,ω,Y0(ω))+(0,z(θtω))⊤, (3.4) whereφ0= (u0,u1)⊤∈EandY0(ω) = (u0,u1−z(ω))⊤. Then φis a continuous random dynamical system associated with the problem (1.1) on E. Recall that the eigenvalues of Care given by (see [14, 20] for example) µ± i=−α±√ α2−4Kλi 2, i= 0,1,...,Nd−1. (3.5) By (3.5), Chas at least two real eigenvalues 0 and −αwith eigenvalues η0= (1,...,1,0,...,0)⊤, η−1= (1,...,1,−α,...,−α)⊤∈E, respectively. Let E1= span{η0},E−1= span{η−1},E11= E1+E−1andE22=E⊥ 11, the orthogonal complement space of E11inE, thenE=E11⊕E22. 7To control the unboundedness of solutions in the direction o fη0, we will study (3.1) in the cylindrical space E1/κη0Z×E2, whereE2=E−1⊕E22(see Section 4 for details). Observe that the Lipschitz constant of Fwith respect to YinEwith the standard Euclidean norm is independent of α >0. Butµ+ i→0 asα→ ∞fori≥1, which gives rise to some difficulty for the investigation of (3.1) in Ewith the standard Euclidean norm. To overcome the difficulty, we introduce a new norm which is equivalent to the s tandard Euclidean norm on E. Here, we only collect some results about the new norm (see [15 , 20] for details). Define two bilinear forms on E11andE22, respectively. For Yi= (ui,vi)⊤∈E11,i= 1,2, let /an}bracketle{tY1,Y2/an}bracketri}htE11=α2 4/an}bracketle{tu1,u2/an}bracketri}ht+/an}bracketle{tα 2u1+v1,α 2u2+v2/an}bracketri}ht, (3.6) where/an}bracketle{t·,·/an}bracketri}htdenotes the inner product on RNd, and for Yi= (ui,vi)⊤∈E22, i= 1,2, let /an}bracketle{tY1,Y2/an}bracketri}htE22=/an}bracketle{tKAu1,u2/an}bracketri}ht+(α2 4−δKλ1)/an}bracketle{tu1,u2/an}bracketri}ht+/an}bracketle{tα 2u1+v1,α 2u2+v2/an}bracketri}ht,(3.7) whereδ∈(0,1]. It is easy to check that the Poincar´ e-type inequality /an}bracketle{tAu,u/an}bracketri}ht ≥λ1/bardblu/bardbl2,∀Y= (u,v)⊤∈E22 holds (see [20] for example), where /bardbl·/bardblis the standard Euclidean norm on RNd. Thus (3.7) is positive definite. For any Yi=Y(1) i+Y(2) i∈E,i= 1,2, where Y(1) 1,Y(1) 2∈E11,Y(2) 1,Y(2) 2∈E22, we define /an}bracketle{tY1,Y2/an}bracketri}htE=/an}bracketle{tY(1) 1,Y(1) 2/an}bracketri}htE11+/an}bracketle{tY(2) 1,Y(2) 2/an}bracketri}htE22. (3.8) Lemma 3.1 ([20]).(1)(3.6)and(3.7)define inner products on E11andE22, respectively. (2)(3.8)defines an inner product on E, and the corresponding norm /bardbl·/bardblEis equivalent to the standard Euclidean norm on E. (3) In terms of the inner product /an}bracketle{t·,·/an}bracketri}htE,E1andE11are orthogonal to E−1andE22, respec- tively. (4) In terms of the norm /bardbl·/bardblE, the Lipschitz constant LFofFwith respect to Ysatisfies LF=2c2|β| α, (3.9) wherec2is as in(HG). Note that E2is orthogonal to E1andE=E1⊕E2. Denote by PandQ(=I−P) the projections from EintoE1andE2, respectively. Set a=α 2−/vextendsingle/vextendsingle/vextendsingleα 2−δKλ1 α/vextendsingle/vextendsingle/vextendsingle. (3.10) Lemma 3.2. (1) For any Y∈E2,/an}bracketle{tCY,Y/an}bracketri}htE≤ −a/bardblY/bardbl2 E. (2)/bardbleCtQ/bardblE≤e−atfort≥0. (3)eCtPY=PYforY∈E,t≥0. 8Proof.(1) and (2) follow from similar arguments as in Lemma 2.3 and C orollary 2.4 in [19]. Let us show (3). For Y∈E, sincePY∈E1andd dteCtPY=eCtCPY= 0, we have eCtPY= eC0PY=PY. By Lemma 3.2 (2), the constant ain (3.10) describes the exponential decay rate of eCt|QE in the new norm. By Lemma 3.1 (4), LFtends to 0 as α→ ∞with respect to the new norm, which essentially helps to overcome the difficulty induced fr om the fact that µ+ i→0 asα→0 fori≥1. The following lemma will be needed to take care unboundednes s of the solutions along the direction of the eigenvectors corresponding to µ+ 0. Lemma 3.3. Letp0=κη0∈E(κis the smallest positive period of g). The random dynamical systemYdefined in (3.3)isp0-translation invariant in the sense that Y(t,ω,Y0+p0) =Y(t,ω,Y0)+p0, t≥0, ω∈Ω, Y0∈E. Proof.SinceCp0= 0 and F(t,ω,Y) isp0-periodic in Y,Y(t,ω,Y0) +p0is a solution of (3.1) with initial data Y0+p0. Thus,Y(t,ω,Y0)+p0=Y(t,ω,Y0+p0). By (3.3) and Lemma 3.3, φis alsop0-translation invariant. Lemma 3.4. For any ǫ >0, there is tempered random variable ˜r(ω)>0such that /bardblz(θtω)/bardbl ≤eǫ|t|˜r(ω)for allt∈R, ω∈Ω, (3.11) where˜r(ω)satisfies e−ǫ|t|˜r(ω)≤˜r(θtω)≤eǫ|t|˜r(ω), t∈R, ω∈Ω. (3.12) Proof.Forj∈Zd N, since|zj(ω)|is a tempered random variable and the mapping t/ma√sto→ln|zj(θtω)| isP-a.s. continuous, itfollowfromProposition4.3.3in[1]th atforany ǫj>0thereisantempered random variable rj(ω)>0 such that 1 rj(ω)≤ |zj(ω)| ≤rj(ω), whererj(ω) satisfies, for P-a.e.ω∈Ω, e−ǫj|t|rj(ω)≤rj(θtω)≤eǫj|t|rj(ω), t∈R. (3.13) Letr(ω) = (rj(ω))j∈Zd N,ω∈Ω and take ǫj=ǫ,j∈Zd N, then we have /bardblz(θtω)/bardbl ≤/parenleftBigg/summationdisplay j∈Zd Ne2ǫ|t|r2 j(ω)/parenrightBigg1 2 =eǫ|t|/bardblr(ω)/bardbl, t∈R, ω∈Ω. Let ˜r(ω) =/bardblr(ω)/bardbl,ω∈Ω. Then (3.11) is satisfied and (3.12) is trivial from (3.13). 94 Existence of Random Attractor In this section, we study the existence of a random attractor . We assume that p0=κη0∈E1 andδ∈(0,1] is such that a >0, where ais as in (3.10). We remark in the end of this section that such δalways exists. By Lemma 3.3 and the fact that Chas a zero eigenvalue, we will define a random dynamical systemYon some cylindrical space induced from the random dynamical systemYonE. Then by properties of Yrestricted on E2, we can prove the existence of a global random attractor of Y. Thus, we can say that Yhas a global random attractor which is unbounded along E1and bounded along E2. Now, we define Y. LetT1=E1/p0ZandE=T1×E2, wherep0Z={kp0:k∈Z}. ForY0∈E, letY0:= Y0(modp0), whichisan element of E. Notethat, byLemma3.3, Y(t,ω,Y0+kp0) =Y(t,ω,Y0)+ kp0,∀k∈Zfort≥0,ω∈Ω andY0∈E. With this, we define Y:R+×Ω×E→Eby setting Y(t,ω,Y0) =Y(t,ω,Y0) (modp0), (4.1) whereY0=Y0(modp0). ThenY:R+×Ω×E→Eis a random dynamical system. Similarly, the random dynamical system φdefined in (3.4) also induces a random dynamical system Φon E. By (3.3), (3.4) and (4.1), Φis defined by Φ(t,ω,Φ0) =Y(t,ω,Y0)+ ˜z(θtω) (modp0), t≥0, ω∈Ω, (4.2) whereΦ0=φ0(modp0), ˜z(θtω) = (0,z(θtω))⊤andY0=Φ0−˜z(ω) (modp0). Recall that PandQ(=I−P) are the projections from EintoE1andE2, respectively. Definition 4.1. Letω∈ΩandR: Ω→R+be a random variable. A random pseudo-ball ω/ma√sto→B(ω) inEwith random radius ω/ma√sto→R(ω)is a set of the form ω/ma√sto→B(ω) ={b∈E:/bardblQb/bardblE≤R(ω)}. Furthermore, a random set ω/ma√sto→B(ω)∈Eis called pseudo-tempered provided that ω/ma√sto→QB(ω) is a tempered random set in E, i.e., for P-a.e.ω∈Ω, lim t→∞e−βtsup{/bardblQb/bardblE:b∈B(θ−tω)}= 0for allβ >0. Clearly, any random pseudo-ball ω/ma√sto→B(ω) inEhas the form ω/ma√sto→E1×QB(ω), where ω/ma√sto→QB(ω)isarandomball in E2. Thenthemeasurability of ω/ma√sto→B(ω)istrivial. ByDefinition 4.1, ifω/ma√sto→B(ω) is a random pseudo-ball in E, thenω/ma√sto→B(ω) (modp0) is random bounded set inE. And if ω/ma√sto→B(ω) is a pseudo-tempered random set in E, thenω/ma√sto→B(ω) (modp0) is a tempered random set in E. We next show the existence of a global random attractor of the induced random dynamical systemYdefined in (4.1). Theorem 4.2. Letα >0andK >0. Then the induced random dynamical system Ydefined in(4.1)has a global random attractor ω/ma√sto→A0(ω). Proof.Forω∈Ω, we obtain from (3.2) that Y(t,ω,Y0(ω)) =eCtY0(ω)+/integraldisplayt 0eC(t−s)F(θsω,Y(s,ω,Y0(ω)))ds. (4.3) 10The projection of (4.3) on E2is QY(t,ω,Y0(ω)) =eCtQY0(ω)+/integraldisplayt 0eC(t−s)QF(θsω,Y(s,ω,Y0(ω)))ds. (4.4) By replacing ωbyθ−tω, it follows from (4.4) that QY(t,θ−tω,Y0(θ−tω)) =eCtQY0(θ−tω)+/integraldisplayt 0eC(t−s)QF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))ds. It then follows from Lemma 3.2 and Q2=Qthat /bardblQY(t,θ−tω,Y0(θ−tω))/bardblE ≤e−at/bardblQY0(θ−tω)/bardblE+/integraldisplayt 0e−a(t−s)/bardblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/bardblEds.(4.5) By Lemma 3.4 with ǫ=a 2and the equivalence of /bardbl·/bardblEand/bardbl·/bardblonE, there is a M1>0 such that /bardblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/bardblE ≤M1/bardblF(θs−tω,Y(s,θ−tω,Y0(θ−tω)))/bardbl =M1/parenleftBig /bardblz(θs−tω)/bardbl2+/bardblf−βg(Yu(s,θ−tω))+(1−α)z(θs−tω)/bardbl2/parenrightBig1 2 ≤M1/parenleftBig (3α2−6α+4)/bardblz(θs−tω)/bardbl2+3/bardblf/bardbl2+3β2c2 1Nd/parenrightBig1 2 ≤a1ea 2(t−s)˜r(ω)+a2, whereYusatisfies Y(s,θ−tω,Y0(θ−tω)) = (Yu(s,θ−tω),Yv(s,θ−tω))⊤,a1=M1√ 3α2−6α+4 anda2=M1/radicalbig 3/bardblf/bardbl2+3β2c2 1Nd. We then find from (4.5) that /bardblQY(t,θ−tω,Y0(θ−tω))/bardblE≤e−at/bardblQY0(θ−tω)/bardblE+2a1 a(1−e−a 2t)˜r(ω)+a2 a(1−e−at). Now for ω∈Ω, define R0(ω) =4a1 a˜r(ω)+2a2 a. Then, for any pseudo-tempered random set ω/ma√sto→B(ω) inEand any Y0(θ−tω)∈B(θ−tω), there is aTB(ω)>0 such that for t≥TB(ω), /bardblQY(t,θ−tω,Y0(θ−tω))/bardblE≤R0(ω), ω∈Ω, which implies Y(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω, whereω/ma√sto→B0(ω) is the random pseudo-ball centered at origin with random ra diusω/ma√sto→R0(ω). Note that ω/ma√sto→R0(ω) is a random variable since ω/ma√sto→˜r(ω) is a random variable, then the measurability of random pseudo-ball ω/ma√sto→B0(ω) is trivial from Definition 4.1. Forω∈Ω, letB(ω) =B(ω) (modp0) andB0(ω) =B0(ω) (modp0), we then have Y(t,θ−tω,B(θ−tω))⊂B0(ω) for all t≥TB(ω), ω∈Ω, 11whereTB(ω) =TB(ω) forω∈Ω, i.e.,ω/ma√sto→B0(ω) is the random absorbing set of Y. Moreover, ω/ma√sto→B0(ω) is bounded and closed, hence compact in E, it then follows from Theorem 2.3 that Yhas a global random attractor ω/ma√sto→A0(ω), where A0(ω) =/intersectiondisplay t>0/uniondisplay τ≥tY(τ,θ−τω,B0(θ−τω)), ω∈Ω. This completes the proof. Corollary 4.3. Letα >0andK >0. Then the induced random dynamical system Φdefined in(4.2)has a global random attractor ω/ma√sto→A(ω), whereA(ω) =A0(ω)+ ˜z(ω) (modp0)for all ω∈Ω. Proof.It follows from (4.2) and Theorem 4.2. Remark 4.4. (1) For any α >0andK >0, there is a δ∈(0,1]such that α2>2δKλ1which impliesa >0, whereais as in(3.10)andλ1is the smallest positive eigenvalue of A. (2) We say that the random dynamical system Y(orφ) has a global random attractor in the sense that the induced random dynamical system Y(orΦ) has a global random attractor, and we will say that Y(orφ) has a global random attractor directly in the sequel. We denote the global random attractor of Yandφbyω/ma√sto→A0(ω)andω/ma√sto→A(ω)respectively. Indeed,ω/ma√sto→A0(ω)andω/ma√sto→A(ω)satisfy A0(ω) =A0(ω) (modp0),A(ω) =A(ω) (modp0), ω∈Ω. Hence a global random attractor of Y(orφ) is unbounded along the one-dimensional space E1and bounded along the one-codimensional space E2. (3) Observe the global attractors of many dissipative syste ms related to (1.1)is one-dimensional (see [8, 17, 18, 19, 20, 21, 23, 24, 26]). Similarly, we expect that the random attractor ω/ma√sto→A(ω)ofφis one-dimensional for each ω∈Ωprovided that αis sufficiently large. We prove that this is true in the next section. (4) By (2), the system (1.1)is dissipative along E2(i.e. it possesses a global random attractor which is bounded along E2). In section 6, we will show that (1.1)with sufficiently large αandKalso has a rotation number and hence all the solutions tend to oscillate with the same frequency eventually. 5 One-dimensional Random Attractor In this section, we apply the invariant and inertial manifol d theory, in particular, the theory establishedin[7]toshowthat therandomattractor ofthera ndomdynamicalsystem φgenerated by (1.1) is one-dimensional (more precisely, is a horizonta l curve) provided that αandKare sufficiently large (see Remark 4.4 (2) for the random attracto r). This method has been applied by Chow, Shenand Zhou[8] to systems of coupled firstorder noi sy oscillators and by Shen, Zhou and Shen [24] to the stochastic damped sine-Gordon equation . The reader is referred to [3, 5] for the theory and application of inertial manifold theory f or stochastic evolution equations. Assume that p0=κη0anda >4LF(see (3.10) for aand (3.9) for LF). Note that the condition a >4LFindicates that the exponential decay rate of eCt|QEin the norm /bardbl · /bardblEis 12larger than four times the Lipschitz constant of Fin the norm /bardbl·/bardblE. It will be seen at the end of this section that the condition a >4LFcan be satisfied provided that αandKare sufficiently large. Definition 5.1. Suppose {Φω}ω∈Ωis a family of maps from E1toE2andn∈N. A family of graphsω/ma√sto→ℓ(ω)≡ {(p,Φω(p)) :p∈E1}is said to be a randomnp0-period horizontal curve if ω/ma√sto→ℓ(ω)is a random set and {Φω}ω∈Ωsatisfy the Lipshitz condition /bardblΦω(p1)−Φω(p2)/bardblE≤ /bardblp1−p2/bardblEfor allp1,p2∈E1, ω∈Ω and the periodic condition Φω(p+np0) = Φω(p)for allp∈E1, ω∈Ω. Note that for any ω∈Ω,ℓ(ω) is a deterministic np0-period horizontal curve. When n= 1, we simply call it a horizontal curve. Lemma 5.2. Leta >4LF. Suppose that ω/ma√sto→ℓ(ω)is a random np0-period horizontal curve in E. Then, ω/ma√sto→Y(t,ω,ℓ(ω))is also a random np0-period horizontal curve in Efor allt >0. Moreover, ω/ma√sto→Y(t,θ−tω,ℓ(θ−tω))is a random np0-period horizontal curve for all t >0. The proof of Lemma 5.2 is similar to that of Lemma 4.2 in [24]. W e hence omit it here. Chooseγ∈(0,a 2) such that 2c2|β| α/parenleftBigg 1 γ+1 a−2γ/parenrightBigg <1, (5.1) where2c2|β| αis the Lipschitz constant of F(see (3.9)). We remark at the end of this section that such aγexists provided that αandKare sufficiently large. The main result in this section is as follows. Theorem 5.3. Assume that a >4LFand there is γ∈(0,a 2)such that (5.1)holds. Then the global random attractor ω/ma√sto→A0(ω)of the random dynamical system Yis a random horizontal curve. Proof.Since equation (3.1) can be viewed as a deterministic system with a random parameter ω∈Ω, we write it here as (3.1) ωforω∈Ω. We first show that for any fixed ω∈Ω, (3.1) ωhas a one-dimensional attracting invariant manifold W(ω). In order to do so, for fixed ω∈Ω, let W(ω) ={Y0∈E|Y(t,ω,Y0) exists for t≤0 and sup t≤0/bardbleγtY(t,ω,Y0)/bardblE<∞}. We prove that W(ω) is a one-dimensional attracting invariant manifold of (3. 1)ω. First of all, by the definition of W(ω), it is clear that for any t∈R, Y(t,ω,W(ω)) =W(θtω), 13that is,{W(ω)}ω∈Ωis invariant. By the variation of constant formula, Y0∈W(ω) if and only if there is˜Y(t) with˜Y(0) =Y0, supt≤0/bardbleγt˜Y(t)/bardblE<∞, ˜Y(t) =eCtξ+/integraldisplayt 0eC(t−s)PFω(s,˜Y(s))ds+/integraldisplayt −∞eC(t−s)QFω(s,˜Y(s))ds, t≤0,(5.2) andY(t,ω,Y0) =˜Y(t), where Fω(t,Y) =F(θtω,Y) andξ=P˜Y(0)∈E1. ForH: (−∞,0]→E such that supt≤0/bardbleγtH(t)/bardblE<∞, define (LH)(t) =/integraldisplayt 0eC(t−s)PH(s)ds+/integraldisplayt −∞eC(t−s)QH(s)ds, t≤0. Then sup t≤0/bardbleγt(LH)(t)/bardblE≤(1 γ+1 a−γ)sup t≤0/bardbleγtH(t)/bardblE≤/parenleftBigg 1 γ+1 a−2γ/parenrightBigg sup t≤0/bardbleγtH(t)/bardblE, which means that /bardblL/bardbl ≤1 γ+1 a−2γ. Thus, Theorem 3.3 in [7] shows that for any ξ∈E1, equation (5.2) has a unique solution ˜Yω(t,ξ) satisfying supt≤0/bardbleγt˜Yω(t,ξ)/bardblE<∞. Let h(ω,ξ) =Q˜Yω(0,ξ) =/integraldisplay0 −∞e−CsQFω(s,˜Yω(s,ξ))ds, ω∈Ω. Then, W(ω) ={ξ+h(ω,ξ) :ξ∈E1} andW(ω) is a one dimensional invariant manifold of (3.1) ω. Furthermore, for any ǫ∈(0,γ), by Lemma 3.4, we have /bardblh(θ−tω,ξ)/bardblE≤a1 a−ǫ˜r(ω)eǫt+a2 a, t≥0, (5.3) wherea1,a2is the same as in the proof Theorem 4.2. To show the attracting property of W(ω), we prove for each given ω∈Ω the existence of a stable foliation {Ws(Y0,ω) :Y0∈W(ω)}of the invariant manifold W(ω) of (3.1) ω. Consider the following integral equation ˆY(t) =eCtη+/integraldisplayt 0eC(t−s)Q/parenleftBig Fω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds +/integraldisplayt ∞eC(t−s)P/parenleftBig Fω(s,ˆY(s)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds, t≥0,(5.4) whereξ+h(ω,ξ)∈W(ω),η=QˆY(0)∈E2andYω(t,ξ+h(ω,ξ)) :=Y(t,ω,ξ+h(ω,ξ)),t≥0 is the solution of (3.1) with initial data ξ+h(ω,ξ) for fixed ω∈Ω. Theorem 3.4 in [7] shows that for any ξ∈E1andη∈E2, equation (5.4) has a unique solution ˆYω(t,ξ,η) satisfying supt≥0/bardbleγtˆYω(t,ξ,η)/bardblE<∞and for any ξ∈E1,η1, η2∈E2, sup t≥0eγt/bardblˆYω(t,ξ,η1)−ˆYω(t,ξ,η2)/bardblE≤M2/bardblη1−η2/bardblE, (5.5) 14whereM2=1 1−2c2|β| α/parenleftbig 1 γ+1 a−2γ/parenrightbig. Let ˆh(ω,ξ,η) =ξ+PˆYω(0,ξ,η) =ξ+/integraldisplay0 ∞e−CsP/parenleftBig Fω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds. Then,Ws(ω,ξ+h(ω,ξ)) ={η+h(ω,ξ)+ˆh(ω,ξ,η) :η∈E2}is a foliation of W(ω) atξ+h(ω,ξ). Observe that ˆYω(t,ξ,η)+Yω(t,ξ+h(ω,ξ))−Yω(t,ξ+h(ω,ξ)) =ˆYω(t,ξ,η) =eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ)) +/integraldisplayt 0eC(t−s)/parenleftBig Fω(s,ˆYω(s,ξ,η)+Yω(s,ξ+h(ω,ξ))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds(5.6) and Yω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ)) =eCt(η+h(ω,ξ)+ˆh(ω,ξ,η)−ξ−h(ω,ξ)) +/integraldisplayt 0eC(t−s)/parenleftBig Fω(s,Yω(s,η+h(ω,ξ)+ˆh(ω,ξ,η))) −Fω(s,Yω(s,ξ+h(ω,ξ)))/parenrightBig ds.(5.7) Comparing (5.6) with (5.7), we find that ˆYω(t,ξ,η) =Yω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ)), t≥0.(5.8) In addition, if η= 0, then by the uniqueness of solution of (5.4), ˆYω(t,ξ,0)≡0 fort≥0, which together with (5.5) and (5.8) shows that sup t≥0eγt/bardblYω(t,η+h(ω,ξ)+ˆh(ω,ξ,η))−Yω(t,ξ+h(ω,ξ))/bardblE≤M2/bardblη/bardblE (5.9) for anyξ∈E1andη∈E2. Therefore, {Ws(Y0,ω) :Y0∈W(ω)}is a stable foliation of the invariant manifold W(ω) of (3.1) ωand then W(ω) is a one-dimensional attracting invariant manifold of (3.1) ω. Next we show that A0(ω) =W(ω) andA0(ω) is a random horizontal curve. Let ω/ma√sto→B(ω) be any pseudo-tempered random set in E. For any t >0 andY0∈B(θ−tω), there is ξ(θ−tω,Y0)∈ E1such that Y0∈Ws(θ−tω,ξ(θ−tω,Y0)+h(θ−tω,ξ(θ−tω,Y0))). 15Letη(θ−tω) = supY0∈B(θ−tω)/bardblQY0−h(θ−tω,ξ(θ−tω,Y0))/bardblE. Then by (5.3) and (5.9), /bardblY(t,θ−tω,Y0)−Y(t,θ−tω,ξ(θ−tω,Y0)+h(θ−tω,ξ(θ−tω,Y0)))/bardblE ≤M2e−γtη(θ−tω) →0 ast→ ∞, which implies that for ω∈Ω, dH(Y(t,θ−tω,B(θ−tω)),W(ω))→0 ast→ ∞. Therefore, A0(ω) =W(ω) forω∈Ω. Moreover, for any random horizontal curve ω/ma√sto→ℓ(ω) inEcontained in some pseudo-tempered random set, dH(Y(t,θ−tω,ℓ(θ−tω)),A0(ω))→0 ast→ ∞ for every ω∈Ω, which means that lim t→∞Y(t,θ−tω,ℓ(θ−tω))⊂A0(ω). Since A0(ω) is one- dimensional, we have for ω∈Ω, A0(ω) = lim t→∞Y(t,θ−tω,ℓ(θ−tω)). It then follows from Lemma 5.2 that ω/ma√sto→A0(ω) is a random horizontal curve. Corollary 5.4. Assume that a >4LFand there is γ∈(0,a 2)such that (5.1)holds. Then the random attractor ω/ma√sto→A(ω)of the random dynamical system φis a random horizontal curve. Proof.It follows from Corollary 4.3, Remark 4.4 and Theorem 5.3. Remark 5.5. At the beginning of this section, we assume that a >4LF. Sincea=α 2−|α 2−δKλ1 α| andLF=2c2|β| α, we can take α,Ksatisfyingα 2−/vextendsingle/vextendsingle/vextendsingleα 2−δKλ1 α/vextendsingle/vextendsingle/vextendsingle>8c2|β| α, whereλ1is the smallest positive eigenvalue of A. On the other hand, we need some γ∈(0,a 2)such that (5.1)holds. Note that min γ∈(0,a 2)/parenleftBigg 1 γ+1 a−2γ/parenrightBigg =/parenleftBigg 1 γ+1 a−2γ/parenrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle γ=a 2+√ 2=(√ 2+1)2 a, which implies that there exist α,Ksatisfying α 2−/vextendsingle/vextendsingle/vextendsingleα 2−δKλ1 α/vextendsingle/vextendsingle/vextendsingle>2c2|β|(√ 2+1)2 α>8c2|β| α. (5.10) Indeed, let c= 2c2|β|(√ 2+1)2, then for any α >√ 2candK >c λ1, there is a δ >0satisfying c Kλ1< δ <min/braceleftBigα2−c Kλ1,1/bracerightBig such that (5.10)holds. 166 Rotation Number In this section, we study the phenomenon of frequency lockin g, i.e., the existence of a rotation number of the coupled second order oscillators with white no ises (1.1). Definition 6.1. The coupled second order system with white noises (1.1)is said to have a rota- tion number ρ∈Rif, forP-a.e.ω∈Ωand each φ0= (u0,u1)⊤∈E, the limit limt→∞Pφ(t,ω,φ0) t exists and lim t→∞Pφ(t,ω,φ0) t=ρη0, whereη0is the basis of E1. Note that the rotation number is considered here by restrict ingφonE1, sinceφis dissipative onE2and limits likewise in Definition 6.1 vanish. From (3.4), we h ave Pφ(t,ω,φ0) t=PY(t,ω,Y0(ω)) t+P(0,z(θtω))⊤ t, (6.1) whereφ0= (u0,u1)⊤andY0(ω) = (u0,u1−z(ω))⊤. By Lemma 2.1 in [9], it is easy to prove that limt→∞P(0,z(θtω))⊤ t= (0,0)⊤. Thus, it sufficient to prove the existence of the rotation num ber of the random system (3.1). Let us show a simple lemma which will be used. For any pi=siη0∈E1,i= 1,2, we define p1≤p2ifs1≤s2. Then we have Lemma 6.2. Suppose that a >4LF. Letℓbe any deterministic np0-periodic horizontal curve (ℓsatisfies the Lipschitz and periodic condition in Definition 5 .1). For any Y1, Y2∈ℓwith PY1≤PY2, there holds PY(t,ω,Y1)≤PY(t,ω,Y2)fort >0, ω∈Ω. (6.2) The proof of this lemma is similar to that of Lemma 6.3 in [24]. We then omit it here. We now have the main result in this section. Theorem 6.3. Leta >4LF. Then the rotation number of (3.1)exists. Proof.By the random dynamical system Ydefined in (4.1), we define the corresponding skew- product semiflow Θt: Ω×E→Ω×Efort≥0 by setting Θt(ω,Y0) = (θtω,Y(t,ω,Y0)). Obviously, (Ω ×E,F ×B,(Θt)t≥0) is a measurable dynamical system, where B=B(E) is the Borelσ-algebra of E. It also can be verified that there is a measure µon Ω×Esuch that (Ω×E,F ×B, µ,(Θt)t≥0) becomes an ergodic metric dynamical system (see [6]). Note that PY(t,ω,Y0) t=PY0 t+1 t/integraldisplayt 0PF(θsω,Y(s,ω,Y0))ds. 17SinceF(θsω,Y(s,ω,Y0)+kp0) =F(θsω,Y(s,ω,Y0)),∀k∈Z, wecanidentify F(θsω,Y(s,ω,Y0)) withF(θsω,Y(s,ω,Y0)) and write F(θsω,Y(s,ω,Y0)) =F(θsω,Y(s,ω,Y0)). Thus, PY(t,ω,Y0) t=PY0 t+1 t/integraldisplayt 0PF(θsω,Y(s,ω,Y0))ds =PY0 t+1 t/integraldisplayt 0F(Θs(ω,Y0))ds,(6.3) whereF=P◦F∈L1(Ω×E,F × B, µ). Lett→ ∞in (6.3), lim t→∞PY0 t= (0,0)⊤and by Ergodic Theorems in [1], there exist a constant ρ∈Rsuch that lim t→∞1 t/integraldisplayt 0F(Θs(ω,Y0))ds=ρη0, which means lim t→∞PY(t,ω,Y0) t=ρη0. forµ-a.e.(ω,Y0)∈Ω×E. Thus, there is Ω∗⊂Ω withP(Ω∗) = 1 such that for any ω∈Ω∗, there isY∗ 0(ω)∈Esuch that lim t→∞PY(t,ω,Y∗ 0(ω)) t=ρη0. By Lemma 3.3, we have that for any n∈Nandω∈Ω∗, lim t→∞PY(t,ω,Y∗ 0(ω)±np0) t= lim t→∞PY(t,ω,Y∗ 0(ω))±np0 t=ρη0. (6.4) Now for any ω∈Ω∗and any Y∈E, there is n0(ω)∈Nsuch that PY∗ 0(ω)−n0(ω)p0≤PY≤PY∗ 0(ω)+n0(ω)p0 and there is a n0(ω)p0-periodic horizontal curve l0(ω) such that Y∗ 0(ω)−n0(ω)p0,Y,Y∗ 0(ω)+ n0(ω)p0∈l0(ω). Then by Lemma 6.2, we have PY(t,ω,Y∗ 0(ω)−n0(ω)p0)≤PY(t,ω,Y)≤PY(t,ω,Y∗ 0(ω)+n0(ω)p0), which together with (6.4) implies that for any ω∈Ω∗and any Y∈E, lim t→∞PY(t,ω,Y) t=ρη0. Consequently, for any a.e. ω∈Ω and any Y∈E, lim t→∞PY(t,ω,Y) t=ρη0. The theorem is thus proved. Corollary 6.4. Assume that a >4LF. Then the rotation number of the coupled second order system with white noises (1.1)exists. Proof.It follows from (6.1) and Theorem 6.3. 18References [1] L.Arnold, Random dynamical systems, Springer Monograp hs in Mathematics, Springer- Verlag, Berlin, 1998. [2] P.W. Bates, K. Lu and B. Wang, Random attractors for stoch astic reaction-diffusion equa- tions on unbounded domains, J. Diff. Eq. 246(2009), 845-869. [3] A. Bensoussan, and F. 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Appl. Dyn. Syst. 4(2005), 883-903. 20

2012-07-17

This paper is devoted to the study of the asymptotic dynamics of a class of coupled second order oscillators driven by white noises. It is shown that any system of such coupled oscillators with positive damping and coupling coefficients possesses a global random attractor. Moreover, when the damping and the coupling coefficients are sufficiently large, the global random attractor is a one-dimensional random horizontal curve regardless of the strength of the noises, and the system has a rotation number, which implies that the oscillators in the system tend to oscillate with the same frequency eventually and therefore the so called frequency locking is successful. The results obtained in this paper generalize many existing results on the asymptotic dynamics for a single second order noisy oscillator to systems of coupled second order noisy oscillators. They show that coupled damped second order oscillators with large damping have similar asymptotic dynamics as the limiting coupled first order oscillators as the damping goes to infinite and also that coupled damped second order oscillators have similar asymptotic dynamics as their proper space continuous counterparts, which are of great practical importance.

Asymptotic Dynamics of a Class of Coupled Oscillators Driven by White Noises

1207.3864v1

1 Nutation Resonance in Ferromagnets Mikhail Cherkasskii1,*, Michael Farle2,3, and Anna Semisalova2 1 Department of General Physics 1 , St. Petersburg State University , St. Petersburg , 199034, Russia 2 Faculty of Physics and Center of Nanointegration (CENIDE), University of Duisburg -Essen, Duisburg, 47057, Germany 3 Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Russia * Corresponding author: macherkasskii@hotmail.co m The inertial dynamic s of magnetization in a ferromagnet is investigated theoreticall y. The analytically derived dynamic response upon microwave excitation shows two pea ks: ferromagnetic and nutation resonances. The exact analytical expressions of frequency and linewidth of the magnetic nutation resonance are deduced from the frequency dependent susceptibility determined by the i nertial Landau -Lifshitz -Gilbert equation. The study shows that the dependence of nutation linewidth on the Gilbert precess ion damping has a minimum , which becomes more expressive with increas e of the applied magnetic field. PACS numbers: 76.50.+g, 78.47.jp, 75.50. -y I. INTRODUCTION Recently, the effects of inertia in the spin dynamics of ferromag nets were reported to cause nutation resonance [1- 12] at frequencies higher than the conventional ferromagnetic resonance . It was shown that inertia is responsible for the nutation , and that this type of motion should be considered together with magnetization precession in the applied magnetic field . Nutation in ferromagnets was confirmed experimentally only recently [2], since nutation and precession operate at substantially different time scales , and conventi onal microwave ferromagnetic resonance (FMR) spectroscopy techniques do not easily reach the high - frequency (sub-Terahertz) regime re quired to observe the inertia effect which in addition yields a much weaker signal . Similar to any other oscillatory system, t he magnetiza tion in a ferromagnet has resonant frequencies usually studied by ferromagnetic resonance [13,14]. The resonant eigenfrequency is determined by the magnetic parameters of the material and applied magnetic field . Including inertia of the magnetization in th e model description shows that nutation and precession are complementary to each other and several resonances can be generated . In this Letter , we concentrate on the investigation of the resonance characteristics of nutation. The investigation of nutation is connected to the progress made in studies of the spin dynamics at ultrashort time scales [15,16] . These successes led to the rapi d development of a new scientific field , the so -called ultrafast magnetism [17- 25]. The experimental as well as theoretical investigation of the inertial spin dynamics is at the very beginning , although it might be of significance for future high speed spintronics applications including ultrafast magnetic switching . Besides nutation driven by magnetization inertia, several other origins of nutation have been reported . Transient nutations (Rabi oscillations) have been widely investigate d in nuclear magnetic resonance [26] and electron spin resonance [27-29], they were recently addressed in ferromagnets [30]. A complex dynamics and Josephson nutation of a local spin 1/ 2s as well as large spin cluster embedded in the tunnel junction between ferromagnetic leads was shown to occur due to a coupling to Josephson current [31-33]. Low-frequency nutation was observ ed in nanomagnets exhibiting a non -linear FMR with the large - angle precession of magnetization where the onset of spin wave instabilities can be delay ed due to geometric confinement [34]. Nutation dynamics due to inertia of magnetization in ferromagnetic thin films was observed for the first time by Neeraj et al. [2]. The microscopic derivation of the magnetization inertia was performed in ref. [3-7]. A relation between the Gilbert damping constant and the inertia l regime characteristic time was elaborated in ref. [3]. The exchange interaction, damping, and moment of inertia can be calculated from first principles as shown in [7]. The study of inertia spin dynamics with a quantum approach in metallic ferromagnets was performed in [8]. In addition, nutation was theoretically analyzed as a part of magnetization dynamics in ferromagnetic nanostructure [9,10] and nanoparticles [11]. Despite these advances, exact analytical expressions for the high-frequency susceptibility including inertia had not been derived yet. In [35], the inertial regime was introduced in the framework of the mesoscopic nonequilibrium thermodynamics theory , and it was shown to be responsible for the nutation superimposed on the precession of magnetization . Wegrowe and Ciornei [1] discussed the 2 equivalence between the inertia in the dynamics of uniform precession and a spinning top within the framework of the Landau –Lifshitz –Gilbert equation generalized to the inertial regime. This equation was studied analytical ly and numerical ly [12,36]. Although the se reports provide numerical tools for obtaining resonance characteristics, the complexity of the numerical solution of differential equations did not allow to estimate the nutation frequency and linewidth accurately . Also in a recent remarkable paper [37] a novel collective excitation – the nutation wave – was reported, and the dispersion characteristics were derived wit hout discussion of the nutation resonance lineshapes and intensities. Thus, at present, there is a necessity to study the resonance properties of nutation in ferromagnet s, and this paper is devoted to this study. We performed the investigation based on the Landau -Lifshitz -Gilbert equation with the addition al inertia term and provide an analytical solution. It is well known that the Landau -Lifshitz -Gilbert equation allow s finding the susceptibility as the ratio between the time- varying magnetization and the time-varying driving magnetic field (see for exampl e [38,39] and references therein). This susceptibility describes well the magnetic response of a ferromagnet in the linear regime, that is a small cone angle of the precession . In this description , the ferromagnet usually is placed in a magnetic field big enough to align all atomic magnetic moments along the field , i.e., the ferromagnet is in the saturated state and the magnetization precess es. The applied driving magnetic field allows one to obser ve FMR as soon as the driving field frequency coincides with eigenfrequency of precession. Using the expression for susceptibility, one can elaborate such resonance characteristics as eigenfrequency and linewidth. We will present similar expressions for the dynamic susceptibility, taking nutation into account. II. SUSCEPTIBILITY The ferromagnet is subjected to a uniform bias magnetic field 0H acting along the z -axis and being strong enough to initiate the magnetic saturation state. The small time -varying magnetic field h is superimposed on the bias field. The coupling between impact and response, taking into account precession, damping, and nutation, is given by the Inertial Landau -Lifshitz -Gilbert ( ILLG) equation 2 2 0,effd d d dt M dt dt M M MMH (1) where is the gyromagnetic ratio, M the magnetization vector , 0M the magnetization at saturation, effH the vector sum of all magnetic fields, external and internal, acting upon the magnetization , the Gilbert damping , and the inertial relaxatio n time. For simplicity, we assume that the ferromagnet is infinite, i.e. there is no demagnetization correction , with negligible magnetocrystalline anisotropy , and only the externally applied field s contribute to the total field. Thus, the bias magnetic field 0H and signal field h are included in effH . We assume that the signal is small 0,hH hence the magnetization is directed along 0.H Our interest is to study the correlated dynamics of nutation and precession simultaneously; therefore we write the magnetization and magnetic field in the general ized form using the Fourier transformation 01ˆ , 2itt M z d e Mm (2) 01ˆ , 2it efft H z d e Hh (3) where ˆz is the unit vector along the z -axis. If we substitute these expressions in the ILLG equation and neglect the small terms, it leads to 00 211 22 ˆˆ ˆˆ .i t i td i e d e M z H z i z z m hm mm (4) By performing the Fourier transform and changing the order of integration , equation (4) becomes 00 21 2 1 2 ˆˆ ˆˆ ,it itd dt i e d dt e M z H z i z z m hm mm (5) where the integral representation of the Dirac delta function can be found. With the delta function, the equation (5) simplifies to 00 2ˆˆ ˆˆ .i M z H z i z z m h m mm (6) By projecting to Cartesian coordinates and introducing the circular variables for positive and negative circular polarization ,xy m m im ,xy h h ih one obtains 2 20, 0,HM HMm m i m m h m m i m m h (7) where 0 H H is the precession frequency and 0.M M The small -signal susceptibility follows from these equations : 3 2 2, , .M H M Him ih (8) It is seen that the susceptibility (8) is identical with the susceptibility for LLG equation , if one drops the inertial term , that is 0. Let us separate dispersive and d issipative parts of the susceptibility ,i 2 2, , , ,MH M MH MD D D D (9) 2 2 4 3 2 2 22 22 , 1 H H HD (10) 2 2 4 3 2 2 22 22 . 1 H H HD (11) The frequency dependence of the dissipative parts of susceptibilit ies and is shown in the Fig. 1. The plus and minus subscripts correspond to right -hand and left -hand direction of rotation. Since the denominators D and D are quartic polynomials, four critical points for either or can be expected . Two of them that are extrema with a clear physical meaning are plotted. In Fig. 1(a) the extremum , corresponding to FMR at 0 H H is shown . Due to the contribution of nutation , the frequency and linewidth of this resonance are slightly different from the ones of usual FMR . The resonance occurs for right -hand precession, i.e. positive polarization. In Fig. 1(b) the nutation resonance possessing negative polarization is presented. Note that the polarizations of ferromagnetic and nutation resonances are reverse d. III. APPROXI MAT ION FOR NUTATION FREQUENCY Let us turn to the description of an approximation of the nutation resonance frequency. If we equate the denominator D to zero, solve the resulting equation, we obtain the approximation from the real part of the roots. This is reasonable , since the numerator of is the linear function of , and we are interested in 1. Indeed , the equation 2 2 4 3 2 2 202 2 1 2H HH (12) has four roots that are complex conjugate in pairs 1,221 1 4 2,2H FMRiiw (13) 1,221 1 4 2.2H Niiw (14) FIG. 1. (Color online) (a) The FMR peak with nutation. (b) The nutation resonance. The calculation was performed for 1/ 2 28 GHz T , 00 1 T, M 00 100 mT, H 0.0065 and 1110 s. One should choose the same sign from the symbol in each formula , simultaneously . The real part of expression (13) gives the approximate frequency for FMR , but in negative numbers, so the sign should be inversed . The approximate frequency of FMR in positive numbers can be derived from equation 0. D The approximate nutation frequency is obtained by the real part of the expression (14). One takes half the sum of two conjugate roots 1,2,Nw neglect s the high terms , and obtains the nutation resonance frequency 1 1 2 .2NH w (15) Note that the expression of Nw is close to the approximation given in [36] at 1/ , H namely weak nu1 .H (16) The similarity of both approximations b ecomes clear , if we perform a Taylor series expansion and return to the notation ,H 2 2 2 3 3 3 2 weak nu 2 2 3 3 31 1 2 1 2 2 4 1,4 1 1 2 1.18 6H HH N H H HH Hw O O 4 IV. PRECISE EXPRESSIONS FOR FREQUENCY AND LINEWIDTH OF NUT ATION The analytical approach proposed in this Letter yields precise values of the frequency of nutation resonance and the full width at half maximum (FWHM) of the peak . The frequency is found by extremum, when the derivative of the dissipative part of susceptibilities (9) is zero 0. (17) It is enough to determine zeros of the n umerator of th e derivative , that are given by 2 2 4 3 2 2 23 4 2 1 0.HH (18) Let us use Ferrari's solution for this quartic equation and introduce the notation: 22 2 2 2 2 3 23 24 343 4, 2 1, 3,8 ,28, 3.16 25, 6rH rH rr r r r r r rr r r r r r r rrr r r rC E C C CEcA B BaA A BBbAA BB A AA (19) In Ferrari's method , one should determine a root of the nested depressed cubic equation . In the investigated case , the root is written 5,6r r r ray U V (20) where 32 3 2 32,27 4 2 ,3 12 1, .3 108 8r r r r r r r r rr rr r r rP Q QU PVU Pc Qaa abc (21) Thus, the precise value of the nutation frequency is given by 2 42 2 13 2 .2 2rr r N r r rr rry A baa ay yB (22) The performed analysis shows that approximate value of nutation resonance frequency is close to precise value. The linewidth of the nutation resonance is found at a half peak height. If one denotes the maximum by ,N X the equation which determines frequencies at half magnitude is 2 2 4 3 2 2 212 2 12 2 0.H H H MX (23) We repeat the procedure for finding solu tions with Ferrari's method introducing the new notations 22 2 2 2 2 3 23 24 3 4 21 2 , 12 1 ,2 1 2 3,8 ,28 3.16 2, 56, 4lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lw lwH HM H lw lw lw lwA B BaA A BBbAA B B B D A AX X CX DX EX C CD A CEcA A (24) A root of the nested depressed cubic equation lwy must be found in the same way as provided in (20) with the corresponding replacement of variables, i.e. subscript r is replaced by lw. The difference between two adjacent roots gives the nutation linewidth 23 2 . 2lw N lw lw lw lwbay a y (25) The explicit expression for the linewidth can be written using the equations (19)-(25). FIG. 2. (Color online) The dependence of the nutation linewidth on the inertial relaxat ion time for 00 100 mT, H 00 1 T, M and 0.0065. 5 The effect of the inertial relaxation time on the nutation linewidth is shown in Fig. 2. One can see that increasing inertial relaxation time leads to narrowing of the linewidth. This behavior is expected and is consistent with the traditional view that decreasing of losses results in narrowing of linewidth. FIG. 3. (Color online) The dependence of nutation resonance linewidth on precession Gilbert damping parameter at different magnetic fields 0H for 00 1 T M and 1110 s. Since the investig ated oscillatory system implemen ts simultaneous two types of motions , it is of interest to study the influence of the Gilbert precession damping parameter on the nutation resonance linewidth. The result is presented in Fig. 3 and is valid for ferromagnets with vanishing anisotropy. One sees that the dependence of N on shows a minimum that becomes more expressive with increasing bias magnetic field. In other words, t he linewidth is parametrized by the magnitude of field. This non-trivial behavior of linewidth relates with the nature of th is oscillatory system, which performs two coupled motions. To elucidate the non-trivial behavior , one can consider the susceptibility (9) in the same way as it is usually performed for the forced harmonic oscillator with damping [40]. For this oscillator , the linewidth can be direct ly calculated from the denominator of the response expression once the driving frequency is equal to eigenfreq uency. In the investigated case of magnetization with inertia , the response expression is (9) with denominator s (10) and (11) written as 2 2 4 3 2 2 22 21 . 2H H HD (26) Since the applied magnetic field is included in this expression as 0,H H the linewi dth depends on the field. The obtained result can be generalized to a fin ite sample with magnetocrystalline anisotropy with method of effective demagnetizing factors [41,42] . In this case the bias magnetic field 0H denotes an external field and in the final expressions this field should be replaced by 0 0 0ˆˆ ,i a d NN H H M where ˆ aN is the anisotropy demagnetizing tensor and ˆ dN is the shape demagnetizing tensor. V. CONCLUSION In summary, we derived a general analytical expression for the linewidth and f requency of nutation resonance in ferromagnets, depending on magnetization, the Gilbert damping, the inertial relaxation time and applied magnetic field. We show the nutation linewidth can be tuned by the applied magnetic field , and this tunability breaks the direct relation between losses and the linewidth. This for example leads to the appearance of a minimum in the nutation resonance linewidth for the damping parameter around 0.15. 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2020-08-27

The inertial dynamics of magnetization in a ferromagnet is investigated theoretically. The analytically derived dynamic response upon microwave excitation shows two peaks: ferromagnetic and nutation resonances. The exact analytical expressions of frequency and linewidth of the magnetic nutation resonance are deduced from the frequency dependent susceptibility determined by the inertial Landau-Lifshitz-Gilbert equation. The study shows that the dependence of nutation linewidth on the Gilbert precession damping has a minimum, which becomes more expressive with increase of the applied magnetic field.

Nutation Resonance in Ferromagnets

2008.12221v3

arXiv:2009.11244v1 [math.AP] 23 Sep 2020REMARK ON THE EXPONENTIAL DECAY OF THE SOLUTIONS OF THE DAMPED WAVE EQUATION GIOVANNI CIMATTI Abstract. A condition which guaranties the exponential decay of the so lu- tions of the initial-boundary value problem for the damped w ave equation is proved. A method for the effective computability of the coeffic ient of expo- nential decay is also presented. 1.introduction Let Ω be a bounded and open subset of RNand Γ its boundary. Consider the initial-boundary value problem (1.1) utt−∆u+aut= 0 in Ω ×(0,∞) (1.2) u= 0 on Γ ×[0,∞) (1.3) u(x,0) =uo(x) forx∈Ω (1.4) ut(x,0) =u1(x) forx∈Ω, whereais a positive constant, uo(x) andu1(x) are given functions satisfying the usual regularity and compatibility conditions implied in the assumption w e make thatu(x,t) is a classical solution of problem (1.1)-(1.4). P.H. Rabinowitz has proven [1] that the ”energy” i.e./integraltext Ω|∇u(x,t)|2dxassociated with the solution of (1.1)-(1.4) decays exponentially as t→ ∞. A different proof of this fact, important in applications, has been given by R. Temam [2], [3] and by R Temam and J.M . Ghidaglia in [4]. In this paper we consider the initial-boundary value prob lem (1.1)-(1.4) with the more general equation (1.5) utt−∆u+σ(x,t)ut= 0 in Ω ×(0,∞), whereσ(x,t) is supposed to be continuous and positive in Ω ×(0,∞). In general there is no exponential decay, as the following example shows. Let N= 1, Ω=(1 ,2) andσ(x,t) =2t x2−3x+2. The solution is now given by u(x,t) =t(x2−3x+ 2) for which there is no exponential decay. We prove, with fully elementary means, and 2010Mathematics Subject Classification. 35B35. Key words and phrases. Damped wave equation, Gronwall’s lemma, Exponential decay , Coef- ficient of exponential decay . 12 GIOVANNI CIMATTI without using the semigroup’s theory, that a sufficient condition for having the exponential decay is (1.6) 0 < σ0≤σ(x,t)≤σ1<∞, whereσ0= infσ(x,t) andσ1= supσ(x,t). We also give an effective way to compute the decay exponent in terms of the parameters σ0,σ1,λ1, whereλ1is the first eigenvalue of the laplacian in Ω with zero boundary conditions. 2.exponential decay Letu(x,t) be a regular solution of (1.5), (1.2), (1.3) and (1.4) with σ(x,t) sat- isfying (1.6). Define (2.1) v(x,t) =ut(x,t)+ǫu(x,t) with 0 < ǫ≤σ0. We have (2.2) utt=vt−ǫv+ǫ2u. Substituting (2.2) in (1.5) we obtain (2.3) vt+(σ−ǫ)v+(ǫ2−ǫσ)u−∆u= 0. Let us multiply (2.3) by v. We arrive at (2.4)/integraldisplay Ωvtvdx+/integraldisplay Ω(σ−ǫ)v2dx−/integraldisplay Ωv∆udx+/integraldisplay Ω(ǫ2−ǫσ)uvdx= 0. Sincev= 0 on Γ ×[0,∞), integrating by parts in (2.4) we have (2.5)1 2d dt/integraldisplay Ωv2dx+/integraldisplay Ω(σ−ǫ)v2dx+/integraldisplay Ω∇v·∇udx−ǫ/integraldisplay Ω(σ−ǫ)uvdx= 0. By (2.1) ∇v=∇ut+ǫ∇u, hence from (2.5), (2.6)1 2d dt/integraldisplay Ω/parenleftbig |∇u|2+v2/parenrightbig dx+/integraldisplay Ω(σ−ǫ)v2dx+ǫ/integraldisplay Ω|∇u|2dx−ǫ/integraldisplay Ω(σ−ǫ)uvdx= 0. Recalling (2.1) we have (2.7)/integraldisplay Ω(σ−ǫ)v2dx≥/integraldisplay Ω(σ0−ǫ)v2dx≥0. To control from below the last integral in (2.6) we estimate from abo veǫ/integraltext Ω(σ− ǫ)uvdx. Letλ1>0 be the first eigenvalue of the laplacian with zero boundary condition on Γ. Using the Cauchy-Schwartz and Poincar inequalities w e obtain, withη >0, (2.8) ǫ/integraldisplay Ω(σ−ǫ)uvdx≤ǫ(σ1−ǫ) 2η/integraldisplay Ω|v|2dx+ǫ(σ1−ǫ)η 2λ1/integraldisplay Ω|∇u|2dx. Changing sign in (2.8) and substituting in (2.6) we obtainTHE EXPONENTIAL DECAY FOR THE DAMPED WAVE EQUATION 3 (2.9) 1 2d dt/integraldisplay Ω/parenleftbig |∇u|2+v2/parenrightbig dx+/bracketleftbigg ǫ+ǫ(ǫ−σ1)η 2λ1/bracketrightbigg/integraldisplay Ω|∇u|2dx+/bracketleftbigg σ0−ǫ+ǫ(ǫ−σ1) 2η/bracketrightbigg/integraldisplay Ω|v|2dx≤0. Defining (2.10) f(ǫ,η,σ1,λ1) =ǫ+ǫ(ǫ−σ1)η 2λ1, g(ǫ,η,σ0,σ1) =σ0−ǫ+ǫ(ǫ−σ1) 2η (2.9) becomes (2.11) 1 2d dt/integraldisplay Ω/parenleftbig |∇u|2+v2/parenrightbig dx+f(ǫ,η,σ1,λ1)/integraldisplay Ω|∇u|2dx+g(ǫ,η,σ0,σ1)/integraldisplay Ω|v|2dx≤0. We wish to find couples ( ǫ,η) satisfying the conditions 0 < ǫ < σ 0and 0< ηsuch that for every choice of the parameters σ0,σ1,λ1,f(ǫ,η,σ,λ1) andg(ǫ,η,s0,σ1) are positive and equal. Among these couples we must find the ( ǫ∗,η∗) which gives to f(andg) the greatest possible value. This will permits to apply to (2.11) the Gronwall’s inequality. The exponential decay with the best decay exp onent then follows easily. Let us consider in the plane ǫ,η, (ǫ >0) the families of curves (2.12) f(ǫ,η,σ1,λ1)−g(ǫ,η,σ0,σ1) = 0. Sinceǫλ1>0, (2.12) is equivalent to (2.13) ǫ(ǫ−σ1)η2+2λ1(2ǫ−σ0)η+λ1ǫ(σ1−ǫ) = 0. Solving (2.13) with respect to ηwe obtain two branches of solutions. The one of interest to us is (2.14) η=/radicalbig (σ0−2ǫ)2λ2 1+ǫ2(σ1−ǫ)2λ1+(2ǫ−σ0)λ1 ǫ(σ1−ǫ). Inserting (2.14) in f(ǫ,η,σ1,λ1) (or ing(ǫ,η,σ0,σ1)) we obtain (2.15) F(ǫ,σ0,σ1,λ1) =σ0 2−/radicalbig (σ0−2ǫ)2λ2 1+ǫ2(σ1−ǫ)2λ1 2λ1. We study (2.16) α=F(ǫ,σ0,σ1,λ1) as a function of ǫdepending of the three parameters σ0,σ1,λ1. We have Lemma 2.1. Letσ1> σ0>0andλ1>0. If¯ǫis a solution of the equation F(ǫ,σ0,σ1,λ1) = 0, then¯ǫ < σ0. Proof.By contradiction, suppose ¯ ǫ=σ0+γ2,γ/negationslash= 0 is a solution. We have (2.17)σ2 0λ2 1= (−σ0−2γ2)2λ2 1+(σ0+γ2)2(σ1−σ0−γ2)2λ1>(σ0+2γ2)2λ2 1> σ2 0λ2 1. On the other hand, if ¯ ǫ=σ0, we have4 GIOVANNI CIMATTI (2.18) σ2 0λ2 1=σ0λ2 1+σ2 0(σ1−σ0)2λ1> σ2 0λ2 1. /square For every values of the parameters ǫ= 0 is a solution of F(ǫ,σ0,σ1,λ1) = 0. Moreover, lim ǫ→±∞F(ǫ,σ0,σ1,λ1) =−∞and (2.19) F′(ǫ,σ0,σ1,λ1) =2(σ0−2ǫ)λ1−ǫ(σ1−ǫ)2+ǫ2(σ1−ǫ) 2/radicalbig (σ0−2ǫ)2λ2 1+ǫ2(σ1−ǫ)2λ1. We have F′(0,σ0,σ1,λ1) = 1 for every value of the parameters σ0,σ1andλ1. Therefore, in a small interval (0 ,β),β >0Fis positive for every value of the parameters. To study completely F′we make the following elementary discussion. The cubic expression (2.20) −2ǫ3+3σ1ǫ2−(4λ1+σ2 1)ǫ+2λ1σ0 has the same sign of F′(ǫ,σ0,σ1,λ1). On the other hand, D= 3σ2 1−24λ1is the discriminant of the derivative of (2.20) Thus, if D <0, (2.19) is always strictly decreasing. As a consequence, F′(ǫ,σ0,σ1,λ1) vanishes in exactly one point ¯ ǫand 0<¯ǫ. This implies that F(ǫ,σ0,σ1,λ1), which always vanishes for ǫ= 0 and is positive immediately to the right of ǫ= 0, has a positive absolute maximum in ǫ∗and vanishes in ǫ= ¯ǫ >¯ǫ∗. IfD= 0 a bifurcation occurs, and, when D >0, F(ǫ,σ0,σ1,λ1)hastworelativemaxima’sandonerelativeminimum. Themaximum immediately to the right of ǫ= 0 is certainly positive, the second maximum may or may not be positive. Let ǫ∗be the point of absolute maximum of F(ǫ,σ0,σ1). Define (2.21) α∗=F(ǫ∗,σ0,σ1,λ1) and (2.22) η∗=/radicalbig (σ0−2ǫ∗)2λ2 1+ǫ∗2(σ1−ǫ∗)2λ1+(2ǫ∗−σ0)λ1 ǫ∗(s1−ǫ∗). With this choice of ǫandηwe havef(ǫ∗,η∗,σ1,λ1) =g(ǫ∗,η∗,σ0,σ1) =α∗>0. Therefore, (2.11) becomes (2.23)1 2d dt/integraldisplay Ω/parenleftbig |∇u|2+v2/parenrightbig dx+α∗/integraldisplay Ω(|∇u|2+v2)dx≤0. Using the Gronwall’s inequality, we obtain (2.24)/integraldisplay Ω/parenleftbig |∇u|2+v2/parenrightbig dx≤/integraldisplay Ω(|∇u(x,0)|2+|v(x,t)|2)dx e−2α∗t. The right hand side of (2.24) can be computed in terms of the initial an d bound- ary data satisfied by u(x,t). For, from (1.3) we obtain ∇u(x,0) =∇u0(x). More- over, since v(x,t) =ut(x,t)+ǫ∗u(x,t), we have v(x,0) =u1(x)+ǫ∗u0(x). HenceTHE EXPONENTIAL DECAY FOR THE DAMPED WAVE EQUATION 5 /integraldisplay Ω|∇u|2dx≤/integraldisplay Ω|∇u0(x)|2+|u1(x)+ǫ∗u0(x)|2dx e−2α∗t. This proves the exponential decay. Remark . For the regular solutions of the non-linear equation (2.25) utt−∆u+m(u)ut= 0 in Ω ×(0,∞) the exponential decay can be obtained, with minor changes, if we ke ep the initial and boundary conditions (1.2), (1.3) and (1.4) and assume m∈C0(R1) and 0< m0≤m(u)≤m1. To see that, we simply define σ(x,t) =m(u(x,t)) where u(x,t) is the regular solution of the non linear problem (2.25), (1.2), (1.3) an d (1.4) References 1. P.H. Rabinowitz, Periodic solutions of nonlinear hyperb olic partial differential equations, Comm. Pure Appl. Math., 22, 145-205, (1967). 2. R. Temam, Infinite-Dimensional Dynamical Systems in Mech anics and Physics, Springer- Verlag, (1988). 3. R. Temam, Behaviour at time t= 0 of the solutions of semilinear evolutions equations, Jou r. Diff. Eqs., 43, 73-92, (1982). 4. Ghidaglia and R. Temam, Attractors for damped nonlinear h yperbolic equations, J. Math. Pures Appl., 66, 273-319, (1987). Department of Mathematics, Largo Bruno Pontecorvo 5, 56127 Pisa Italy E-mail address :cimatti@dm.unipi.it

2020-09-23

A condition which guaranties the exponential decay of the solutions of the initial-boundary value problem for the damped wave equation is proved. A method for the effective computability of the coefficient of exponential decay is also presented.

Remark on the exponential decay of the solutions of the damped wave equation

2009.11244v1

arXiv:1510.03571v2 [cond-mat.mtrl-sci] 19 Nov 2015Nonlocal torque operators in ab initio theory of the Gilbert damping in random ferromagnetic alloys I. Turek∗ Institute of Physics of Materials, Academy of Sciences of th e Czech Republic, ˇZiˇ zkova 22, CZ-616 62 Brno, Czech Republic J. Kudrnovsk´ y†and V. Drchal‡ Institute of Physics, Academy of Sciences of the Czech Repub lic, Na Slovance 2, CZ-182 21 Praha 8, Czech Republic (Dated: July 5, 2018) We present an ab initio theory of theGilbert dampingin substitutionally disorder ed ferromagnetic alloys. The theory rests on introduced nonlocal torques whi ch replace traditional local torque operators in the well-known torque-correlation formula an d which can be formulated within the atomic-sphereapproximation. Theformalism is sketchedin asimpletight-bindingmodel andworked out in detail in the relativistic tight-binding linear muffin -tin orbital (TB-LMTO) method and the coherent potential approximation (CPA). The resulting nonlocal torques are represented by nonrandom, non-site-diagonal and spin-independent matri ces, which simplifies the configuration averaging. The CPA-vertex corrections play a crucial role f or the internal consistency of the theory and for its exact equivalence to other first-principles appr oaches based on the random local torques. This equivalence is also illustrated by the calculated Gilb ert damping parameters for binary NiFe and FeCo random alloys, for pure iron with a model atomic-lev el disorder, and for stoichiometric FePt alloys with a varying degree of L1 0atomic long-range order. PACS numbers: 72.10.Bg, 72.25.Rb, 75.78.-n I. INTRODUCTION The dynamics of magnetization of bulk ferromagnets, utrathin magnetic films and magnetic nanoparticles rep- resents an important property of these systems, espe- cially in the context of high speed magnetic devices for data storage. While a complete picture of magnetization dynamics including, e.g., excitation ofmagnonsand their interaction with other degrees of freedom, is still a chal- lenge for the modern theory of magnetism, remarkable progresshas been achieved during the last years concern- ing the dynamics of the total magnetic moment, which can be probed experimentally by means of the ferromag- netic resonance1or by the time-resolved magneto-optical Kerr effect.2Time evolution of the macroscopic magne- tization vector Mcan be described by the well-known Landau-Lifshitz-Gilbert (LLG) equation3,4 dM dt=Beff×M+M M×/parenleftbigg α·dM dt/parenrightbigg ,(1) whereBeffdenotes an effective magnetic field (with the gyromagnetic ratio absorbed) acting on the magnetiza- tion,M=|M|, and the quantity α={αµν}denotes a symmetric 3 ×3 tensor of the dimensionless Gilbert damping parameters ( µ,ν=x,y,z). The first term in Eq. (1) defines a precession of the magnetization vector around the direction of the effective magnetic field and the second term describes a damping of the dynamics. The LLG equation in itinerant ferromagnets is appropri- ate for magnetization precessions very slow as compared to precessions of the single-electron spin due to the ex- change splitting and to frequencies of interatomic elec- tron hoppings.A large number of theoretical approaches to the Gilbert damping has been workedout during the last two decades; here we mention only schemes within the one- electron theory of itinerant magnets,5–20where the most important effects of electron-electron interaction are cap- tured by means of a local spin-dependent exchange- correlation (XC) potential. These techniques can be naturally combined with existing first-principles tech- niques based on the density-functional theory, which leads to parameter-freecalculations of the Gilbert damp- ing tensor of pure ferromagnetic metals, their ordered and disordered alloys, diluted magnetic semiconductors, etc. One part of these approaches is based on a static limit of the frequency-dependent spin-spin correlation function of a ferromagnet.5–8,15,16Other routes to the Gilbert damping employ relaxations of occupation num- bers of individual Bloch electron states during quasi- static nonequilibrium processes or transition rates be- tween different states induced by the spin-orbit (SO) interaction.9–12,14,20The dissipation of magnetic energy accompanying the slow magnetization dynamics, evalu- ated within a scattering theory or the Kubo linear re- sponse formalism, leads also to explicit expressions for the Gilbert damping tensor.13,17–19Most of these formu- lations yield relations equivalent to the so-called torque- correlation formula αµν=−α0Tr{Tµ(G+−G−)Tν(G+−G−)},(2) inwhich thetorqueoperators Tµareeither duetothe XC or SO terms of the one-electron Hamiltonian. In Eq. (2), which has a form of the Kubo-Greenwood formula and is valid for zero temperature of electrons, the quantity α0is related to the system magnetization (and to fundamental2 constants and units used, see Section IIB), the trace is taken over the whole Hilbert space of valence electrons, andthesymbols G±=G(EF±i0)denotetheone-particle retarded and advanced propagators (Green’s functions) at the Fermi energy EF. Implementation of the above-mentioned theories in first-principles computational schemes proved opposite trends of the intraband and interband contributions to the Gilbert damping parameter as functions of a phe- nomenological quasiparticle lifetime broadening.7,11,12 These qualitative studies have recently been put on a more solid basis by considering a particular mechanism of the lifetime broadening, namely, a frozen temperature- induced structural disorder, which represents a realistic model for a treatment of temperature dependence of the Gilbert damping.21,22This approach explained quanti- tatively the low-temperature conductivity-like and high- temperature resistivity-like trends of the damping pa- rameters of iron, cobalt and nickel. Further improve- mentsofthemodel, includingstatictemperature-induced randomorientationsoflocalmagneticmoments, haveap- peared recently.23 Theab initio studies have also been successful in re- production and interpretation of values and concentra- tion trends of the Gilbert damping in random ferromag- netic alloys, such as the NiFe alloy with the face-centered cubic (fcc) structure (Permalloy)17,22and Fe-based al- loys with the body-centered cubic (bcc) structure (FeCo, FeV,FeSi).19,22,24Otherstudiesaddressedalsotheeffects of doping the Permalloy and bcc iron by 5 dtransition- metal elements19,20,22and of the degree of atomic long- range order in equiconcentration FeNi and FePt alloys with the L1 0-type structures.20Recently, an application to halfmetallic Co-based Heusler alloys has appeared as well.25The obtained results revealed correlations of the damping parameter with the density of states at the Fermienergyandwiththesizeofmagneticmoments.22,24 In a one-particle mean-field-like description of a ferro- magnet, the total spin is not conserved due to the XC field and the SO interaction. The currently employed formsofthetorqueoperators Tµinthe torque-correlation formula (2) reflect these two sources; both the XC- and the SO-induced torques are local and their equivalence for the theory of Gilbert damping has been discussed by several authors.15,16,26In the case of random alloys, this equivalence rests on a proper inclusion of vertex cor- rections in the configuration averaging of the damping parameters αµνas two-particle quantities. The purpose of the present paper is to introduce an- other torque operator that can be used in the torque- correlationformula(2) andto discussits properties. This operatoris due to intersiteelectronhopping andit is con- sequently nonlocal; in contrast to the local XC- and SO- induced torques which are random in random crystalline alloys, the nonlocal torque is nonrandom, i.e., indepen- dent on the particular configuration of a random alloy, which simplifies the configuration averaging of Eq. (2). We show that a similar nonlocal effective torque appearsin the fully relativistic linear muffin-tin orbital (LMTO) method in the atomic-sphere approximation (ASA) used recently for calculations of the conductivity tensor in spin-polarized random alloys.27,28Here we discuss theo- retical aspects of the averaging in the coherent-potential approximation (CPA)29,30and illustrate the developed ab initio scheme byapplicationsto selected binaryalloys. We also compare the obtained results with those of the LMTO-supercell technique17and with other CPA-based techniques, the fully relativisticKorringa-Kohn-Rostoker (KKR) method19,22and the LMTO method with a sim- plified treatment of the SO interaction.20 The paper is organized as follows. The theoretical for- malism is contained in Section II, with a general discus- sion of various torque operators and results of a simple tight-binding model presented in Section IIA. The fol- lowingSection IIB describes the derivation of the LMTO torque-correlation formula with nonlocal torques; tech- nical details are left to Appendix A concerning linear- response calculations with varying basis sets and to Ap- pendix B regarding the LMTO method for systems with a tilted magnetization direction. Selected formal proper- ties of the developed theory are discussed in Section IIC. Applications of the developed formalism can be found in Section III. Details of numerical implementation are listed in Section IIIA followed by illustrating examples forsystemsofthreedifferent kinds: binarysolidsolutions of 3dtransition metals in Section IIIB, pure iron with a simple model of random potential fluctuations in Section IIIC, and stoichiometric FePt alloys with a partial long- range order in Section IIID. The main conclusions are summarized in Section IV. II. THEORETICAL FORMALISM A. Torque-correlation formula with alternative torque operators The torque operators Tµentering the torque- correlation formula (2) are closely related to compo- nents of the time derivative of electron spin. For spin- polarized systems described by means of an effective Schr¨ odinger-Paulione-electronHamiltonian H, actingon two-componentwavefunctions, thecompletetimederiva- tive of the spin operator is given by the commutation re- lationtµ=−i[σµ/2,H], where ¯ h= 1 is assumed and σµ (µ=x,y,z) denote the Pauli spin matrices. Let us write the Hamiltonian as H=Hp+Hxc, whereHpincludes all spin-independent terms and the SO interaction (Hamil- tonian of a paramagnetic system) while Hxc=Bxc(r)·σ denotes the XC term due to an effective magnetic field Bxc(r). The complete time derivative (spin torque) can then be written as tµ=tso µ+txc µ, where tso µ=−i[σµ/2,Hp], txc µ=−i[σµ/2,Hxc].(3) As discussed, e.g., in Ref. 15, the use of the complete torquetµinthetorque-correlationformula(2)leadsiden-3 tically to zero; the correct Gilbert damping coefficients αµνfollow from Eq. (2) by using either the SO-induced torquetso µ, or the XC-induced torque txc µ. Note that only transverse components (with respect to the easy axis of the ferromagnet)of the vectors tsoandtxcare needed for the relevant part of the Gilbert damping tensor (2). The equivalence of both torque operators (3) for the Gilbert damping can be extended. Let us consider a sim- ple system described by a model tight-binding Hamilto- nianH, written now as H=Hloc+Hnl, where the first termHlocis a lattice sum of local atomic-like terms and the nonlocal second term Hnlincludes all intersite hop- ping matrix elements. Let us assume that all effects of the SO interaction and XC fields are contained in the local term Hloc, so that the hopping elements are spin- independent and [ σµ,Hnl] = 0. (Note that this assump- tion, often used in model studies, is satisfied only ap- proximatively in real ferromagnets with different widths of the majority and minority spin bands.) Let us write explicitly Hloc=/summationtext R(Hp R+Hxc R), whereRlabelsthe lat- tice sites and where Hp Rcomprises the spin-independent part and the SO interaction of the Rth atomic poten- tial while Hxc Ris due to the local XC field of the Rth atom. The operators Hp RandHxc Ract only in the sub- space of the Rth site; the subspaces of different sites are orthogonal to each other. The total spin operator can be written as σµ/2 = (1/2)/summationtext RσRµ, where the local operator σRµis the projection of σµon theRth sub- space. Let us assume that each term Hp Ris spherically symmetric and that Hxc R=Bxc R·σR, where the effec- tive field Bxc Rof theRth atom has a constant size and direction. Let us introduce local orbital-momentum op- eratorsLRµand their counterparts including the spin, JRµ=LRµ+ (σRµ/2), which are generators of local infinitesimal rotations with respect to the Rth lattice site, and let us define the corresponding lattice sums Lµ=/summationtext RLRµandJµ=/summationtext RJRµ=Lµ+(σµ/2). Then the local terms Hp RandHxc Rsatisfy, respectively, commu- tation rules [ JRµ,Hp R] = 0 and [ LRµ,Hxc R] = 0. By using the above assumptions and definitions, the XC-induced spin torque (3) due to the XC term Hxc=/summationtext RHxc Rcan be reformulated as txc µ=−i/summationdisplay R[σRµ/2,Hxc R] =−i/summationdisplay R[JRµ,Hxc R] (4) =−i/summationdisplay R[JRµ,Hp R+Hxc R] =−i[Jµ,Hloc]≡tloc µ. The last commutator defines a local torque operator tloc µ due to the local part of the Hamiltonian Hlocand the op- eratorJµ,incontrasttothespinoperator σµ/2inEq.(3). Let us define the complementary nonlocal torque tnl µdue to the nonlocal part of the Hamiltonian Hnl, namely, tnl µ=−i[Jµ,Hnl] =−i[Lµ,Hnl], (5) and let us employ the fact that the complete time deriva- tive of the operator Jµ, i.e., the torque ˜tµ=−i[Jµ,H] = tloc µ+tnl µ, leads identically to zero when used in Eq. (2).This fact implies that the Gilbert damping parame- ters can be also obtained from the torque-correlation formula with the nonlocal torques tnl µ. These torques are equivalent to the original spin-dependent local XC- or SO-induced torques; however, the derived nonlocal torques are spin-independent, so that commutation rules [tnl µ,σν] = 0 are satisfied. Inordertoseetheeffect ofdifferent formsofthe torque operators, Eqs. (3) and (5), we have studied a tight- binding model of p-orbitals on a simple cubic lattice with the ground-state magnetization along zaxis. The local (atomic-like) terms of the Hamiltonian are specified by the XC term bσRzand the SO term ξLR·σR, which are added to a random spin-independent p-level at en- ergyǫ0+DR, whereǫ0denotes the nonrandom center of thep-band while the random parts DRsatisfy configu- ration averages /an}bracketle{tDR/an}bracketri}ht= 0 and /an}bracketle{tDR′DR/an}bracketri}ht=γδR′Rwith the disorder strength γ. The spin-independent nonlocal (hopping) part of the Hamiltonian has been confined to nonrandom nearest-neighbor hoppings parametrized by twoquantities, W1(ppσhopping) and W′ 1(ppπhopping), see, e.g., page 36 of Ref. 31. The particular values have been set to b= 0.3,ξ= 0.2,EF−ǫ0= 0.1,γ= 0.05, W1= 0.3 andW′ 1=−0.1 (the hoppings were chosen such that the band edges for ǫ0=b=ξ=γ= 0 are±1). Theconfigurationaverageofthe propagators /an}bracketle{tG±/an}bracketri}ht=¯G± and of the torque correlation (2) was performed in the self-consistentBornapproximation(SCBA)includingthe vertex corrections. Since all three torques, Eqs. (3) and (5), are nonrandom operators in our model, the only rel- evant component of the Gilbert damping tensor, namely αxx=αyy=α, could be unambiguously decomposed in the coherent part αcohand the incoherent part αvcdue to the vertex corrections. The results are summarized in Fig. 1 which displays the torque correlation α/α0as a function of the SO cou- plingξ(Fig. 1a) and the XC field b(Fig. 1b). The total valueα=αcoh+αvcis identical for all three forms of the torque operator, in contrast to the coherent parts αcohwhich exhibit markedly different values and trends as compared to each other and to the total α. This re- sult is in line with conclusions drawn by the authors of Ref. 15, 16, and 26 proving the importance of the ver- tex corrections for obtaining the same Gilbert damping parameters from the SO- and XC-induced torques. The only exception seems to be the case of the SO splitting much weaker than the exchange splitting, where the ver- tex corrections for the SO-induced torque can be safely neglected, see Fig. 1a. This situation, encountered in 3dtransition metals and their alloys, has been treated with the SO-induced torque on an ab initio level with ne- glectedvertexcorrectionsinRef. 11and12. Onthe other hand, the use of the XC-induced torque calls for a proper evaluation of the vertex corrections; their neglect leads toquantitativelyandphysicallyincorrectresultsasdocu- mented by recent first-principles studies.19,22The vertex correctionsareindispensablealsoforthe nonlocaltorque, in particular for correct vanishing of the total torque cor-4 0 2 4 0 0.1 0.2torque correlation spin-orbit coupling(a) totcoh-nl coh-xc coh-so 0 2 4 0 0.1 0.2 0.3torque correlation exchange field(b) totcoh-nl coh-xc coh-so FIG. 1. (Color online) The torque correlation α/α0, Eq. (2), in a tight-binding p-orbital model treated in the SCBA as a function of the spin-orbit coupling ξ(a) and of the ex- change field b(b). The full diamonds display the total torque correlation (tot) and the open symbols denote the coherent contributions αcoh/α0calculated with the SO-induced torque (coh-so), the XC-induced torque (coh-xc), Eq. (3), and the nonlocal torque (coh-nl), Eq. (5). relation both in the nonrelativistic limit ( ξ→0, Fig. 1a) and in the nonmagnetic limit ( b→0, Fig. 1b). Finally, let us discuss briefly the general equivalence of the SO- and XC-induced spin torques, Eq. (3), in the fully relativistic four-component Dirac formalism.32,33 The Kohn-Sham-Dirac Hamiltonian can be written as H=Hp+Hxc, whereHp=cα·p+mc2β+V(r) and Hxc=Bxc(r)·βΣ, wherecis the speed of light, mde- notes the electron mass, p={pµ}refers to the momen- tum operator, V(r) is the spin-independent part of the effective potential and the α={αµ},βandΣ={Σµ} arethe well-known4 ×4matricesofthe Diractheory.34,35Then the XC-induced torque is txc=Bxc(r)×βΣ, which is currently used in the KKR theory of the Gilbert damping.19,22The SO-induced torque is tso=p×cα, i.e., it is given directly by the relativistic momentum ( p) and velocity ( cα) operators. One can see that the torque tsoislocalbutindependent oftheparticularsystemstud- ied. A comparison of both alternatives, concerning the total damping parameters as well as their coherent and incoherent parts, would be desirable; however, this task is beyond the scope of the present study. B. Effective torques in the LMTO method In ourab initio approach to the Gilbert damping, we employ the torque-correlation formula (2) with torques derived from the XC field.15,19,22The torque operators are constructed by considering infinitesimal deviations of the direction of the XC field of the ferromagnet from its equilibrium orientation, taken asa reference state. These deviations result from rotations by small angles around axesperpendiculartothe equilibrium direction ofthe XC field; componentsofthetorqueoperatorarethengivenas derivatives of the one-particle Hamiltonian with respect to the rotation angles.36 For practical evaluation of Eq. (2) in an ab initio tech- nique (such as the LMTO method), one has to consider a matrix representation of all operators in a suitable or- thonormal basis. The most efficient techniques of the electronic structure theory require typically basis vectors tailored to the system studied; in the present context, this leads naturally to basis sets depending on the angu- lar variables needed to define the torque operators. Eval- uation of the torque correlation using angle-dependent bases is discussed in Appendix A, where we prove that Eq. (2) can be calculated solely from the matrix ele- ments of the Hamiltonian and their angular derivatives, see Eq. (A7), whereas the angular dependence of the ba- sis vectorsdoes not contribute directly to the final result. The relativistic LMTO-ASA Hamiltonian matrix for the reference system in the orthogonal LMTO represen- tation is given by37–39 H=C+(√ ∆)+S(1−γS)−1√ ∆, (6) where the C,√ ∆ andγdenote site-diagonal matrices of the standard LMTO potential parameters and Sis the matrix of canonical structure constants. The change of the Hamiltonian matrix Hdue to a uniform rotation of the XC field is treated in Appendix B; it is sum- marized for finite rotations in Eq. (B7) and for angu- lar derivatives of Hin Eqs. (B8) and (B9). The resol- ventG(z) = (z−H)−1of the LMTO Hamiltonian (6) for complex energies zcan be expressed using the auxil- iary resolvent g(z) = [P(z)−S]−1, which represents an LMTO-counterpart of the scattering-path operator ma- trix of the KKR method.32,33The symbol P(z) denotes the site-diagonal matrix of potential functions; their an- alytic dependence on zand on the potential parameters5 can be found elsewhere.27,37The relation between both resolvents leads to the formula28 G+−G−=F(g+−g−)F+, (7) where the same abbreviation F= (√ ∆)−1(1−γS) as in Eq. (B8) was used and g±=g(EF±i0) . The torque-correlation formula (2) in the LMTO-ASA method follows directly from relations (A7), (B8), (B9) and (7). The components of the Gilbert damping tensor {αµν}in the LLG equation (1) can be obtained from a basic tensor {˜αµν}given by ˜αµν=−α0Tr{τµ(g+−g−)τν(g+−g−)},(8) where the quantities τµ=−i[Jµ,S] =−i[Lµ,S] (9) define components of an effective torque in the LMTO- ASA method. The site-diagonal matrices JµandLµ (µ=x,y,z) are Cartesian components of the total and orbital angular momentum operator, respectively, see text aroundEqs.(B8) and (B9). The tracein (8) extends over all orbitals of the crystalline solid and the prefactor can be written as α0= (2πMspin)−1, where Mspinde- notes the spin magnetic moment of the whole crystal in units of the Bohr magneton µB.15,19,22 Let us discuss properties of the effective torque (9). Its form is obviously identical to the nonlocal torque (5). The matrix τµis non-site-diagonal, but—for a random substitutional alloy on a nonrandom lattice—it is non- random (independent on the alloy configuration). More- over, it is given by a commutator of the site-diagonal nonrandom matrix Jµ(orLµ) and the LMTO structure- constantmatrix S. Thesepropertiespointtoacloseanal- ogy between the effective torque and the effective veloci- ties in the LMTO conductivity tensor based on a concept of intersite electron hopping.27,28,40Let us mention that existing ab initio approaches employ random torques, either the XC-induced torque in the KKR method19,22 or the SO-induced torque in the LMTO method.20An- other interesting property of the effective torque τµ(9) is its spin-independence which follows from the spin- independence of the matrices LµandS. The explicit relation between the symmetric tensors {αµν}and{˜αµν}canbeeasilyformulatedfortheground- state magnetization along zaxis; then it is given simply byαxx= ˜αyy,αyy= ˜αxx, andαxy=−˜αxy. These relations reflect the fact that an infinitesimal deviation towards xaxis results from an infinitesimal rotation of the magnetization vector around yaxis and vice versa. Note that the other components of the Gilbert damp- ing tensor ( αµzforµ=x,y,z) are not relevant for the dynamics of small deviations of magnetization direction described by the LLG equation (1). For the ground- state magnetization pointing along a general unit vector m= (mx,my,mz), one has to employ the Levi-Civita symbolǫµνλin order to get the Gilbert damping tensorαas αµν=/summationdisplay µ′ν′ηµµ′ηνν′˜αµ′ν′, (10) whereηµν=/summationtext λǫµνλmλ. The resultingtensor(10) satis- fies the condition α·m= 0 appropriate for the dynamics of small transverse deviations of magnetization. The application to random alloys requires configura- tion averaging of ˜ αµν(8). Since the effective torques τµ are nonrandom, one can write a unique decomposition of the average into the coherent and incoherent parts, ˜αµν= ˜αcoh µν+ ˜αvc µν, where the coherent part is expressed by means of the averaged auxiliary resolvents ¯ g±=/an}bracketle{tg±/an}bracketri}ht as ˜αcoh µν=−α0Tr{τµ(¯g+−¯g−)τν(¯g+−¯g−)}(11) and the incoherent part (vertex corrections) is given as a sum of four terms, namely, ˜αvc µν=−α0/summationdisplay p=±/summationdisplay q=±sgn(pq)Tr/an}bracketle{tτµgpτνgq/an}bracketri}htvc.(12) In this work, the configuration averaging has been done in the CPA. Details concerning the averaged resolvents can be found, e.g., in Ref. 39 and the construction of the vertex corrections for transport properties was described in Appendix to Ref. 30. C. Properties of the LMTO torque-correlation formula The damping tensor (8) has been formulated in the canonical LMTO representation. In the numerical im- plementation, the well-known transformation to a tight- binding(TB)LMTOrepresentation41,42isadvantageous. The TB-LMTO representation is specified by a diag- onal matrix βof spin-independent screening constants (βR′ℓ′m′s′,Rℓms=δR′Rδℓ′ℓδm′mδs′sβRℓin a nonrelativis- tic basis) and the transformation of all quantities be- tween both LMTO representations has been discussed in the literature for pure crystals42as well as for random alloys.28,39,43The same techniques can be used in the present case together with an obvious commutation rule [Jµ,β] = [Lµ,β] = 0. Consequently, the conclusions drawn are the same as for the conductivity tensor:28the total damping tensor (8) as well as its coherent (11) and incoherent (12) parts in the CPA are invariant with re- spect to the choice of the LMTO representation. It should be mentioned that the central result, namely the relations(8) and (9), is not limited to the LMTO the- ory, but it can be translatedinto the KKRtheory aswell, similarly to the conductivity tensor in the formalism of intersite hopping.40The LMTO structure-constant ma- trixSandtheauxiliaryGreen’sfunction g(z)willbethen replacedrespectivelybytheKKRstructure-constantma- trix and by the scattering-path operator.32,33Note, how-6 ever, that the total ( Jµ) and orbital ( Lµ) angular mo- mentum operatorsin the effective torques (9) will be rep- resented by the same matrices as in the LMTO theory. Let us mention for completeness that the present LMTO-ASA theory allows one to introduce effective lo- cal (but random) torques as well. This is based on the fact that only the Fermi-level propagators g±defined by the structure constant matrix Sand by the potential functions at the Fermi energy, P=P(EF), enter the zero-temperature expression for the damping tensor ˜ αµν (8). Since the equation of motion ( P−S)g±= 1 implies immediately S(g+−g−) =P(g+−g−) and, similarly, (g+−g−)S= (g+−g−)P, one can obviously replace the nonlocal torques τµ(9) in the torque-correlation formula (8) by their local counterparts τxc µ= i[P,Jµ], τso µ= i[P,Lµ].(13) These effective torques are represented by random, site- diagonal matrices; the τxc µandτso µcorrespond, respec- tively, to the XC-induced torque used in the KKR method22and to the SO-induced torque used in the LMTO method with a simplified treatment of the SO- interaction.20In the case of random alloys treated in the CPA, the randomness of the local torques (13) calls for the approach developed by Butler44for the averaging of the torque-correlationcoefficient (8). One can provethat the resulting damping parameters ˜ αµνobtained in the CPA with the local and nonlocal torques are fully equiv- alent to each other; this equivalence rests heavily on a proper inclusion of the vertex corrections45and it leads to further important consequences. First, the Gilbert damping tensor vanishes exactly for zero SO interaction, which follows from the use of the SO-induced torque τso µ and from the obvious commutation rule [ P,Lµ] = 0 valid for the spherically symmetric potential functions (in the absence of SO interaction). This result is in agreement with thenumericalstudyofthe toymodel inSectionIIA, see Fig. 1a for ξ= 0. On an ab initio level, this prop- erty has been obtained numerically both in the KKR method22and in the LMTO method.26Second, the XC- andSO-inducedlocaltorques(13)withintheCPAareex- actly equivalent as well, as has been indicated in a recent numerical study for a random bcc Fe 50Co50alloy.26In summary, the nonlocaltorques(9) andboth localtorques (13) can be used as equivalent alternatives in the torque- correlation formula (8) provided that the vertex correc- tions are included consistently with the CPA-averaging of the single-particle propagators. III. ILLUSTRATING EXAMPLES A. Implementation and numerical details The numerical implementation of the described the- ory and the calculations have been done with similar tools as in our recent studies of ground-state46and 3 6 9 0 10 20 30103 α ε (µRy)fcc Ni80Fe20bcc Fe80Co20 (x 10) FIG. 2. The Gilbert damping parameters αof random fcc Ni80Fe20(full circles) and bcc Fe 80Co20(open squares) alloys as functions of the imaginary part of energy ε. The values of αfor the Fe 80Co20alloy are magnified by a factor of 10. transport27,28,47properties. The ground-state magne- tization was taken along zaxis and the selfconsistent XC potentials were obtained in the local spin-density ap- proximation (LSDA) with parametrization according to Ref. 48. The valence basis comprised s-,p-, andd-type orbitalsand the energyargumentsforthe propagators¯ g± and the CPA-vertex corrections were obtained by adding a tiny imaginary part ±εto the real Fermi energy. We have found that the dependence of the Gilbert damping parameter on εis quite smooth and that the value of ε= 10−6Ry is sufficient for the studied systems, see Fig. 2 for an illustration. Similar smooth dependences havebeenobtainedalsoforotherinvestigatedalloys,such as Permalloy doped by 5 delements, Heusler alloys, and stoichiometric FePt alloys with a partial atomic long- range order. In all studied cases, the number Nofk vectors needed for reliable averaging over the Brillouin zone (BZ) was properly checked; as a rule, N∼108in the full BZ was sufficient for most systems, but for di- luted alloys (a few percent of impurities), N∼109had to be taken. B. Binary fcc and bcc solid solutions The developed theory has been applied to random bi- nary alloys of 3 dtransition elements Fe, Co, and Ni, namely, to the fcc NiFe and bcc FeCo alloys. The most important results, including a comparison to other exist- ingab initio techniques, are summarized in Fig. 3. One can see a good agreement of the calculated concentration trends of the Gilbert damping parameter α=αxx=αyy with the results of an LMTO-supercell approach17and of the KKR-CPA method.22The decrease of αwith in-7 0 4 8 12 0 0.2 0.4 0.6103 α Fe concentrationfcc NiFe(a) this work LMTO-SC 0 2 4 6 0 0.2 0.4 0.6103 α Co concentrationbcc FeCo(b) this work KKR-CPA FIG. 3. (Color online) The calculated concentration depen- dences of the Gilbert damping parameter αfor random fcc NiFe (a) and bcc FeCo (b) alloys. The results of this work are marked by the full diamonds, whereas the open circles depict the results of other approaches: the LMTO supercell (LMTO-SC) technique17and the KKR-CPA method.22 creasingFecontentintheconcentratedNiFealloyscanbe relatedto the increasingalloymagnetization17andto the decreasing strength of the SO-interaction,20whereas the behaviorinthedilutelimitcanbeexplainedbyintraband scattering due to Fe impurities.11,12,14In the case of the FeCo system, the minimum of αaround 20% Co, which is also observed in room-temperature experiments,49,50 is related primarily to a similar concentration trend of the density of states at the Fermi energy,22though the maximum of the magnetization at roughly the same alloy composition51might partly contribute as well. A more detailed comparison of all ab initio results is presented in Table I for the fcc Ni 80Fe20random alloy (Permalloy). The differences in the values of αfrom the differenttechniquescanbe ascribedtovarioustheoretical features and numericaldetails employed, such asthe sim-TABLE I. Comparison of the Gilbert damping parameter α for the fcc Ni 80Fe20random alloy (Permalloy) calculated by the present approach and by other techniques using the CPA or supercells (SC). The last column displays the coherent pa rt αcohof the total damping parameter according to Eq. (11). The experimental value corresponds to room temperature. Method α αcoh This work, ε= 10−5Ry 4 .9×10−31.76 This work, ε= 10−6Ry 3 .9×10−31.76 KKR-CPAa4.2×10−3 LMTO-CPAb3.5×10−3 LMTO-SCc4.6×10−3 Experimentd8×10−3 aReference 22. bReference 20. cReference 17. dReference 49. plified treatment of the SO-interaction in Ref. 20 instead of the fully relativistic description, or the use of super- cells in Ref. 17 instead of the CPA. Taking into account that calculated residual resistivities for this alloy span a wide interval between 2 µΩcm, see Ref. 27 and 52, and 3.5µΩcm, see Ref. 17, one can consider the scatter of the calculated values of αin Table I as little important. The theoretical values of αare smaller systematically than the measured values, typically by a factor of two. This discrepancy might be partly due to the effects of finite temperatures as well as due to additional structural de- fects of real samples. A closer look at the theoretical results reveals that the total damping parameters αareappreciablysmallerthan themagnitudesoftheircoherentandvertexparts,seeTa- ble I for the case of Permalloy. This is in agreement with theresultsofthemodelstudyinSectionIIA;similarcon- clusions about the importance of the vertex corrections have been done with the XC-induced torques in other CPA-based studies.19,22,26The present results prove that this unpleasant feature of the nonlocal torques does not represent a serious obstacle in obtaining reliable values of the Gilbert damping parameter in random alloys. We note that the vertex corrections can be negligible in ap- proaches employing the SO-induced torques, at least for systems with the SO splittings much weaker than the XC splittings,12such as the binary ferromagnetic alloys of 3 d transition metals,26see also Section IIA. C. Pure iron with a model disorder As it has been mentioned in Section I, the Gilbert damping of pure ferromagnetic metals exhibits non- trivial temperature dependences, which have been re- produced by means of ab initio techniques with vari- ous levels of sophistication.11,12,21,23In this study, we have simulated the effect of finite temperatures by intro-8 0 3 6 9 0123402040103 α ρ (µΩcm) 103 δ2 (Ry2)bcc Fe FIG. 4. (Color online) The calculated Gilbert damping pa- rameter α(full squares) and the residual resistivity ρ(open circles) of pure bcc iron as functions of δ2, where δis the strength of a model atomic-level disorder. ducing static fluctuations of the one-particle potential. The adopted model of atomic-level disorderassumes that random spin-independent shifts ±δ, constant inside each atomic sphere and occurring with probabilities 50% of bothsigns,areaddedtothenonrandomselfconsistentpo- tential obtained at zero temperature. The Fermi energy iskeptfrozen,equaltoitsselfconsistentzero-temperature value. This model can be easily treated in the CPA; the resulting Gilbert damping parameter αof pure bcc Fe as a function of the potential shift δis plotted in Fig. 4. Thecalculateddependence α(δ) isnonmonotonic, with a minimum at δ≈30 mRy. This trend is in a qualita- tive agreement with trends reported previously by other authors, who employed phenomenological models of the electron lifetime11,12as well as models for phonons and magnons.21,23The origin of the nonmonotonic depen- denceα(δ) has been identified on the basis of the band structure of the ferromagnetic system as an interplay be- tween the intraband contributions to α, dominating for small values of δ, and the interband contributions, domi- nating for large values of δ.7,11,12Since the present CPA- based approach does not use any bands, we cannot per- form a similar analysis. The obtained minimum value of the Gilbert damping, αmin≈10−3(Fig. 4), agrees reasonably well with the values obtained by the authors of Ref. 11, 12, 21, and 23. This agreement indicates that the atomic-level dis- order employed here is equivalent to a phenomenological lifetime broadening. For a rough quantitative estimation of the temperature effect, one can employ the calculated resistivity ρof the model, which increases essentially lin- early with δ2, see Fig. 4. Since the metallic resistivity due to phonons increases linearly with the temperature T(for temperatures not much smaller than the Debye temperature), one can assume a proportionality between 10 20 30 40 0 0.5 1152535103 α DOS(EF) (states/Ry) LRO parameter SL10 FePt FIG. 5. (Color online) The calculated Gilbert damping pa- rameter α(full squares) and the total DOS (per formula unit) at the Fermi energy (open circles) of stoichiometric L1 0FePt alloys as functions of the LRO parameter S. δ2andT. The resistivity of bcc iron at the Curie tem- perature TC= 1044 K due to lattice vibrations can be estimated around 35 µΩcm,23,53which sets an approx- imate temperature scale to the data plotted in Fig. 4. However, a more accurate description of the temperature dependence of the Gilbert damping parameter cannot be obtained, mainly due to the neglected true atomic dis- placements and the noncollinearity of magnetic moments (magnons).23 D. FePt alloys with a partial long-range order Since important ferromagnetic materials include or- dered alloys, we address here the Gilbert damping in sto- ichiometric FePt alloys with L1 0atomic long-range order (LRO). Their transport properties47and the damping parameter20have recently been studied by means of the TB-LMTO method in dependence on a varying degree of the LRO. These fcc-based systems contain two sublat- ticeswith respectiveoccupationsFe 1−yPtyandPt 1−yFey wherey(0≤y≤0.5) denotes the concentration of anti- site atoms. The LRO parameter S(0≤S≤1) is then defined as S= 1−2y, so that S= 0 corresponds to the random fcc alloy and S= 1 corresponds to the perfectly ordered L1 0structure. The resulting Gilbert damping parameter is displayed in Fig. 5 as a function of S. The obtained trend with a broadmaximumat S= 0andaminimumaround S= 0.9 agrees very well with the previous result.20The values of αin Fig. 5 are about 10% higher than those in Ref. 20, which can be ascribed to the fully relativistic treatment in the present study in contrast to a simplified treatment of the SO interaction in Ref. 20. The Gilbert damping9 in the FePt alloys is an order of magnitude stronger than in the alloys of 3 delements (Section IIIB) owing to the stronger SO interaction of Pt atoms. The origin of the slow decrease of αwith increasing S(for 0≤S≤0.9) can be explained by the decreasing total density of states (DOS) at the Fermi energy, see Fig. 5, which represents an analogy to a similar correlation observed, e.g., for bcc FeCo alloys.22 All calculated values of αshown in Fig. 5, correspond- ing to 0 ≤S≤0.985, are appreciably smaller than the measured one which amounts to α≈0.06 reported for a thin L1 0FePt epitaxial film.54The high measured value of αmight be thus explained by the present cal- culations by assuming a very small concentration of an- tisites in the prepared films, which does not seem too realistic. Another potential source of the discrepancy lies in the thin-film geometry used in the experiment. Moreover, the divergence of αin the limit of S→1 (Fig. 5) illustrates a general shortcoming of approaches based on the torque-correlation formula (2), since the zero-temperature Gilbert damping parameter of a pure ferromagnet should remain finite. A correct treatment of this case, including the dilute limit of random alloys (Fig. 3), must take into account the full interacting sus- ceptibility in the presence of SO interaction.15,55Pilotab initiostudies in this direction have recently appeared for nonrandom systems;56,57however, their extension to dis- ordered systems goes far beyond the scope of this work. IV. CONCLUSIONS We have introduced nonlocal torques as an alterna- tive to the usual local torque operators entering the torque-correlation formula for the Gilbert damping ten- sor. Within the relativistic TB-LMTO-ASA method, this idea leads to effective nonlocal torques as non-site- diagonal and spin-independent matrices. For substitu- tionally disordered alloys, the nonlocal torques are non- random, which allows one to develop an internally con- sistent theory in the CPA. The CPA-vertex corrections proved indispensable for an exact equivalence of the non- local nonrandom torques with their local random coun- terparts. The concept of the nonlocal torques is not lim- ited to the LMTO method and its formulation both in a semiempirical TB theory and in the KKR theory is straightforward. The numerical implementation and the results for bi- nary solid solutions show that the total Gilbert damping parameters from the nonlocal torques are much smaller than magnitudes of the coherent parts and of the ver- tex corrections. Nevertheless, the total damping param- eters for the studied NiFe, FeCo and FePt alloys compare quantitatively very well with results of other ab initio techniques,17,20,22which indicates a fair numerical sta- bility of the developed theory. The performed numerical study of the Gilbert damp- ing in pure bcc iron as a function of an atomic-level dis-order yields a nonmonotonic dependence in a qualitative agreementwith the trends consisting of the conductivity- like and resistivity-like regions, obtained from a phe- nomenological quasiparticle lifetime broadening7,11,12or from the temperature-induced frozen phonons21,22and magnons.23Future studies should clarify the applicabil- ity of the introduced nonlocal torques to a full quanti- tative description of the finite-temperature behavior as well as to other torque-related phenomena, such as the spin-orbit torques due to applied electric fields.58,59 ACKNOWLEDGMENTS The authors acknowledge financial support by the Czech Science Foundation (Grant No. 15-13436S). Appendix A: Torque correlation formula in a matrix representation In this Appendix, evaluation of the Kubo-Greenwood expression for the torque-correlation formula (2) is dis- cussed in the case of the XC-induced torque operators using matrix representations of all operators in an or- thonormal basis that varies due to the varying direc- tion of the XC field. All operators are denoted by a hat, in order to be distinguished from matrices repre- senting these operators in the chosen basis. Let us con- sider a one-particle Hamiltonian ˆH=ˆH(θ1,θ2) depend- ing on two real variables θj,j= 1,2, and let us denote ˆT(j)(θ1,θ2) =∂ˆH(θ1,θ2)/∂θj. In our case, the variables θjplay the role of rotation angles and the operators ˆT(j) are the corresponding torques. Let us denote the resol- vents of ˆH(θ1,θ2) at the Fermi energy as ˆG±(θ1,θ2) and let us consider a special linear response coefficient (argu- mentsθ1andθ2are omitted here and below for brevity) c= Tr{ˆT(1)(ˆG+−ˆG−)ˆT(2)(ˆG+−ˆG−)} (A1) = Tr{(∂ˆH/∂θ1)(ˆG+−ˆG−)(∂ˆH/∂θ2)(ˆG+−ˆG−)}. This torque-correlation coefficient equals the Gilbert damping parameter (2) with the prefactor ( −α0) sup- pressed. For its evaluation, we introduce an orthonormal basis|χm(θ1,θ2)/an}bracketri}htand represent all operators in this ba- sis. This leads to matrices H(θ1,θ2) ={Hmn(θ1,θ2)}, G±(θ1,θ2) ={(G±)mn(θ1,θ2)}andT(j)(θ1,θ2) = {T(j) mn(θ1,θ2)}, where Hmn=/an}bracketle{tχm|ˆH|χn/an}bracketri}ht,(G±)mn=/an}bracketle{tχm|ˆG±|χn/an}bracketri}ht, T(j) mn=/an}bracketle{tχm|ˆT(j)|χn/an}bracketri}ht=/an}bracketle{tχm|∂ˆH/∂θj|χn/an}bracketri}ht,(A2) and, consequently, to the response coefficient (A1) ex- pressed by using the matrices (A2) as c= Tr{T(1)(G+−G−)T(2)(G+−G−)}.(A3) However, in evaluation of the last expression, atten- tion has to be paid to the difference between the ma- trixT(j)(θ1,θ2) and the partial derivative of the matrix10 H(θ1,θ2) with respect to θj. This difference follows from the identity ˆH=/summationtext mn|χm/an}bracketri}htHmn/an}bracketle{tχn|, which yields T(j) mn=∂Hmn/∂θj+/summationdisplay k/an}bracketle{tχm|∂χk/∂θj/an}bracketri}htHkn +/summationdisplay kHmk/an}bracketle{t∂χk/∂θj|χn/an}bracketri}ht, (A4) where we employed the orthogonality relations /an}bracketle{tχm(θ1,θ2)|χn(θ1,θ2)/an}bracketri}ht=δmn. Their partial derivatives yield /an}bracketle{tχm|∂χn/∂θj/an}bracketri}ht=−/an}bracketle{t∂χm/∂θj|χn/an}bracketri}ht ≡Q(j) mn,(A5) where we introduced elements of matrices Q(j)={Q(j) mn} forj= 1,2. Note that the matrices Q(j)(θ1,θ2) reflect explicitlythe dependenceofthebasisvectors |χm(θ1,θ2)/an}bracketri}ht onθ1andθ2. The relation (A4) between the matrices T(j)and∂H/∂θ jcan be now rewritten compactly as T(j)=∂H/∂θ j+[Q(j),H]. (A6) Since the last term has a form of a commutator with the Hamiltonianmatrix H, theuseofEq.(A6)intheformula (A3) leads to the final matrix expression for the torque correlation, c= Tr{(∂H/∂θ 1)(G+−G−)(∂H/∂θ 2)(G+−G−)}.(A7) The equivalence of Eqs. (A3) and (A7) rests on the rules [Q(j),H] = [EF−H,Q(j)] and (EF−H)(G+−G−) = (G+−G−)(EF−H) = 0 and on the cyclic invariance of the trace. It is also required that the matrices Q(j)are compatible with periodic boundary conditions used in calculations of extended systems, which is obviously the case for angular variables θjrelated to the global changes (uniform rotations) of the magnetization direction. The obtained result means that the original response coefficient (A1) involving the torques as angular deriva- tives of the Hamiltonian can be expressed solely by us- ing matrix elements of the Hamiltonian in an angle- dependent basis; theangulardependence ofthebasisvec- tors does not enter explicitly the final torque-correlation formula (A7). Appendix B: LMTO Hamiltonian of a ferromagnet with a tilted magnetic field Here we sketch a derivation of the fully relativis- tic LMTO Hamiltonian matrix for a ferromagnet with the XC-field direction tilted from a reference direction along an easy axis. The derivation rests on the form of the Kohn-Sham-Dirac Hamiltonian in the LMTO-ASA method.37–39The symbols with superscript 0 refer to the referencesystem,thesymbolswithoutthissuperscriptre- fer to the system with the tilted XC field. The operators (Hamiltonians, rotation operators) are denoted by sym- bols with a hat. The spin-dependent parts of the ASApotentials due to the XC fields are rigidly rotated while the spin-independent parts are unchanged, in full anal- ogy to the approach employed in the relativistic KKR method.19,22 The ASA-Hamiltonians of both systems are given by lattice sums ˆH0=/summationtext RˆH0 RandˆH=/summationtext RˆHR, where the individual site-contributions are coupled mutually by ˆHR=ˆURˆH0 RˆU+ R, whereˆURdenotesthe unitaryoperator of a rotation (in the orbital and spin space) around the Rth lattice site which brings the local XC field from its reference direction into the tilted one. Let |φ0 RΛ/an}bracketri}htand |˙φ0 RΛ/an}bracketri}htdenote, respectively, the phi and phi-dot orbitals of the reference Hamiltonian ˆH0 R, then |φRΛ/an}bracketri}ht=ˆUR|φ0 RΛ/an}bracketri}ht,|˙φRΛ/an}bracketri}ht=ˆUR|˙φ0 RΛ/an}bracketri}ht(B1) define the phi and phi-dot orbitals of the Hamiltonian ˆHR. The orbital index Λ labels all linearly indepen- dentsolutions(regularattheorigin)ofthespin-polarized relativistic single-site problem; the detailed structure of Λ can be found elsewhere.37–39Let us introduce further the well-known empty-space solutions |K∞,0 RN/an}bracketri}ht(extending over the whole real space), |Kint,0 RN/an}bracketri}ht(extending over the interstitial region), and |K0 RN/an}bracketri}htand|J0 RN/an}bracketri}ht(both trun- cated outside the Rth sphere), needed for the definition of the LMTOs of the reference system.41,42,60Their in- dexN, which defines the spin-spherical harmonics of the large component of each solution, can be taken either in the nonrelativistic ( ℓms) form or in its relativistic ( κµ) counterpart. We define further |ZRN/an}bracketri}ht=ˆUR|Z0 RN/an}bracketri}htforZ=K∞, K, J. (B2) Isotropyofthe emptyspaceguaranteesrelations(for Z= K∞,K,J) |ZRN/an}bracketri}ht=/summationdisplay N′|Z0 RN′/an}bracketri}htUN′N, |Z0 RN/an}bracketri}ht=/summationdisplay N′|ZRN′/an}bracketri}htU+ N′N, (B3) whereU={UN′N}denotes a unitary matrix represent- ing the rotation in the space of spin-spherical harmonics and where U+ N′N≡(U+)N′N= (UNN′)∗= (U−1)N′N; the matrix Uis the same for all lattice sites Rsince we consider only uniform rotations of the XC-field direction inside the ferromagnet. The expansion theorem for the envelope orbital |K∞,0 RN/an}bracketri}htis |K∞,0 RN/an}bracketri}ht=|Kint,0 RN/an}bracketri}ht+|K0 RN/an}bracketri}ht −/summationdisplay R′N′|J0 R′N′/an}bracketri}htS0 R′N′,RN,(B4) whereS0 R′N′,RNdenote elements of the canonical structure-constant matrix (with vanishing on-site ele- ments,S0 RN′,RN= 0) of the reference system. The use of relations (B3) in the expansion (B4) together with an abbreviation |Kint RN/an}bracketri}ht=/summationdisplay N′|Kint,0 RN′/an}bracketri}htUN′N (B5)11 yields the expansion of the envelope orbital |K∞ RN/an}bracketri}htas |K∞ RN/an}bracketri}ht=|Kint RN/an}bracketri}ht+|KRN/an}bracketri}ht −/summationdisplay R′N′|JR′N′/an}bracketri}ht(U+S0U)R′N′,RN,(B6) whereUandU+denote site-diagonal matrices with el- ementsUR′N′,RN=δR′RUN′Nand (U+)R′N′,RN= δR′RU+ N′N. Note the same form of expansions (B4) and (B6), with the orbitals |Z0 RN/an}bracketri}htreplaced by the rotated or- bitals|ZRN/an}bracketri}ht(Z=K∞,K,J), with the interstitial parts |Kint,0 RN/an}bracketri}htreplacedbytheirlinearcombinations |Kint RN/an}bracketri}ht, and with the structure-constant matrix S0replaced by the product U+S0U. The non-orthogonal LMTO |χ0 RN/an}bracketri}htfor the reference system is obtained from the expansion (B4), in which all orbitals|K0 RN/an}bracketri}htand|J0 RN/an}bracketri}htare replaced by linear com- binations of |φ0 RΛ/an}bracketri}htand|˙φ0 RΛ/an}bracketri}ht. A similar replacement of the orbitals |KRN/an}bracketri}htand|JRN/an}bracketri}htby linear combinations of |φRΛ/an}bracketri}htand|˙φRΛ/an}bracketri}htin the expansion (B6) yields the non- orthogonal LMTO |χRN/an}bracketri}htfor the system with the tilted XC field. The coefficients in these linear combinations— obtained from conditions of continuous matching at the sphere boundaries and leading directly to the LMTO po- tentialparameters—areidenticalforboth systems, asfol- lows from the rotationrelations (B1) and (B2). For these reasons, the only essential difference between both sys- tems in the construction of the non-orthogonal and or- thogonal LMTOs (and of the accompanying Hamiltonian and overlap matrices in the ASA) is due to the difference between the matrices S0andU+S0U. As a consequence, the LMTO Hamiltonian matrix in the orthogonalLMTO representationfor the system with a tilted magnetizationis easilyobtained fromthat forthe reference system, Eq. (6), and it is given by H=C+(√ ∆)+U+SU(1−γU+SU)−1√ ∆,(B7) where the C,√ ∆ andγare site-diagonal matrices of the potential parameters of the reference system and where we suppressed the superscript 0 at the structure- constant matrix Sof the reference system. Note that the dependence of Hon the XC-field direction is con- tained only in the similarity transformation U+SUof the original structure-constant matrix Sgenerated by the rotation matrix U. For the rotation by an angle θaround an axis along a unit vector n, the rotation matrix is given by U(θ) = exp( −in·Jθ), where the site-diagonal matrices J≡(Jx,Jy,Jz) with matrix elements Jµ R′N′,RN=δR′RJµ N′N(µ=x,y,z) reduce to usual matrices of the total (orbital plus spin) angu- lar momentum operator. The limit of small θyields U(θ)≈1−in·Jθ, which leads to the θ-derivative of the Hamiltonian matrix (B7) at θ= 0: ∂H/∂θ= i(F+)−1[n·J,S]F−1, (B8) where we abbreviated F= (√ ∆)−1(1−γS) andF+= (1−Sγ)[(√ ∆)+]−1. Since the structure-constant matrixSis spin-independent, the total angular momentum op- eratorJin (B8) canbe replacedbyits orbitalmomentum counterpart L≡(Lx,Ly,Lz), so that ∂H/∂θ= i(F+)−1[n·L,S]F−1.(B9) The relations (B8) and (B9) are used to derive the LMTO-ASA torque-correlation formula (8). Appendix C: Equivalence of the Gilbert damping in the CPA with local and nonlocal torques (Supplemental Material) 1. Introductory remarks The problem of equivalence of the Gilbert damping tensor expressed with the local (loc) and nonlocal (nl) torques can be reduced to the problem of equivalence of these two expressions: αloc=α0Tr/an}bracketle{t(g+−g−)[P,K](g+−g−)[P,K]/an}bracketri}ht =α0/summationdisplay p=±/summationdisplay q=±sgn(pq)Tr/an}bracketle{tgp[P,K]gq[P,K]/an}bracketri}ht =α0/summationdisplay p=±/summationdisplay q=±sgn(pq)βloc pq, (C1) and αnl=α0Tr/an}bracketle{t(g+−g−)[K,S](g+−g−)[K,S]/an}bracketri}ht =α0/summationdisplay p=±/summationdisplay q=±sgn(pq)Tr/an}bracketle{tgp[K,S]gq[K,S]/an}bracketri}ht =α0/summationdisplay p=±/summationdisplay q=±sgn(pq)βnl pq. (C2) The symbols Tr and /an}bracketle{t.../an}bracketri}htand the quantities α0,g±,P andShavethe samemeaning as in the main text and the quantity Ksubstitutes any of the operators (matrices) JµorLµ. Note that owing to the symmetric nature of theoriginaldampingtensors,theanalysiscanbeconfined to scalar quantities αlocandαnldepending on a general site-diagonal nonrandom operator K. The choice of K= Kµin (C1) and (C2) produces the diagonal elements of both tensors, whereas the choice of K=Kµ±Kνfor µ/ne}ationslash=νleads to all off-diagonal elements. The quantities βloc pqandβnl pqare expressions of the form βloc= Tr/an}bracketle{tg1(P1K−KP2)g2(P2K−KP1)/an}bracketri}ht, βnl= Tr/an}bracketle{tg1[K,S]g2[K,S]/an}bracketri}ht, (C3) where the g1andg2replace the gpandgq, respectively. For an internal consistency of these and following expres- sions, we have also introduced P1=P2=P. This supplement contains a proof of the equivalence ofβlocandβnland, consequently, of αlocandαnl. The CPA-average in βnlwith a nonlocal nonrandom torque has been done using the theory by Velick´ y29as worked out in detail within the present LMTO formalism by Carva et al.30whereas the averaging in βlocinvolving a local but random torque has been treated using the approach by Butler.4412 2. Auxiliary quantities and relations SincethenecessaryformulasoftheCPAinmultiorbital techniques30,44are little transparent, partly owing to the complicated indices of two-particle quantities, we employ here a formalism with the lattice-site index Rkept but with all orbital indices suppressed. The Hilbert spaceis a sum ofmutually orthogonalsub- spaces of individual lattice sites R; the corresponding projectors will be denoted by Π R. A number of rele- vant operators are site-diagonal, i.e., they can be written asX=/summationtext RXR, where the site contributions are given byXR= ΠRX=XΠR= ΠRXΠR. Such operators are, e.g., the random potential functions, Pj=/summationtext RPj R, and the nonrandom coherent potential functions Pj=/summationtext RPj R, wherej= 1,2. The operator Kin (C3) is site- diagonal as well, but its site contributions KRwill not be used explicitly in the following. Among the number ofCPA-relationsfor single-particle properties, we will use the equation of motion for the average auxiliary Green’s functions ¯ gj(j= 1,2), ¯gj(Pj−S) = (Pj−S)¯gj= 1, (C4) as well as the definition of random single-site t-matrices tj R(j= 1,2) with respect to the effective CPA-medium, given by tj R= (Pj R−Pj R)[1+ ¯gj(Pj R−Pj R)]−1.(C5) The operators tj Rare site-diagonal, being non-zero only in the subspace of site R. The last definition leads to identities (1−t1 R¯g1)P1 R=P1 R+t1 R(1−¯g1P1 R), P2 R(1−¯g2t2 R) =P2 R+(1−P2 R¯g2)t2 R,(C6) which will be employed below together with the CPA- selfconsistency conditions /an}bracketle{ttj R/an}bracketri}ht= 0 (j= 1,2). For the purpose of evaluation of the two-particle aver- ages in (C3), we introduce several nonrandom operators: f12= ¯g1K−K¯g2, ζ12= ¯g1[K,S]¯g2,(C7) and a site-diagonal operator γ12=/summationtext Rγ12 R, where γ12=P1K−KP2, γ12 R=P1 RK−KP2 R.(C8) By interchanging the superscripts 1 ↔2 in (C7) and (C8), one can also get quantities f21,ζ21,γ21andγ21 R; this will be implicitly understood in the relations below as well. The three operators f12,ζ12andγ12satisfy a relation f12+ζ12+¯g1γ12¯g2= 0, (C9) which can be easily proved from their definitions (C7) and (C8) and from the equation of motion (C4). An- other quantityto be used in the followingis a nonrandomsite-diagonal operator ϑ12related to the local torque and defined by ϑ12 R=/an}bracketle{t(1−t1 R¯g1)(P1 RK−KP2 R)(1−¯g2t2 R)/an}bracketri}ht, ϑ12=/summationdisplay Rϑ12 R. (C10) Its site contributions can be rewritten explicitly as ϑ12 R=γ12 R+/an}bracketle{tt1 R(f12+ ¯g1γ12 R¯g2)t2 R/an}bracketri}ht.(C11) The last relation follows from the definition (C10), from theidentities(C6)andfromtheCPA-selfconsistencycon- ditions. Moreover,the site contributions ϑ12 Randγ12 Rsat- isfy a sum rule γ12 R=/summationdisplay R′′/an}bracketle{tt1 R¯g1γ12 R′¯g2t2 R/an}bracketri}ht+/an}bracketle{tt1 Rζ12t2 R/an}bracketri}ht+ϑ12 R,(C12) where the prime at the sum excludes the term with R′= R. This sum rule can be proved by using the definitions ofζ12(C7) and γ12 R(C8) and by employing the previous relation for ϑ12 R(C11) and the equation of motion (C4). The treatment of two-particle quantities requires the use of a direct product a⊗bof two operators aandb. This is equivalent to the concept of a superoperator, i.e., a linear mapping defined on the vector space of all linear operators. In this supplement, superoperators are de- noted by an overhat, e.g., ˆ m. In the present formalism, the direct product of two operators aandbcan be iden- tified with a superoperator ˆ m=a⊗b, which induces a mapping x/ma√sto→ˆmx= (a⊗b)x=axb, (C13) wherexdenotes an arbitrary usual operator. This defi- nition leads, e.g., to a superoperator multiplication rule (a⊗b)(c⊗d) = (ac)⊗(db). (C14) In the CPA, the most important superoperators are ˆw12=/summationdisplay R/an}bracketle{tt1 R⊗t2 R/an}bracketri}ht (C15) and ˆχ12=/summationdisplay RR′′ ΠR¯g1ΠR′⊗ΠR′¯g2ΠR(C16) where the prime at the double sum excludes the terms withR=R′. The quantity ˆ w12represents the irre- ducible CPA-vertex and the quantity ˆ χ12corresponds to arestrictedtwo-particlepropagatorwithexcludedon-site terms. By using these superoperators, the previous sum rule (C12) can be rewritten compactly as (ˆ1−ˆw12ˆχ12)γ12= ˆw12ζ12+ϑ12,(C17) whereˆ1 = 1⊗1 denotes the unit superoperator.13 Let us introduce finally a symbol {x;y}, wherexand yare arbitrary operators, which is defined by {x;y}= Tr(xy). (C18) Thissymbolissymmetric, {x;y}={y;x}, linearinboth arguments and it satisfies the rule {(a⊗b)x;y}={x;(b⊗a)y},(C19) which follows from the cyclic invariance of the trace. An obvious consequence of this rule are relations {ˆw12x;y}={x; ˆw21y}, {ˆχ12x;y}={x; ˆχ21y}, (C20) where ˆw21and ˆχ21are defined by (C15) and (C16) with the superscript interchange 1 ↔2. 3. Expression with the nonlocal torque The configuration averaging in βnl(C3), which con- tains the nonrandom operator [ K,S], leads to two terms βnl=βnl,coh+βnl,vc, (C21) where the coherent part is given by βnl,coh= Tr{¯g1[K,S]¯g2[K,S]}(C22) and the vertex corrections can be compactly written as30 βnl,vc={(ˆ1−ˆw12ˆχ12)−1ˆw12ζ12;ζ21},(C23) with all symbols and quantities defined in the previous section. The coherent part can be written as a sum of four terms, βnl,coh=βnl,coh A+βnl,coh B+βnl,coh C+βnl,coh D, βnl,coh A= Tr{S¯g1KS¯g2K}, βnl,coh B= Tr{¯g1SK¯g2SK}, βnl,coh C=−Tr{¯g1KS¯g2SK}, βnl,coh D=−Tr{S¯g1SK¯g2K}, (C24) which can be further modified using the equation of mo- tion (C4) and its consequences, e.g., S¯gj=Pj¯gj−1. For the first term βnl,coh A, one obtains: βnl,coh A= Tr{P1¯g1KP2¯g2K}+Tr{KK} −Tr{KP2¯g2K}−Tr{P1¯g1KK}.(C25) The last three terms do not contribute to the sum over four pairs of indices ( p,q), where p,q∈ {+,−}, in Eq. (C2). For this reason, they can be omitted for the present purpose, which yields expressions ˜βnl,coh A= Tr{P1¯g1KP2¯g2K}, ˜βnl,coh B= Tr{¯g1P1K¯g2P2K}, (C26)where the second relation is obtained in the same way from the original term βnl,coh B. A similar approach can be applied to the third term βnl,coh C, which yields βnl,coh C=−Tr{¯g1KP2¯g2P2K} +Tr{¯g1KP2K}+Tr{¯g1KSK}.(C27) The last term does not contribute to the sum over four pairs (p,q) in Eq. (C2), which leads to expressions ˜βnl,coh C= Tr{¯g1KP2K}−Tr{¯g1KP2¯g2P2K}, ˜βnl,coh D= Tr{P1K¯g2K}−Tr{P1¯g1P1K¯g2K},(C28) where the second relation is obtained in the same way from the original term βnl,coh D. The sum of all four con- tributions in (C26) and (C28) yields ˜βnl,coh=˜βnl,coh A+˜βnl,coh B+˜βnl,coh C+˜βnl,coh D = Tr{¯g1KP2K}+Tr{P1K¯g2K} +Tr{¯g1γ12¯g2γ21}, (C29) where weused the operators γ12andγ21defined by (C8). The total quantity βnl(C21) is thus equivalent to ˜βnl=˜βnl,coh+βnl,vc = Tr{¯g1KP2K}+Tr{P1K¯g2K} +Tr{¯g1γ12¯g2γ21}+βnl,vc,(C30) where the tildes mark omission of terms irrelevant for the summation over ( p,q) in Eq. (C2). 4. Expression with the local torque The configuration averagingin βloc(C3), involving the random local torque, leads to a sum of two terms:44 βloc=βloc,0+βloc,1, (C31) where the term βloc,0is given by a simple lattice sum βloc,0=/summationdisplay Rβloc,0 R, βloc,0 R= Tr/angbracketleftbig ¯g1(1−t1 R¯g1)(P1 RK−KP2 R) ׯg2(1−t2 R¯g2)(P2 RK−KP1 R)/angbracketrightbig ,(C32) see Eq. (76) of Ref. 44, and the term βloc,1can be written in the present formalism as βloc,1={ˆχ12(ˆ1−ˆw12ˆχ12)−1ϑ12;ϑ21},(C33) which corresponds to Eq. (74) of Ref. 44. The definitions of ˆw12and ˆχ12aregivenby(C15)and(C16), respectively, and ofϑ12andϑ21by (C10). The quantity βloc,0 R(C32) gives rise to four terms, βloc,0 R=QR,A+QR,B+QR,C+QR,D, (C34) QR,A= Tr/an}bracketle{t¯g1(1−t1 R¯g1)P1 RK¯g2(1−t2 R¯g2)P2 RK/an}bracketri}ht, QR,B= Tr/an}bracketle{tP1 R¯g1(1−t1 R¯g1)KP2 R¯g2(1−t2 R¯g2)K/an}bracketri}ht, QR,C=−Tr/an}bracketle{tP1 R¯g1(1−t1 R¯g1)P1 RK¯g2(1−t2 R¯g2)K/an}bracketri}ht, QR,D=−Tr/an}bracketle{t¯g1(1−t1 R¯g1)KP2 R¯g2(1−t2 R¯g2)P2 RK/an}bracketri}ht,14 which will be treated separately. The term QR,Acan be simplified by employing the identities (C6) and the CPA-selfconsistency conditions. This yields: QR,A=UR,A+VR,A, (C35) UR,A= Tr{¯g1P1 RK¯g2P2 RK}, VR,A= Tr/an}bracketle{t¯g1t1 R(1−¯g1P1 R)K¯g2t2 R(1−¯g2P2 R)K/an}bracketri}ht =VR,A1+VR,A2+VR,A3+VR,A4, VR,A1= Tr/an}bracketle{t¯g1t1 RK¯g2t2 RK/an}bracketri}ht, VR,A2= Tr/an}bracketle{t¯g1t1 R¯g1P1 RK¯g2t2 R¯g2P2 RK/an}bracketri}ht, VR,A3=−Tr/an}bracketle{t¯g1t1 R¯g1P1 RK¯g2t2 RK/an}bracketri}ht, VR,A4=−Tr/an}bracketle{t¯g1t1 RK¯g2t2 R¯g2P2 RK/an}bracketri}ht. A similar procedure applied to QR,Byields: QR,B=UR,B+VR,B, (C36) UR,B= Tr{P1 R¯g1KP2 R¯g2K}, VR,B= Tr/an}bracketle{t(1−P1 R¯g1)t1 R¯g1K(1−P2 R¯g2)t2 R¯g2K/an}bracketri}ht =VR,B1+VR,B2+VR,B3+VR,B4, VR,B1= Tr/an}bracketle{tt1 R¯g1Kt2 R¯g2K/an}bracketri}ht, VR,B2= Tr/an}bracketle{tP1 R¯g1t1 R¯g1KP2 R¯g2t2 R¯g2K/an}bracketri}ht, VR,B3=−Tr/an}bracketle{tt1 R¯g1KP2 R¯g2t2 R¯g2K/an}bracketri}ht, VR,B4=−Tr/an}bracketle{tP1 R¯g1t1 R¯g1Kt2 R¯g2K/an}bracketri}ht. The term QR,Crequires an auxiliary relation P1 R¯g1(1−t1 R¯g1)P1 R=P1 R(¯g1P1 R−1) +P1 R−(1−P1 R¯g1)t1 R(1−¯g1P1 R),(C37) that follows from a repeated use of the identities (C6). This relation together with the CPA-selfconsistency lead to the form: QR,C=UR,C+VR,C, (C38) UR,C= Tr{P1 R(1−¯g1P1 R)K¯g2K} −Tr/an}bracketle{tP1 RK¯g2(1−t2 R¯g2)K/an}bracketri}ht, VR,C=−Tr/an}bracketle{t(1−P1 R¯g1)t1 R(1−¯g1P1 R)K¯g2t2 R¯g2K/an}bracketri}ht =VR,C1+VR,C2+VR,C3+VR,C4, VR,C1=−Tr/an}bracketle{tt1 RK¯g2t2 R¯g2K/an}bracketri}ht, VR,C2=−Tr/an}bracketle{tP1 R¯g1t1 R¯g1P1 RK¯g2t2 R¯g2K/an}bracketri}ht, VR,C3= Tr/an}bracketle{tt1 R¯g1P1 RK¯g2t2 R¯g2K/an}bracketri}ht, VR,C4= Tr/an}bracketle{tP1 R¯g1t1 RK¯g2t2 R¯g2K/an}bracketri}ht. A similar procedure applied to QR,Dyields: QR,D=UR,D+VR,D, (C39) UR,D= Tr{¯g1KP2 R(1−¯g2P2 R)K} −Tr/an}bracketle{t¯g1(1−t1 R¯g1)KP2 RK/an}bracketri}ht, VR,D=−Tr/an}bracketle{t¯g1t1 R¯g1K(1−P2 R¯g2)t2 R(1−¯g2P2 R)K/an}bracketri}ht =VR,D1+VR,D2+VR,D3+VR,D4, VR,D1=−Tr/an}bracketle{t¯g1t1 R¯g1Kt2 RK/an}bracketri}ht, VR,D2=−Tr/an}bracketle{t¯g1t1 R¯g1KP2 R¯g2t2 R¯g2P2 RK/an}bracketri}ht, VR,D3= Tr/an}bracketle{t¯g1t1 R¯g1KP2 R¯g2t2 RK/an}bracketri}ht, VR,D4= Tr/an}bracketle{t¯g1t1 R¯g1Kt2 R¯g2P2 RK/an}bracketri}ht,Let us focus now on U-terms in Eqs. (C35 – C39). The second terms in UR,C(C38) and UR,D(C39) do not con- tribute to the sum over four pairs ( p,q) in Eq. (C1), so that the original UR,CandUR,Dcan be replaced by equivalent expressions ˜UR,C= Tr{P1 R(1−¯g1P1 R)K¯g2K}, ˜UR,D= Tr{¯g1KP2 R(1−¯g2P2 R)K}.(C40) The sum of all U-terms for the site Ris then equal to ˜UR=UR,A+UR,B+˜UR,C+˜UR,D = Tr{P1 RK¯g2K}+Tr{¯g1KP2 RK} +Tr{¯g1γ12 R¯g2γ21 R}, (C41) whereγ12 Randγ21 Rare defined in (C8), and the lattice sum of all U-terms can be written as /summationdisplay R˜UR= Tr{P1K¯g2K}+Tr{¯g1KP2K} +/summationdisplay RTr{¯g1γ12 R¯g2γ21 R}. (C42) The summation of V-terms in Eqs. (C35 – C39) can be done in two steps. First, we obtain VR,1=VR,A1+VR,B1+VR,C1+VR,D1 = Tr/an}bracketle{tt1 Rf12t2 Rf21/an}bracketri}ht, VR,2=VR,A2+VR,B2+VR,C2+VR,D2 = Tr/an}bracketle{tt1 R¯g1γ12 R¯g2t2 R¯g2γ21 R¯g1/an}bracketri}ht, VR,3=VR,A3+VR,B3+VR,C3+VR,D3 = Tr/an}bracketle{tt1 R¯g1γ12 R¯g2t2 Rf21/an}bracketri}ht, VR,4=VR,A4+VR,B4+VR,C4+VR,D4 = Tr/an}bracketle{tt1 Rf12t2 R¯g2γ21 R¯g1/an}bracketri}ht, (C43) where the operators f12andf21have been defined in (C7). Second, one obtains the sum of all V-terms for the siteRas VR=VR,1+VR,2+VR,3+VR,4 (C44) = Tr/an}bracketle{tt1 R(f12+ ¯g1γ12 R¯g2)t2 R(f21+ ¯g2γ21 R¯g1)/an}bracketri}ht. The lattice sums of all U- andV-terms lead to an expres- sion equivalent to the original quantity βloc,0(C32): ˜βloc,0=/summationdisplay R˜UR+/summationdisplay RVR = Tr{P1K¯g2K}+Tr{¯g1KP2K} +/summationdisplay RTr{¯g1γ12 R¯g2γ21 R} +/summationdisplay RTr/angbracketleftbig t1 R(f12+¯g1γ12 R¯g2) ×t2 R(f21+ ¯g2γ21 R¯g1)/angbracketrightbig ,(C45) where the tildes mark omission of terms not contributing to the summation over ( p,q) in Eq. (C1). Let us turn now to the contribution βloc,1(C33). It can be reformulated by expressing the quantity ϑ12(and15 ϑ21) in terms of the quantities γ12andζ12(andγ21and ζ21) from the sum rule (C17) and by using the identities (C20). The resultingformcan be written compactlywith help of an auxiliary operator ̺12(and̺21) defined as ̺12= ˆχ12γ12+ζ12. (C46) The result is βloc,1=βnl,vc+{ˆχ12γ12;γ21} −{ˆw12̺12;̺21}, (C47) where the first term has been defined in (C23). For the second term in (C47), we use the relation ˆχ12γ12=/summationdisplay RΠR¯g1(γ12−γ12 R)¯g2ΠR,(C48) which follows from the site-diagonal nature of the opera- torγ12(C8) and from the definition of the superoperator ˆχ12(C16). This yields: {ˆχ12γ12;γ21}= Tr{¯g1γ12¯g2γ21} −/summationdisplay RTr{¯g1γ12 R¯g2γ21 R}.(C49) For the third term in (C47), only the site-diagonalblocks of the operator ̺12(and̺21), Eq. (C46), are needed be- causeofthesite-diagonalnatureofthesuperoperator ˆ w12 (C15). These site-diagonal blocks are given by ΠR̺12ΠR= ΠR/bracketleftbig ¯g1(γ12−γ12 R)¯g2+ζ12/bracketrightbig ΠR =−ΠR(f12+ ¯g1γ12 R¯g2)ΠR,(C50) whichfollowsfromthepreviousrelations(C48)and(C9).This yields: {ˆw12̺12;̺21}=/summationdisplay RTr/angbracketleftbig t1 R(f12+ ¯g1γ12 R¯g2) ×t2 R(f21+ ¯g2γ21 R¯g1)/angbracketrightbig .(C51) The term βloc,1(C47) is then equal to βloc,1=βnl,vc+Tr{¯g1γ12¯g2γ21} −/summationdisplay RTr{¯g1γ12 R¯g2γ21 R} −/summationdisplay RTr/angbracketleftbig t1 R(f12+¯g1γ12 R¯g2) ×t2 R(f21+ ¯g2γ21 R¯g1)/angbracketrightbig .(C52) The total quantity βloc(C31) is thus equivalent to the sum of (C45) and (C52): ˜βloc=˜βloc,0+βloc,1 = Tr{P1K¯g2K}+Tr{¯g1KP2K} +Tr{¯g1γ12¯g2γ21}+βnl,vc,(C53) where the tildes mark omission of terms irrelevant for the summation over ( p,q) in Eq. (C1). 5. 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2015-10-13

We present an ab initio theory of the Gilbert damping in substitutionally disordered ferromagnetic alloys. The theory rests on introduced nonlocal torques which replace traditional local torque operators in the well-known torque-correlation formula and which can be formulated within the atomic-sphere approximation. The formalism is sketched in a simple tight-binding model and worked out in detail in the relativistic tight-binding linear muffin-tin orbital (TB-LMTO) method and the coherent potential approximation (CPA). The resulting nonlocal torques are represented by nonrandom, non-site-diagonal and spin-independent matrices, which simplifies the configuration averaging. The CPA-vertex corrections play a crucial role for the internal consistency of the theory and for its exact equivalence to other first-principles approaches based on the random local torques. This equivalence is also illustrated by the calculated Gilbert damping parameters for binary NiFe and FeCo random alloys, for pure iron with a model atomic-level disorder, and for stoichiometric FePt alloys with a varying degree of L10 atomic long-range order.

Nonlocal torque operators in ab initio theory of the Gilbert damping in random ferromagnetic alloys

1510.03571v2

arXiv:1403.3199v1 [math.OC] 13 Mar 2014The best decay rate of the damped plate equation in a square Ka¨ ıs Ammari∗Abdelkader Sa¨ ıdi† September 18, 2021 Abstract. In this paper we study the best decay rate of the solutions of a dam ped plate equation in a square and with a homogeneousDirichlet boundary conditions. We show that the fastest decay rate is given by the supremum of the real p art of the spectrum of the infinitesimal generator of the underlying semigroup, if the damping c oefficient is in L∞(Ω). Moreover, we give some numerical illustrations by spectral comput ation of the spectrum associated to the damped plate equation. The numerical results ob tained for various cases of damping are in a good agreement with theoretical ones. Computa tion of the spectrum and energy of discrete solution of damped plate show that the best decay rate is given by spectral abscissa of numerical solution. Keywords : optimal decay rate, damped plate, spectrum. AMS subject classifications : 35A05, 35B40, 35B37, 93B07. 1 Introduction Let Ω = (0 ,1)×(0,1)⊂IR2,∂Ω = Γ. We consider the plate equation with interior dissipation, more precisely we have the following partial d ifferential equations : ∂2u ∂t2(x,t)+∆2u(x,t)+a(x)∂u ∂t(x,t) = 0,(x,t)∈Ω×(0,+∞),(1.1) with boundary conditions : u(x,t) = 0,∆u(x,t) = 0,(x,t)∈Γ×(0,+∞), (1.2) and initial conditions : u(x,0) =u0(x),∂u ∂t(x,0) =u1(x), x∈Ω, (1.3) wherea(x)∈L∞(Ω) is a nonnegative damping coefficient. ∗UR Analysis and Control of Pde, UR 13ES64, Department of Math ematics, Faculty of Sciences of Monastir, University of Monastir, 5019 Monastir, Tunisi a, email: kais.ammari@fsm.rnu.tn †Institut de Recherche Math´ ematique Avanc´ ee, University of Strasbourg, 7 rue Ren´ e Descartes, F-67084 Strasbourg, France, email: saidi@math.unistra.f r 1Ifuis a solution of (1.1)-(1.3) we define the energy of uat instant tby : E(t) =1 2/integraldisplay Ω/parenleftBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +|∆u|2/parenrightBig dx. (1.4) Simple calculations show that a sufficiently smooth solution of (1.1)-(1.3) satisfies E(t)−E(0) =−/integraldisplayt 0/integraldisplay ωa(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂u ∂s/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dsdx. ∀t≥0. (1.5) In particular (1.5) implies that E(t)≤E(0),∀t≥0. (1.6) Estimate above suggests that the natural wellposedness spa ce for (1.1)-(1.3) is X= V×L2(Ω),with inner product /an}bracketle{t[f,g],[u,v]/an}bracketri}ht=/integraltext Ω(∆f∆¯u+g¯v)dx,whereV= H2(Ω)∩H1 0(Ω). We have the following wellposedness result Proposition 1.1. Suppose that (u0,u1)∈V×L2(Ω). Then the problem (1.1)-(1.3) admits a unique solution u∈C(0,+∞;V)∩C1(0,+∞;L2(Ω)) Moreover usatisfies the energy estimate (1.5). If we denote by U= [u,∂tu], we can rewrite (1.1)-(1.3) in the form : /braceleftbigg∂tU−AaU= 0, Q= Ω×(0,+∞), U(x,0) =U0(x),Ω,(1.7) whereU0= (u0,u1),Aa:D(Aa)⊂X→Xis the operator defined by : Aa=/parenleftbigg0Id −∆2−a(x)/parenrightbigg , (1.8) D(Aa) =/braceleftbig (u,v)∈[H4(Ω)∩V]×V,∆u|Γ= 0/bracerightbig . In order to state the result on the optimal location of the act uator, we define the decay rate, depending on a, as ω(a) = inf{ω|there exists C=C(ω)>0 such that E(t)≤C(ω)e2ωtE(0), for every solution of (1.1)-(1.3) with initial data in V×L2(Ω)},(1.9) and the spectral abscissa of Aa, µ(a) = sup{Reλ:λ∈σ(Aa)}, (1.10) whereE(t) is defined in (1.4) and σ(Aa) denotes the spectrum of Aa. It follows easily that µ(a)≤w(a). (1.11) According to (1.6) we have that ω(a)≤0 for all a∈L∞(Ω) is nonnegative. Moreover, if a∈L∞(Ω) is nonnegative and satisfying the following condition: ∃c >0s.t., a(x)≥c;a.e.,in an open subset ω⊂Ω, meas(ω)/ne}ationslash= 0.(1.12) We have, according to [12, 1, 16] (see Section 2) for more deta ils) that ω(a)<0. The main result, on the optimal decay rate, is 2Theorem 1.1. Ifa∈L∞(Ω)then µ(a) =w(a). (1.13) Moreover, if the assumption in damping coefficient (1.12)is holds, then all finite energy solutions of (1.1)-(1.3)are exponentially stable which implies that the fastest decay rate of the solutions of (1.1)-(1.3)satisfies (1.13). The problem of finding the optimal decay rate for beams with di stributed interior damping is difficult and has not a complete answer in the case of a variable (in space) damping coefficient. We refer to [2], [3], [4], [6], [14 ], [10], [9], [11] and to references therein. Recently C. Castro and S. Cox in [8] made adecisive contribution by showing that one can get an arbitrarily large decay rate by means of appropriate damping. By this way, they answer by the negative to an old con jecture according to which the best decay rate should be provided by the best con stant damping. The main novelties brought in by this paper is that, in the simple r case of a distributed interior damping, we can give the precise optimal decay rate and we illustrate this result numerically. One of the main ingredients of this stud y is a result showing that the eigenfunctions of the associated dissipative oper ator form a Riesz basis with parentheses in the energy space. We remark that the corresponding optimal decay rate problem make sense since, in this case, the system is exponentially stable (see for ins tance [12], [1]). The paperis organized as follows. Section 2 contains some ba ckground on optimal decay rate of dissipative systems. In sectiontheorique we g ive the proof of the main result. Section 4 is devoted to some illustrations of the mai n result an optimal decay rate by numerical spectral analysis strategy. We pres ent numerical results of computation of the spectrum for different cases of damping. Th e discrete energy of solution is computed in each case and compared to the spectra l abscissa. The latest is showed to be the best decay rate in each example presented. 2 Some background on optimal decay rate for dissipative oper ators LetHbe a Hilbert space equipped with the norm ||.||H, and let A:D(A)⊂H→H beself-adjoint, positiveandwithcompactinvertibleoper ator. Then, Ahasadiscrete spectrum, let ( µk)k≥1be its eigenvalues, each taken with its multiplicity. Denot e a complete orthonormal system of eigenvectors of the operat orAthat correspond to these eigenvalues by ( ϕk)k≥1. Moreover, we suppose that ( µk)k≥1satisfies the following generalized uniform gap: ∃p∈N∗, c >0; such that µk+p−µk≥c,∀k∈N∗. (2.14) We introduce the scale of Hilbert spaces Hβ,β∈IR, as follows: for every β≥0, Hβ=D(Aβ), with the norm /bardblz/bardblβ=/bardblAβz/bardblH. The space H−βis defined by duality with respect to the pivot space Has follows: H−β=H∗ βforβ >0. The operator A can beextended (or restricted) to each Hβ, such that it becomes a boundedoperator A:Hβ→Hβ−1∀β∈IR. (2.15) 3Let a bounded linear operator B:U→H, whereUis another Hilbert space which will be identified with its dual. The systems we consider are described by ¨x(t)+Ax(t)+BB∗˙x(t) = 0, (2.16) x(0) =x0,˙x(0) =x1. (2.17) We can rewrite the system (2.16)-(2.17) as a first order differe ntial equation, by puttingz(t) =/parenleftbiggx(t) ˙x(t)/parenrightbigg : ˙z(t)−Az(t)+BB∗z(t) = 0,z(0) =z0=/parenleftbiggx0 x1/parenrightbigg , (2.18) where A=/parenleftbigg0I −A0/parenrightbigg :D(A) =H1×H1 2⊂ H=H1 2×H→ H,B=/parenleftbigg0 B/parenrightbigg ∈ L(U,H). By the same way the system (2.16)-(2.17) can be also rewritte n by : ˙z(t)+Adz(t) = 0,z(0) =z0, (2.19) where Ad=A−BB∗:D(Ad) =H1×H1 2⊂ H → H . It is clear that the operator Ais skew-adjoint on Hand hence, it generates a strongly continous group of unitary operators on H, denoted by ( S(t))t∈IR. SinceAdis dissipative and onto, it generates a contraction semigro up onH, denoted by ( Sd(t))t∈IR+. The system (2.16)-(2.17) is well-posed. More precisely, th e following classical result, holds. Proposition 2.1. Suppose that (x0,x1)∈H1 2×H. Then the problem (2.16)-(2.17) admits a unique solution (x,˙x)∈C([0,∞);H1 2×H). Moreover wsatisfies, for all t≥0, the energy estimate E(0)−E(t) =/integraldisplayt 0/bardblB∗˙x(s)/bardbl2 U, ds, (2.20) whereE(t) =1 2/bardbl(x(t),˙x(t))/bardbl2 H. From (2.20) it follows that the mapping t/ma√sto→ /bardbl(x(t),˙x(t))/bardbl2 His non increasing. In many applications it is important to know if this mapping d ecays exponentially whent→ ∞, i.e. if the system (2.16)-(2.17) is exponentially stable. One of the methods currently used for proving such exponential stabil ity results is based on 4an observability inequality for the undamped system associ ated to the initial value problem ¨φ(t)+Aφ(t) = 0, (2.21) φ(0) =x0,˙φ(0) =x1. (2.22) Itiswellknownthat(1.3)-(2.22)iswell-posedin H1×H1 2andinH. Theresultbelow, proved in [12, 1], shows that the exponential stability of (2 .16)-(2.17) is equivalent to an observability inequality for (2.21)-(2.22). Theorem 2.1. The system described by (2.16)-(2.17)is exponentially stable in Hif and only if there exists T,CT>0such that CT/integraldisplayT 0||B∗S(t)z0||2 Udt≥ ||z0||2 H∀z0∈ H1. (2.23) In order to state the result on the optimal decay rate, we defin e the decay rate, depending on B, as ω(B) = inf{ω|there exists C=C(ω)>0 such that E(t)≤C(ω)e2ωtE(0), for every solution of (2.16)-(2.17) with initial data in H1 2×H}(2.24) and the spectral abscissa as µ(B) = sup{Reλ:λ∈σ(Ad)}, (2.25) whereσ(Ad) denotes the spectrum of Ad. It follows easily that µ(B)≤ω(B). (2.26) We recall that a Riesz basis in a Hilbert space, is by definitio n, isomorphic to an orthonormal basis. Definition 2.1. A system (φk)k≥1of a space His called a basis with parentheses if the series f=/summationdisplay k≥1ckφkconverges in the norm of Hfor anyf∈Hafter some arrangement of parentheses that does not depend on f. If a system remains a ba- sis after any permutation of the sets of its vectors correspo nding to the terms of the series enclosed in parentheses, then such a system is cal led a Riesz basis with parentheses. The operator Adis a perturbed self-adjoint operator, with the self-adjoin t part is with discrete spectrumwhich satisfies thegap condition (2. 14) and the perturbation, BB∗∈ L(H) is bounded. Then, according to [18, Theorem 2], we have that the eigenvectors of Adforms a Riesz basis with parentheses and according to [5] we obtain on estimation of optimal decay rate. We have the follo wing: Theorem 2.2. ([5, Ammari-Dimassi-Zerzeri]) If the observability inequa lity(2.23) is holds then, ω(B) =µ(B)<0. (2.27) In other words if all finite energy solutions of (2.16)-(2.17)are exponentially stable then the fastest decay rate of the solutions of (2.16)-(2.17)satisfies (2.27). 53 Proof of Theorem 1.1 The eigenvalue problem for the non self-adjoint, quadratic operator pencil generated by (1.1)-(1.3) is obtained by replacing uin (1.1) by u(x,t) =eλtφ(x). We obtain from (1.1) the standard form (Aa−λId)Φ = 0;Φ = [ φ,λφ] =φ[1,λ]. The condition for the existence of non trivial solutions is t hatλ∈σ(Aa) (the spectrumof Aa). SinceD(Aa) is compactly embeddedin theenergy space V×L2(Ω) then the spectrum σ(Aa) is discrete and the eigenvalues of Aahave a finite algebraic multiplicity. On the other hand, since Aais a bounded monotone perturbation of a skew-adjoint operator (undamped A0), it follows from the Hille-Yosida theorem thatAagenerates a C0-semigroup of contractions on the energy space V×L2(Ω). According to [16, Proposition A.1] the spectra of Aasatisfies the gap assumption (2.14). So, by Theorem 2.2 we end the proof. /square 4 Discretization of the eigenvalue problem The domain Ω is approximated by a net of equidistant discrete points Ω h=Mij withi,j= 1,2,...,NandN= (1/h)+1. The Laplacian operator is approximated in a standard way by a second order centered difference scheme : ∆hu= (ui+1,j+ui−1,j+ui,j+1+ui,j−1−4ui,j)/h2(4.28) whereui,j=u(xi,yj). The discrete approximation of the operator Aadefined by (1.8) is then given by a nonsymmetric block matrix, Ah, of the form : Ah=/bracketleftbigg0−IdN ∆2 hah/bracketrightbigg (4.29) whereIdNis the identity matrix of order Nand ∆2 his the discrete version of the bilaplacian. ahis a diagonal matrix with akk=a(xi,yj). The value of kis deter- mined by the numbering of the discrete points Mij. We use the implicitly restarted Arnodi-Lanszos method of Sorenson [19] to compute the eigen values of matrix Ah. This method is a generalization of an inverse power method wi th subspace iteration [6],[19]. 5 The case of square We consider a domain Ω = (0 ,1)×(0,1) with a set of normalized eigenfunctions Φn,m(x,y) =√ 2sin(nπx)√ 2sin(mπy), for all ( x,y)∈Ω the solution u(x,y,t) of the problem (1.1)-(1.2) is given by : u(x,y,t) =/summationdisplay n,mαn,m(t)Φn,m(x,y) (5.30) 6replacing (6.32) in equation (1.1) and multiplying by a test eigenfunction Φ k,l(x,y) the solution for the low frequencies is solution of the equat ion : d2αn,m(t) dt2+((nπ)2+(mπ)2)2αn,m(t)+dαn,m(t) dt= 0 (5.31) In the first example we consider a damped plate with constant c oefficient a(x) = 1 and a damping region covering all the domain ω= Ω. We compute the spec- trum of the damped plate (figure 1) and the energy of the first ei genmodes : n=m=3 and n=m=12 (figure 2 ). The energy of the solution (line 1 ) is com- pared to E0eRe(λ0)t(line 2), where E0=E(0) is the energy of initial conditions andRe(λ0) =inf{Re(λ)/λ∈σ(A)}, (σ(A) : the spectrum of operator A). In the other examples we consider a damping in different domain sω. The spectrum of the damped plate is computed and energy for different eigenm odes is compared toE0eRe(λ0)t(line 2 : dashed line). −0.55 −0.54 −0.53 −0.52 −0.51 −0.5 −0.49 −0.48 −0.47 −0.46 −0.45−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.22 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 1 : spectrum of the damped plate a(x) = 1 ,ω= Ω (left) , a(x) = 1 ,ω= (0,1 2)×(0,1 2) (right). 0 2 4 6 8 10 12 14 16 18 2000.511.522.533.544.55 timeenergyline1 line2 0 5 10 15 20 25 30 35 40020004000600080001000012000 timeenergyline1 line2 figure 2 : energy of the damped plate, a(x) = 1 ,ω= Ω,n=m= 3 (left), and n=m= 12 (right). 70 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55 timeenergyline1 line2 0 5 10 15 20 25 30 35 40 45 50020040060080010001200 timeenergyline1 line2 figure 3 : energy of the damped plate, a(x) = 1,ω= (0,1 2)×(0,1 2),n=m= 3 (left), and n=m= 8 (right). In figures 4. 5. and 6. we consider a damping in the domain ω= (0,2 5)×(0,2 5) and ω= (0,3 5)×(0,3 5) : −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.26 −0.24 −0.22 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 4 : spectrum of the damped plate, a(x) = 1 ,ω= (0,0.4)×(0,0.4) (left), ω= (0,3 5)×(0,3 5) (right). 0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.7 timeenergyline1 line2 0 5 10 15 20 25 30 35 40 45 50020040060080010001200 timeenergyline1 line2 figure 5 : energy of the damped plate, a(x) = 1,ω= (0,0.4)×(0,0.4),n=m= 2 (left), and n=m= 8 (right). 80 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55 timeenergyline1 line2 0 5 10 15 20 25 30 35 40 45 50020040060080010001200 timeenergyline1 line2 figure 6 : energy of the damped plate, a(x) = 1,ω= (0,0.6)×(0,0.6),n=m= 3 (left), and n=m= 8 (right). In figures 7., 8. and 9. we consider a damping in the domain ω= (0,1 4)×(0,1 4) and ω= (0,1 5)×(0,1 5) : −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 7 : spectrum of the damped plate, a(x) = 1,ω= (0,1 4)×(0,1 4) (left) and ω= (0,1 5)×(0,1 5) (right). 0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.7 timeenergyline1 line2 0 5 10 15 20 25 30 35 40 45 5001020304050607080 timeenergyline1 line2 figure 8 : energy of the damped plate, a(x) = 1,ω= (0,0.25)×(0,0.25),n=m= 2 (left), and n=m= 5 (right). 90 10 20 30 40 50 60 70 80 90 10000.511.522.533.544.55 timeenergyline1 line2 0 100 200 300 400 500 600 700 800020040060080010001200 timeenergyline1 line2 figure 9 : energy of the damped plate, a(x) = 1,ω= (0,0.2)×(0,0.2),n=m= 3 (left), and n=m= 8 (right). In figures 10., 11. and 12. we consider a damping in the domain ω= (0,1)×(0,1 2) andω= (0,1)×(0,1 4) : −0.3 −0.29 −0.28 −0.27 −0.26 −0.25 −0.24 −0.23 −0.22 −0.21 −0.2−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 10 : spectrum of the damped plate, a(x) = 1 and ω= (0,1)×(0,1 2) (left) , a(x) = 1 and ω= (0,1)×(0,1 4) (right). 0 5 10 15 20 25 30 35 40 45 5001020304050607080 timeenergyline1 line2 0 5 10 15 20 25 30 35 40 45 5005001000150020002500300035004000 timeenergyline1 line2 figure 11 : energy of the damped plate, a(x) = 1,ω= (0,1)×(0,0.5),n=m= 5 (left), and n=m= 10 (right). 100 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55 timeenergyline1 line2 0 5 10 15 20 25 30 35 40 45 5005001000150020002500300035004000 timeenergyline1 line2 figure 12 : energy of the damped plate, a(x) = 1,ω= (0,1)×(0,1 4),n=m= 3 (left), and n=m= 10 (right). −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.2 −0.15 −0.1 −0.05 0−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 13 : spectrum of the damped plate, a(x) = 1,ω= (0.4,0.6)×(0.4,0.6) (left), a(x) = 1,ω= (0.35,0.65)×(0.35,0.65) (right). 0 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100050100150200250 timeenergyline1 line2 figure 14 : energy of the damped plate, a(x) = 1,ω= (0.4,0.6)×(0.4,0.6), n=m= 2 (left), and n=m= 6 (right). 110 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100050100150200250 timeenergyline1 line2 figure 15 : energy of the damped plate, a(x) = 1,ω= (0.35,0.65)×(0.35,0.65), n=m= 2 (left), and n=m= 6 (right). −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 16 : spectrum of the damped plate, a(x) = 1,ω= (0.25,0.75)×(0.25,0.75) (left),a(x) = 1,ω= (0.5,0.75)×(0.3,0.6) (right). 0 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100020040060080010001200 timeenergyline1 line2 figure 17 : energy of the damped plate, a(x) = 1,ω= (0.25,0.75)×(0.25,0.75), n=m= 2 (left), and n=m= 8 (right). In figures 13., 14. and 15., for the value a(x) = 1 we consider a damping in the domainω= (0.4,0.6)×(0.4,0.6) andω= (0.35,0.65)×(0.35,0.65). In figures 16., 17. and 18. , we consider a damping in the domain ω= (0.25,0.75)×(0.25,0.75), andω= (0.5,0.75)×(0.3,0.6). 120 10 20 30 40 50 60 70 80 90 10000.10.20.30.40.50.60.7 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100020040060080010001200 timeenergyline1 line2 figure 18 : energy of the damped plate, a(x) = 1,ω= (0.5,0.75)×(0.3,0.6), n=m= 2 (left), and n=m= 8 (right). In the following examples the damping is a function given by a(x1,x2) =x1x2. In the case of a square the spectrum of the damped plate is comput ed for three position of damping domain ω= Ω,ω= (0,0.5)×(0,0.5) andω= (0.35,0.75)×(0.35,0.75) −0.125 −0.12 −0.115 −0.11 −0.105 −0.1 −0.095 −0.09 −0.085 −0.08−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 19 : spectrum of the damped plate, a(x1,x2) =x1x2,ω= Ω . 0 1 2 3 4 5 6 7 8 9 1001020304050607080 timeenergyline1 line2 0 1 2 3 4 5 6 7 8 9 100.60.811.21.41.61.822.22.4x 105 timeenergyline1 line2 figure 20 : energy of the damped plate, a(x1,x2) =x1x2, damping in all the domainn=m= 5 (left), and n=m= 20 (right). 13−14 −12 −10 −8 −6 −4 −2 x 10−3−4000−3000−2000−100001000200030004000 real axisimaginary axis 0 10 20 30 40 50 60 70 80 90 10020304050607080 timeenergyline1 line2 figure 21 : a(x1,x2) =x1x2,ω= (0,0.5)×(0,0.5) spectrum of the damped plate, (left), energy of the damped plate, n=m= 5, (right) . −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0−4000−3000−2000−100001000200030004000 real axisimaginary axis 0 10 20 30 40 50 60 70 80 90 10001020304050607080 timeenergyline1 line2 figure 22 : a(x1,x2) =x1x2,ω= (0.35,0.75)×(0.35,0.75) , spectrum of the damped plate, (left) , energy of the damped plate, n=m= 5, (right). In the next example we consider a damping defined by a(x1,x1) =sin(x1)cos(x2) for three region ω= Ω,ω= (0,0.5)×(0,0.5) andω= (0.3,0.7)×(0.3,0,7). We plot the spectrum in the three situations and the energy of the dam ped plate compared toE0eRe(λ0)t(line 2 : dashed line). figures 23., 24., and figure 25. 14−0.195 −0.19 −0.185 −0.18 −0.175 −0.17−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 23 : spectrum of the damped plate, a(x1,x2) =sin(x1)cos(x2),ω= Ω (left) , ω= (0,0.5)×(0,0.5) (right). 0 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100020040060080010001200 timeenergyline1 line2 figure 24 : energy of the damped plate, a(x1,x2) =sin(x1)cos(x2),ω= Ω, n=m= 3 (left), and n=m= 8 (right). 0 5 10 15 20 25 30 35 40 45 5000.511.522.533.544.55 timeenergyline1 line2 0 5 10 15 20 25 30 35 40 45 50050100150200250 timeenergyline1 line2 figure 25: energy of the damped plate, a(x1,x2) =sin(x1)cos(x2),ω= (0,0.5)×(0,0.5),n=m= 3(left), and n=m= 6 (right). 156 The case of rectangle In this case we consider rectangular a domain Ω = (0 ,a)×(0,b) with a set of normalized eigenfunctions Φ n,m(x,y) =/radicalBig 2 asin(nπx a)/radicalBig 2 bsin(mπy b), for all ( x,y)∈Ω the solution u(x,y,t) of the problem (1.1)-(1.2) is given by : u(x,y,t) =/summationdisplay n,mαn,m(t)Φn,m(x,y) (6.32) with : d2αn,m(t) dt2+((nπ a)2+(mπ b)2)2αn,m(t)+dαn,m(t) dt= 0 (6.33) as in the case of square we compute the spectrum and energy of t he first eigenmodes for different domains ω. We compare in each case the energy of the solution (line 1) toE0eRe(λ0)t(line 2), with Re(λ0) =inf{Re(λ)/λ∈σ(A)}. −0.55 −0.54 −0.53 −0.52 −0.51 −0.5 −0.49 −0.48 −0.47 −0.46 −0.45−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 26 : spectrum of the damped plate a(x) = 1,ω= Ω (left), and ω= (0,1 2)×(0,1) (right). 0 2 4 6 8 10 12 14 16 18 2000.511.522.533.544.55 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100020040060080010001200140016001800 timeenergyline1 line2 figure 27 : energy of the damped plate , a(x) = 1,n=m= 3 (left), and n=m= 12 (right). 160 10 20 30 40 50 60 70 80 90 10000.050.10.150.20.25 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 1000500100015002000250030003500400045005000 timeenergyline1 line2 figure 28 : energy of the damped plate, a(x) = 1,ω= (0,1 2)×(0,1),n=m= 2 (left), and n=m= 10 (right). −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08−4000−3000−2000−100001000200030004000 real axisimaginary axis −0.1 −0.09 −0.08 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0−4000−3000−2000−100001000200030004000 real axisimaginary axis figure 29 : spectrum of the damped plate a(x) = 1,,ω= (0,0.6)×(0,1), (left) , ω= (0,1 5)×(0,1) (right). 0 10 20 30 40 50 60 70 80 90 10000.20.40.60.811.21.41.61.82 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100020040060080010001200140016001800 timeenergyline1 line2 figure 30 : energy of the damped plate, a(x) = 1,ω= (0,1 5)×(0,1),n=m= 3 (left), and n=m= 10 (right). 170 10 20 30 40 50 60 70 80 90 10000.050.10.150.20.25 timeenergyline1 line2 0 10 20 30 40 50 60 70 80 90 100020040060080010001200140016001800 timeenergyline1 line2 figure 31 : energy of the damped plate, a(x) = 1,ω= (0,0.6)×(0,1),n=m= 2 (left), and n=m= 10 (right). 7 Optimization of the position of the damped region ω Starting from a fixed damping domain ωwe try to see the influence of the position on decay energy. In the first example we plot the energy for five different damping position : ω= (0.,0.4)×(0,0.4),ω= (0.3,0.7)×(0.3,0.7) (dashed line), ω= (0.5,0.9)×(0.4,0.8) ,ω= (0.1,0.5)×(0.5,0.9),ω= (0.3,0.7)×(0,0.4). The best position in this case is the middle of the plate. 0 5 10 15 20 25 30 35 40020040060080010001200 timeenergy figure 32 : energy of the damped plate, a(x) = 1,n=m= 8. In the second example we plot the energy for five damping regio nω= (0.,0.3)× (0,0.3) (green), ω= (0.35,0.65)×(0.35,0.65) (red), ω= (0.1,0.4)×(0.6,0.9) (blue) ,ω= (0.7,1)×(0.5,0.8) (black), ω= (0.7,1)×(0.1,0.4), yellow. The best position in this case is the corner : ω= (0.,0.3)×(0,0.3). 180 10 20 30 40 50 60 70 80020040060080010001200 timeenergy figure 33 : energy of the damped plate, a(x) = 1,n=m= 8. 0 10 20 30 40 50 60 70 8001020304050607080 timeenergy figure 34 : energy of the damped plate, a(x) = 1,n=m= 5. In the third example we plot the energy for five damping region ω= (0.,0.2)× (0,0.2) (black), ω= (0.1,0.3)×(0.1,0.3) (green), ω= (0.2,0.4)×(0.2,0.4) (blue) ,ω= (0.3,0.5)×(0.3,0.5) (yellow), ω= (0.4,0.6)×(0.4,0.6), (red). for n=m= 5,n=m= 6 and n=m= 7, 0 50 100 150 200 250 30001020304050607080 timeenergy 0 50 100 150 200 250 300050100150200250 timeenergy figure 35 : energy of the damped plate, a(x) = 1,n=m= 5 and n=m= 7. 190 10 20 30 40 50 60 70 80020040060080010001200 timeenergy figure 36 : energy of the damped plate, a(x) = 1,n=m= 7. References [1]K. Ammari and M. Tucsnak, Stabilization of second order evolution equa- tions by a class of unbounded feedbacks, ESAIM Control Optim. Calc. Var, ESAIM Control Optim. Calc. Var, 6(2001), 361-386. [2]K. Ammari, A. Henrot and M. Tucsnak, Optimal location of the actuator for the pointwise stabilization of a string, C. R. Acad. Sci. Paris S´ er I.Math. , 330(2000), 275-280. [3]K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the point wise stabilization of a string, Asymptotic Analysis., 28(2001), 215-240. [4]K. Ammari and A. Sa ¨ıdi,Optimal location of the actuator at high frequency for the pointwise stabilization of a Bernoulli-Euler beam, Control Cybernetics, 31(2002), 57-66. [5]K. Ammari, M. Dimassi and M. Zerzeri, Best decay rate of some dissipatifs systems, preprint. [6]M. Asch and G. Lebeau, The spectrum of the damped wave operator for a bounded domain in IR2,Experimental Mathematics, 12(2003), 227-241. [7]C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the bo undary, SIAM J. Control Optim., 30(1992), 1024-1065. [8]C. Castro and S. Cox, Achieving arbitrarily large decay in the dampedwave equation, SIAM J. Control. Optim. ,39(2001), 1748–1755. [9]S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math.J. ,44(1995), 545-573. [10]S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial Differential Equations ,19(1994), 213-243. 20[11]P. Freitas, Optimizing the rate of decay of solutions of the wave equatio n usinggeneticalgorithms: acounterexampletotheconstant dampingconjecture, SIAM J. Control Optim ,37(1999), 376-387. [12]A. Haraux, Une remarque sur la stabilisation de certains syst` emes du deuxi` eme ordre en temps, Port. Math. ,46(1989), 245-258. [13]P. Freitas, Optimizing the Rate of Decay of Solutions of the Wave Equa- tion Using Genetic Algorithms: A Counterexample to the Cons tant Damping Conjecture SIAM J. Control Optim., ,37, No 2, (1998), 376-387. [14]G. Lebeau, Equation des ondes amorties, Algebraic and geometric methods in mathematical physics. Proceedings of the 1st Ukrainian- French-Romanian summer school, Kluwer Academic Publishers. Math. Phys. Stu d.19(1996), 73- 109. [15]J. L. Lions et E. Magenes, Probl` emes aux limites non homog` enes et appli- cations, Dunod, Paris, 1968. [16]K. Ramadani, T. Takahashi and M. Tucsnak , Internal stabilization of the plate equation in a square: the continuous and the semi-disc etized problems, J. Math. Pures Appl., 85(2006), 17-37. [17]D. L. Russell, Decay rates for weakly damped systems in Hilbert space ob- tained with control theoretic methods, J. Diff. Eq. ,19(1975), 344-370. [18]A. A. Shkalikov, On the basis property of root vectors of a perturbed self-adjoint operator, Proceedings of the Steklov Institute of Mathematics ,269 (2010), 290 - 303. [19]D. C. Sorenson, Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Problems , MATLAB documentation, 1995. 21

2014-03-13

In this paper we study the best decay rate of the solutions of a damped plate equation in a square and with a homogeneous Dirichlet boundary conditions. We show that the fastest decay rate is given by the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup, if the damping coefficient is in $L^\infty(\Omega).$ Moreover, we give some numerical illustrations by spectral computation of the spectrum associated to the damped plate equation. The numerical results obtained for various cases of damping are in a good agreement with theoretical ones. Computation of the spectrum and energy of discrete solution of damped plate show that the best decay rate is given by spectral abscissa of numerical solution.

The best decay rate of the damped plate equation in a square

1403.3199v1

Bayesian interpretation of Backus-Gilbert methods Luigi Del Debbio,𝑎Alessandro Lupo,𝑏,∗Marco Panero𝑐and Nazario Tantalo𝑑 𝑎Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK 𝑏Aix-Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France 𝑐Department of Physics, University of Turin & INFN, Turin Via Pietro Giuria 1, I-20125 Turin, Italy 𝑑University and INFN of Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133, Rome, Italy E-mail: alessandro.lupo@cpt.univ-mrs.fr The extraction of spectral densities from Euclidean correlators evaluated on the lattice is an important problem, as these quantities encode physical information on scattering amplitudes, finite-volume spectra, inclusive decay rates, and transport coefficients. In this contribution, we show that the Bayesian approach to this “inverse” problem, based on Gaussian processes, can be reformulated in a way that yields a solution equivalent, up to statistical uncertainties, to the one obtained in a Backus-Gilbert approach. After discussing this equivalence, we point out its implications for a reliable determination of spectral densities from lattice simulations. The 40th International Symposium on Lattice Field Theory (Lattice 2023) July 31st - August 4th, 2023 Fermi National Accelerator Laboratory ∗Speaker ©Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2311.18125v1 [hep-lat] 29 Nov 2023Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo 1. Introduction ThestudyofthenumericalinversionoftheLaplacetransformhasbecomepopularinthelattice community,duetoitsimportanceindetermininghadronicobservablesfromsimulationsofquantum chromodynamics,andgaugetheoriesingeneral. Theproblem,however,isaparticularlychallenging one, and requires a careful treatment to overcome the difficulties related to the inherently limited information that lattice data can provide, due to systematic and statistical uncertainties. Several approaches have been devised [1–5] as a means to provide a stable and reliable solution to the problem, leading to an increase in applications [6–17]. In this work, we focus on two popular methods to tackle this inverse problem: the variation of the Backus-Gilbert (BG) procedure introduced in Ref. [1] and a Bayesian approach based on Gaussian Processes (GP) [3, 18–20]. Even though these two approaches are based on drastically different philosophies we shall prove, by building on the results of Ref. [18] and expanding the discussion present in Ref. [11], that a suitable choice of the inputs produces a Bayesian solution centredaroundthemodifiedBGprediction. AfterabriefintroductioninSection2wedescribethe Bayesian solution to the inversion problem in Section 3 which we then generalise, in Section 4, to match the results from Ref. [1]. 2. Formulation of the problem Inlatticesimulations,Euclideancorrelators 𝐶𝐿𝑇computedinafinitehypervolume 𝐿3×𝑇are related to the spectral density 𝜌𝐿𝑇(𝐸)via a (generalised) Laplace transform, 𝐶𝐿𝑇(𝑡)=∫∞ 0𝑑𝐸 𝑏𝑇(𝑡,𝐸)𝜌𝐿𝑇(𝐸), (1) that has to be inverted to extract 𝜌𝐿𝑇(𝐸). We define 𝐶𝐿(𝑡)as the correlator in the limit 𝑇→∞, where𝑏𝑇(𝑡,𝐸)→𝑒−𝑡𝐸. Spectraldensitiesareespeciallyhardtomanageinafinitevolume,where they are a sum of Dirac 𝛿distributions across the discrete spectrum of the Hamiltonian. For this reason, numerical methods typically target a smeared version of the spectral density, whereby the finite-volume function 𝜌𝐿(𝐸)is convoluted with a Schwartz function S𝜎(𝜔,𝐸), 𝜌𝐿(𝜎;𝜔)=∫∞ 0𝑑𝐸S𝜎(𝜔,𝐸)𝜌𝐿(𝐸), (2) lim 𝜎→0S𝜎(𝜔,𝐸)=𝛿(𝜔−𝐸). (3) Eq. (2) provides a way to define the infinite-volume limit for the spectral density by the following non-commuting double limit 𝜌(𝜔)=lim 𝜎→0lim 𝐿→∞𝜌𝐿(𝜎;𝜔). (4) Aswelldocumentedintheliterature,theproblemofextracting 𝜌𝐿(𝜎,𝜔)from𝐶𝐿(𝑡)isill-defined. To clarify this point, we begin by noting that if one had access to an infinite set of discrete data that were exact, i.e., unaffected by uncertainties, then the solution could be written as a linear combination of the data 𝜌exact(𝜎;𝜔)=∞∑︁ 𝜏=1𝑔exact 𝜏(𝜎;𝜔)𝐶exact(𝑎𝜏). (5) 2Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo It is important to note that the previous infinite sum is an exactrepresentation of the continuous smeared spectral density even if the data only constitute a discrete set. A first obstruction arises from the fact that, in reality, the correlator is available only at a finite number of data points 0<𝜏≤𝜏max<𝑇/𝑎(where𝜏=𝑡/𝑎), which results into a systematic error 𝜌exact(𝜎;𝜔)=𝜏max∑︁ 𝜏=1𝑔𝜏(𝜎;𝜔)𝐶exact(𝑎𝜏)+𝛿sys(𝜎,𝜏 max). (6) The coefficients 𝑔𝜏(𝜎;𝜔)from Eq. (6) are typically very large and change sign swiftly, a property thatisnecessaryinordertoreproduceasmoothfunctionoutoftheexponentiallydecayingkernels of the correlators. This leads to a major difficulty in inverting Eq. (1). The correlators obtained fromlatticesimulationsare,infact,unavoidablyaffectedbystatisticalandsystematicuncertainties, making Eq. (6) numerically unstable. A way to tackle this problem consists in “regularising” the sum 𝜌(𝜎;𝜔)=𝜏max∑︁ 𝜏=1𝑔𝜏(𝜎;𝜔)𝐶(𝑎𝜏) (7) by reducing the size of the 𝑔𝜏coefficients, so that 𝜌(𝜎;𝜔)does not depend too strongly on the noise of the correlators. At the same time, the “regularised” coefficients should still yield meaningful results with controlled uncertainties. Before describing two of these regularisations in the following section, we remark that any linear combination of correlators necessarily reproduces a smeared spectral density, 𝜏max∑︁ 𝜏=1𝑔𝜏(𝜎;𝜔)𝐶(𝑎𝜏)=∫ 𝑑𝐸 𝜏max∑︁ 𝜏=1𝑔𝜏(𝜎;𝜔)𝑏𝑇(𝑎𝜏,𝐸)! 𝜌𝐿(𝐸), (8) the smearing kernel being S𝜎(𝐸,𝜔)=𝜏max∑︁ 𝜏=1𝑔𝜏(𝜎;𝜔)𝑏𝑇(𝑎𝜏,𝐸). (9) 3. Bayesian Inference with Gaussian Processes LetRbe a stochastic field described by the Gaussian Process centred around 𝜌priorand with covarianceKprior Π[R]=1 Nexp −1 2R−𝜌prior2 Kprior , (10) where we introduced the norm |R−𝜌prior2 Kprior=∫ 𝑑𝐸1∫ 𝑑𝐸2 R(𝐸1)−𝜌prior(𝐸1) K−1 prior(𝐸1,𝐸2) R(𝐸2)−𝜌prior(𝐸2) , (11) and the normalisation N=∫ DRΠ[R]. (12) In the last expressions, D𝑅is the functional integration measure over the field variable R, which we shall use to describe a continuous function such as the spectral density. 3Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo LetC∈R𝜏maxbe a vector of stochastic variables, with components C(𝑡)=∫ 𝑑𝐸𝑏𝑇(𝑡,𝐸)R(𝐸)+𝜂(𝑡). (13) The vector 𝜼∈R𝜏max, which describes the statistical fluctuations of the lattice correlators, is represented as a real-valued stochastic variable, for which we assume a multivariate Gaussian distribution with zero mean and covariance Cov 𝑑, G[𝜼,Cov𝑑]=1√︁ det(2𝜋Cov𝑑)exp −1 2𝜼Cov−1 𝑑𝜼 . (14) Eq. 13 allows the evaluation of the covariance: Cov[C(𝑡1),C(𝑡2)]=Σ𝑡1𝑡2+(Cov𝑑)𝑡1𝑡2, (15) where we defined Σ𝑡1𝑡2=∫ 𝑑𝐸1∫ 𝑑𝐸2𝑏𝑇(𝑡1,𝐸1)Kprior(𝐸1,𝐸2)𝑏𝑇(𝑡2,𝐸2). (16) We are interested in the value of the spectral density at some energy 𝜔, its covariance and its correlation withC; for this purpose we extend the dimensionality of the covariance of Eq. (15) as follows. Let us introduce the vector 𝑭∈R𝜏max, Cov[C(𝑡),R(𝜔)]=𝐹𝑡(𝜔), (17) and the scalar function 𝐹∗, Cov[R(𝜔),R(𝜔)]=𝐹∗(𝜔). (18) The extended covariance is then Σtot= 𝐹∗ 𝑭𝑇 𝑭Σ+Cov𝑑! . (19) Let𝐶obs(𝑡)bethecentralvalueofthecorrelatormeasuredonthelatticebyaveragingoveragauge ensemble, and let us denote 𝑪priorthe vector of components 𝐶prior(𝑡)=∫ 𝑑𝐸𝑏𝑇(𝑡,𝐸)𝜌prior(𝐸). (20) The joint probability density for R(𝜔)andC(𝑡)is the multidimensional Gaussian G R−𝜌prior,C−𝑪prior;Σtot , (21) which can be factorised into the product of two Gaussian distributions by performing a block diagonalisationofthetotalcovarianceinEq.(19). Ofthetworesultingdistributions,onerepresents the posterior probability density for R(𝜔)given its prior distribution and the set of measurements for the correlator, while the other is the likelihood of the data: G R−𝜌prior,C−𝑪prior;Σtot =G R−𝜌post;Kpost G C−𝑪prior;Σ+Cov𝑑 .(22) 4Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo The posterior Gaussian distribution for the spectral density is centred around: 𝜌post(𝜔) C=𝐶obs=𝜌prior(𝜔)+𝑭𝑇 1 Σ+Cov𝑑 𝑪obs−𝑪prior , (23) and has variance Kpost(𝜔,𝜔) C=𝐶obs=Kprior(𝜔,𝜔)−𝑭𝑇 1 Σ+Cov𝑑𝑭. (24) In order to make contact with Eq. (6), we introduce the coefficients 𝒈GP(𝜔)=𝑭𝑇 1 Σ+Cov𝑑. (25) Let us discuss the previous equations. Eq. (19) shows how the inverse problem is regularised once expressed in terms of probability distributions: the numerical instability, which leads to the very largecoefficientsofEq.(5),isduetotheill-conditioningofthemodelcovariance ΣinEq.(16). The covarianceCov 𝑑,addedtothematrix Σ,cutsoffitsnear-zeroeigenvalues. Theresultingsolutionis then stable within statistical uncertainties. The very same regularisation is used in Backus-Gilbert methods[1,11,21],despitethefactthattheycontainnoformulationintermsofstochasticvariables. AspointedoutaroundEqs.(8)and(9),thecentreoftheposteriordescribesasmearedspectral density,evenifthesmearingwasnotoneoftheinitialassumptions. Thesmearingfunction,which canbeinferredfromEq.(9),isdeterminedbythechoiceofthepriorsandthenoiseonthedata,as seeninEq.(25). Consequently,thesmearingkernelobtainedinthissetupisnotknownapriori. In thisregard,thesolutiongiveninthissectionissimilartotheoriginalBackus-Gilbertproposal[21] ratherthanthemethodofRef.[1],wherethesmearingkernelischosen. Thedirectconnectionwith Ref. [1] is given in the next section. 4. Backus-Gilbert methods in the Bayesian framework The first step is to target the probability density for a spectral density smeared with an input kernelS𝜎(𝜔,𝐸). To generalise the results of the previous section, we introduce the stochastic variable R𝜎(𝜔)=∫ 𝑑𝐸S𝜎(𝜔,𝐸)R(𝐸). (26) The same steps described in Section 3 lead to the following extended covariance, Σtot= 𝐹𝜎∗𝑭𝜎 𝑭𝜎Σ+Cov𝑑! , (27) whereΣis as in Eq. (16), while the other functions now include reference to the smearing kernel: 𝐹𝜎 ∗(𝜔)=∫ 𝑑𝐸1∫ 𝑑𝐸2S𝜎(𝜔,𝐸 1)Kprior(𝐸1,𝐸2)S𝜎(𝐸2,𝜔), (28) 𝐹𝜎 𝑡(𝜔)=∫ 𝑑𝐸1∫ 𝑑𝐸2𝑏𝑇(𝑎𝜏,𝐸 1)Kprior(𝐸1,𝐸2)S𝜎(𝐸2,𝜔). (29) To match Ref. [1] one can select [11] a diagonal model covariance, Kprior(𝐸1,𝐸2)=𝑒𝛼𝐸 𝜆𝛿(𝐸1−𝐸2), (30) 5Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo where𝛼 < 2and𝜆∈ (0,∞)are, in this context, hyperparameters, which can be chosen by maximising the data likelihood [3]. They are introduced to match the parameters appearing in Eq. (A3) in Ref. [1]. Letthesmearingkernelbe,forinstance,aGaussian 𝐺𝜎(𝜔,𝐸)=exp[−(𝜔−𝐸)2/2𝜎2]/√ 2𝜋𝜎. The posterior Gaussian distribution for a smeared spectral density, given the prior distribution and the observed data, is centred around 𝜌post 𝜎(𝜔)=𝜌prior 𝜎(𝜔)+𝜏max∑︁ 𝜏=1𝑔BG 𝜏(𝜎,𝜔)𝐶(𝑎𝜏), (31) and has variance Kpost(𝜔,𝜔)=∫ 𝑑𝐸𝐺2 𝜎(𝜔,𝐸)𝑒𝛼𝜔 𝜆 −𝜏max∑︁ 𝜏=1𝑔BG 𝜏(𝜎,𝜔)𝐹𝜎 𝜏(𝜔). (32) Due to the careful choice of model covariance in Eq. (30), the coefficients 𝑔BG 𝜏(𝜎,𝜔)are the same that were derived in Ref. [1]. We recall that in the latter, the coefficients 𝑔BG 𝜏are determined in a drastically different way, i.e. by minimising the following functional: (1−𝜆′)∫∞ 0𝑑𝐸𝑒𝛼𝐸𝜏max∑︁ 𝜏=1𝑔𝜏(𝜎;𝜔)𝑏𝑇(𝑎𝜏,𝐸)−𝐺𝜎(𝜔,𝐸)2 +𝜆′𝜏max∑︁ 𝜏,𝜏′=1𝑔𝜏(𝜎;𝜔)Cov𝜏𝜏′𝑔𝜏′(𝜎;𝜔). (33) The Backus-Gilbert parameter 𝜆′∈(0,1)is related to the 𝜆in Eq. (30) by 𝜆=𝜆′/(1−𝜆′). We identified a setup in which Bayesian and Ref. [1], two frameworks with utterly different philosophies,providethesamecentralvalueforthesmearedspectraldensities. Thereare,however, important differences between the two methods, that we shall now discuss. A first aspect concerns the error on the smeared spectral density. In non-Bayesian methods, including the one introduced in Ref. [1], statistical uncertainties are often propagated from the data by bootstrap. In the context ofGPs,ontheotherhand,byworkingwithGaussiandistributionsweareabletoprovideananalytic expression for the error, which is inherited by Eq. (32), the variance of the probability density that describes the smeared spectral densities. Another important difference is the way algorithmic parameters are determined. For the Backus-Gilbert method, a procedure that has been proven effective [11, 12, 22] was introduced in Ref. [9]: it consists in finding a range of parameters, 𝜆 and𝛼, such that any shift in the smeared spectral density is smaller than statistical fluctuations. In this way, one ensures that the result does not depend on unphysical parameters of the algorithm. In Bayesian inference, 𝜆and𝛼have a different interpretation as they are hyperparameters that determine the prior distribution. The latter, however, are again chosen ad hocas inputs of the procedure, hence they should not affect the final result (up to statistical fluctuations). In Bayesian inference, it is common to determine the hyperparameters by minimising the negative logarithmic likelihood (NLL), 𝜏max 2Log(2𝜋)+1 2Logdet(Σ+Cov𝑑)+1 2(𝑪obs−𝑪prior)1 Σ+Cov𝑑(𝑪obs−𝑪prior).(34) 6Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo Figure 1: Toppanel: spectraldensitysmearedwithaGaussian,fordifferentvaluesofthe 𝜆-parameter. The central value is obtained according to Ref. [1], or equivalently from Eq. (31). The error is computed with a bootstrapintheformercase(BGinthelegend),andisgivenbyEq.(32)inthelatter(GPinthelegend). The figurealsoshowsthestabilityregiondescribedinRef.[9],whichpredictsavalueidentifiedbythehorizontal band. This is consistent with the value obtained by choosing the 𝜆that minimises the NLL (red star). The results are obtained using lattice data from Ref. [12]. Inlightoftheanalogydescribedinthiswork,itisnaturaltoaskhowvaluesof {𝜆,𝛼}specified by the minimum of the NLL relate to values determined according to the non-Bayesian procedure of Refs. [1, 9]. To this end we show, in the top panel of Fig. 1, a comparison between a smeared spectraldensityobtainedinthetwoapproaches,asafunctionof 𝜆. Thecentralvaluesarethesame, as inferred from Eq. (31). The statistical errors are determined with a bootstrap procedure (BG in the legend) or from the square root of half Eq. (32) (GP in the legend). In this example, which usesMonteCarlolatticedata 1fromRef.[12],theuncertaintiesarefoundtobeofthesameorderof magnitude,whichisanon-trivialresult,giventheprofounddifferenceinthewaytheyareobtained. Another striking observation is that the value of 𝜆determined from the minimum of the NLL (red star in the bottom panel), lies in a region in which the smeared spectral density does not change, within statistical noise, by changing the value of 𝜆. The value for the smeared spectral density that onewouldobtainfollowingRef.[9],shownasahorizontalbandinthetoppanelofFig.1,isinfact 1The correlator used is a two-point correlation function of pseudoscalar mesons, in the two-index antisymmetric representation of 𝑆𝑈(4)gauge theory, corresponding to the ensemble 𝐵3of Ref. [12]. 7Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo consistent with the value obtained minimising the NLL. This provides a non-trivial validation for both methods and, eventually, for the predictions they yield. 5. Conclusions The method described in Ref. [1] and Gaussian processes are popular choices to compute smeared spectral densities from lattice correlators. We have shown that the approach proposed in Ref. [1] can be reformulated in a Bayesian framework. This leads to a drastically different interpretation of the variables at hand, which become stochastic variables characterised by their probability distributions. The solution, and its error, are understood as the central value and the width of probability distributions. Nonetheless, for a specific choice of priors, the final prediction is identical, within statistical uncertainties, to the one from Ref. [1]. In our numerical tests, based on lattice correlation functions of meson-like states, we have found that the Bayesian error on the spectral reconstruction, coming from the width of its probability density, is compatible with the statistical error that one obtains by bootstrapping in a frequentist fashion. The analogy extends to the determination of the input parameters, where the prescription of Ref. [9] for the 𝜆parameter is found to be compatible with the minimisation of the NLL. Further details on this comparison will be presented in a forthcoming publication. Acknowledgments AL is funded in part by l’Agence Nationale de la Recherche (ANR), under grant ANR-22- CE31-0011. AL and LDD received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 813942. LDD is also supported by the UK Science and Technology Facility Council (STFC) grant ST/P000630/1. MP was partially supported by the Spoke 1 “FutureHPC & BigData” of the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC)fundedbyMUR(M4C2-19)–NextGenerationEU(NGEU),bytheItalianPRIN“Progetti diRicercadiRilevanteInteresseNazionale–Bando2022”,prot. 2022TJFCYB,andbythe“Simons Collaboration on Confinement and QCD Strings” funded by the Simons Foundation. 8Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo References [1] Martin Hansen, Alessandro Lupo, and Nazario Tantalo. Extraction of spectral densities from lattice correlators. Phys. Rev. D , 99(9):094508, 2019. [2] Lukas Kades, Jan M. Pawlowski, Alexander Rothkopf, Manuel Scherzer, Julian M. Urban, SebastianJ.Wetzel,NicolasWink,andFelixP.G.Ziegler. SpectralReconstructionwithDeep Neural Networks. Phys. Rev. D , 102(9):096001, 2020. [3] Jan Horak, Jan M. Pawlowski, José Rodríguez-Quintero, Jonas Turnwald, Julian M. Urban, NicolasWink,andSavvasZafeiropoulos. ReconstructingQCDspectralfunctionswithGaus- sian processes. Phys. Rev. D , 105(3):036014, 2022. [4] Thomas Bergamaschi, William I. Jay, and Patrick R. Oare. Hadronic structure, conformal maps, and analytic continuation. Phys. Rev. D , 108(7):074516, 2023. [5] Michele Buzzicotti, Alessandro De Santis, and Nazario Tantalo. Teaching to extract spectral densities from lattice correlators to a broad audience of learning-machines. 7 2023. [6] John Bulava and Maxwell T. Hansen. Scattering amplitudes from finite-volume spectral functions. Phys. Rev. D , 100(3):034521, 2019. [7] Paolo Gambino and Shoji Hashimoto. Inclusive Semileptonic Decays from Lattice QCD. Phys. Rev. Lett. , 125(3):032001, 2020. [8] GabrielaBailas,ShojiHashimoto,andTsutomuIshikawa. Reconstructionofsmearedspectral function from Euclidean correlation functions. PTEP, 2020(4):043B07, 2020. [9] JohnBulava,MaxwellT.Hansen,MichaelW.Hansen,AgostinoPatella,andNazarioTantalo. Inclusive rates from smeared spectral densities in the two-dimensional O(3) non-linear 𝜎- model.JHEP, 07:034, 2022. [10] Paolo Gambino, Shoji Hashimoto, Sandro Mächler, Marco Panero, Francesco Sanfilippo, Silvano Simula, Antonio Smecca, and Nazario Tantalo. Lattice QCD study of inclusive semileptonic decays of heavy mesons. JHEP, 07:083, 2022. [11] ConstantiaAlexandrouetal. ProbingtheEnergy-SmearedRRatioUsingLatticeQCD. Phys. Rev. Lett., 130(24):241901, 2023. [12] LuigiDelDebbio,AlessandroLupo,MarcoPanero,andNazarioTantalo.Multi-representation dynamics of SU(4) composite Higgs models: chiral limit and spectral reconstructions. Eur. Phys. J. C , 83(3):220, 2023. [13] JanM.Pawlowski,CoralieS.Schneider,JonasTurnwald,JulianM.Urban,andNicolasWink. Yang-Mills glueball masses from spectral reconstruction. 12 2022. [14] Claudio Bonanno, Francesco D’Angelo, Massimo D’Elia, Lorenzo Maio, and Manuel Nav- iglio. Sphaleron rate from a modified Backus-Gilbert inversion method. 5 2023. 9Bayesian interpretation of Backus-Gilbert methods Alessandro Lupo [15] AntonioEvangelista,RobertoFrezzotti,GiuseppeGagliardi,VittorioLubicz,FrancescoSan- filippo, Silvano Simula, and Nazario Tantalo. Direct lattice calculation of inclusive hadronic decay rates of the 𝜏lepton.PoS, LATTICE2022:296, 2023. [16] R. Frezzotti, G. Gagliardi, V. Lubicz, F. Sanfilippo, S. Simula, and N. Tantalo. Spectral- function determination of complex electroweak amplitudes with lattice QCD. 6 2023. [17] AlessandroBarone,ShojiHashimoto,AndreasJüttner,TakashiKaneko,andRyanKellermann. Approaches to inclusive semileptonic B (𝑠)-meson decays from Lattice QCD. JHEP, 07:145, 2023. [18] Andrew P Valentine and Malcolm Sambridge. Gaussian process models—I. A framework forprobabilisticcontinuousinversetheory. GeophysicalJournalInternational ,220(3):1632– 1647, 11 2019. [19] Luigi Del Debbio, Tommaso Giani, and Michael Wilson. Bayesian approach to inverse problems: an application to NNPDF closure testing. Eur. Phys. J. C , 82(4):330, 2022. [20] Alessandro Candido, Luigi Del Debbio, Tommaso Giani, and Giacomo Petrillo. Inverse Problems in PDF determinations. PoS, LATTICE2022:098, 2023. [21] GeorgeBackusandFreemanGilbert. TheResolvingPowerofGrossEarthData. Geophysical Journal International , 16(2):169–205, 10 1968. [22] Antonio Evangelista, Roberto Frezzotti, Nazario Tantalo, Giuseppe Gagliardi, Francesco Sanfilippo,SilvanoSimula,andVittorioLubicz. Inclusivehadronicdecayrateofthe 𝜏lepton from lattice QCD. Phys. Rev. D , 108(7):074513, 2023. 10

2023-11-29

The extraction of spectral densities from Euclidean correlators evaluated on the lattice is an important problem, as these quantities encode physical information on scattering amplitudes, finite-volume spectra, inclusive decay rates, and transport coefficients. In this contribution, we show that the Bayesian approach to this "inverse" problem, based on Gaussian processes, can be reformulated in a way that yields a solution equivalent, up to statistical uncertainties, to the one obtained in a Backus-Gilbert approach. After discussing this equivalence, we point out its implications for a reliable determination of spectral densities from lattice simulations.

Bayesian interpretation of Backus-Gilbert methods

2311.18125v1

Landau-Lifshitz theory of the magnon-drag thermopower Benedetta Flebus,1, 2Rembert A. Duine,1, 3and Yaroslav Tserkovnyak2 1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 3Department of Applied Physics, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands Metallic ferromagnets subjected to a temperature gradient exhibit a magnonic drag of the electric current. We address this problem by solving a stochastic Landau-Lifshitz equation to calculate the magnon-drag thermopower. The long-wavelength magnetic dynamics result in two contributions to the electromotive force acting on electrons: (1) An adiabatic Berry-phase force related to the solid angle subtended by the magnetic precession and (2) a dissipative correction thereof, which is rooted microscopically in the spin-dephasing scattering. The rst contribution results in a net force pushing the electrons towards the hot side, while the second contribution drags electrons towards the cold side, i.e., in the direction of the magnonic drift. The ratio between the two forces is proportional to the ratio between the Gilbert damping coecient and the coecient parametrizing the dissipative contribution to the electromotive force. The interest in thermoelectric phenomena in ferromag- netic heterostructures has been recently revived by the discovery of the spin Seebeck eect [1]. This eect is now understood to stem from the interplay of the thermally- driven magnonic spin current in the ferromagnet and the (inverse) spin Hall voltage generation in an adjacent nor- mal metal [2]. Lucassen et al. [3] subsequently proposed that the thermally-induced magnon ow in a metallic ferromagnet can also produce a detectable (longitudinal) voltage in the bulk itself, due to the spin-transfer mecha- nism of magnon drag. Specically, smooth magnetization texture dynamics induce an electromotive force [4], whose net average over thermal uctuations is proportional to the temperature gradient. In this Letter, we develop a Landau-Lifshitz theory for this magnon drag, which generalizes Ref. [3] to include a heretofore disregarded Berry-phase contribution. This additional magnon drag can reverse the sign of the thermopower, which can have potential utility for designing scalable thermopiles based on metallic ferromagnets. Electrons propagating through a smooth dynamic tex- ture of the directional order parameter n(r;t) [such that jn(r;t)j1, with the self-consistent spin density given bys=sn] experience the geometric electromotive force of [4] Fi=~ 2(n
@tn@in@tn
@in) (1) for spins up along nandFifor spins down. The resul- tant electric current density is given by ji="# ehFii=~P 2ehn
@tn@in@tn
@ini;(2) where="+#is the total electrical conductivity, P= ("#)=is the conducting spin polarization, and eis the carrier charge (negative for electrons). The averaging h:::iin Eq. (2) is understood to be taken over the steady- state stochastic uctuations of the magnetic orientation.The latter obeys the stochastic Landau-Lifshitz-Gilbert equation [5] s(1 +n)@tn+n(Hz+h) +X i@iji= 0;(3) whereis the dimensionless Gilbert parameter [6], H parametrizes a magnetic eld (and/or axial anisotropy) along thezaxis, and ji=An@inis the magnetic spin- current density, which is proportional to the exchange stinessA. ForH > 0, the equilibrium orientation is n! z, which we will suppose in the following. The Langevin eld stemming from the (local) Gilbert damp- ing is described by the correlator [7] hhi(r;!)h j(r0;!0)i=2s~!ij(rr0)(!!0) tanh~! 2kBT(r); (4) upon Fourier transforming in time: h(!) =R dtei!th(t). At temperatures much less than the Curie tempera- ture,Tc, it suces to linearize the magnetic dynamics with respect to small-angle uctuations. To that end, we switch to the complex variable, nnxiny, parametriz- ing the transverse spin dynamics. Orienting a uniform thermal gradient along the xaxis,T(x) =T+x@xT, we Fourier transform the Langevin eld (4) also in real space, with respect to the yandzaxes. Linearizing Eq. (3) for small-angle dynamics results in the Helmholtz equation: A(@2 x2)n(x;q;!) =h(x;q;!); (5) where2q2+[H(1+i)s!]=A,hhxihy, and qis the two-dimensional wave vector in the yzplane. Solving Eq. (5) using the Green's function method, we substitute the resultant ninto the expression for the charge current density (2), which can be appropriately rewritten in thearXiv:1605.06578v1 [cond-mat.mes-hall] 21 May 20162 following form (for the nonzero xcomponent): jx=~P 2eZd2qd! (2)3! Re(1 +i)hn(x;q;!)@xn(x;q0;!0)i (2)3(qq0)(!!0): (6) Tedious but straightforward manipulations, using the correlator (4), nally give the following thermoelectric current density: jx=sP@xT 4eA2kBT2Zd2qd! (2)3(~!)3 sinh2~! 2kBTRe [(1 +i)I]; (7) whereI()=jj2(Re)2, having made the convention that Re>0. To recast expression (7) in terms of magnon modes, we incorporate the integration over qxby noticing that, in the limit of low damping, !0, I=2 Z dqx1 +iq2 x=~! (~!q2xq22)2+ (~!)2:(8) Here, we have introduced the magnetic exchange length p A=H and dened ~ !s!=A . After approximating the Lorentzian in Eq. (8) with the delta function when 1, Eq. (7) can nally be expressed in terms of a dimensionless integral J(a)Z1 a=p 2dxx5p 2x2a2 sinh2x2; (9) as j= 1 3 J kBP 2eT Tc3=2 rT: (10) Here,Tis the ambient temperature, kBTcA(~=s)1=3 estimates the Curie temperature, and p ~A=skBT is the thermal de Broglie wavelength in the absence of an applied eld. We note that ;1 while, in typical transition-metal ferromagnets [8]. For temperatures much larger than the magnon gap (typically of the order of 1 K in metallic ferromagnets), and we can approximate J(=)J(0) 1. This limit eectively corresponds to the gapless magnon dispersion of q~!q~Aq2=s. Within the Boltzmann phenomenology, the magnonic heat cur- rent induced by a uniform thermal gradient is given by jQ=rTR [d3q=(2)3](@qx!q)2(!q)q@TnBE, where 1(!q) = 2!qis the Gilbert-damping decay rate of magnons (to remain within the consistent LLG phe- nomenology) and nBE= [exp(q=kBT)1]1is the Bose- Einstein distribution function. By noticing that q@TnBE=kB~!q=2kBT sinh( ~!q=2kBT)2 ; (11) rThydrodynamic r⌦<0geometricˆxˆzˆy ⌦ eeeeeeee FIG. 1. Schematics for the two contributions to the electron- magnon drag. In the absence of decay (i.e., !0), magnons drifting from the hot (left) side to the cold (right) side drag the charge carriers viscously in the same direction, inducing a thermopower /. The (geometric) Berry-phase drag gov- erned by the magnon decay is proportional to and acts in the opposite direction. It is illustrated for a spin wave that is thermally emitted from the left. As the spin wave propagates to the right, the solid angle subtended by the spin preces- sion shrinks, inducing a force oriented to the left for spins parallel to n. it is easy to recast the second, /contribution to Eq. (10) in the form j()=~P 2eAjQ; (12) which reproduces the main result of Ref. [3]. The magnon-drag thermopower (Seebeck coecient), S=@xV @xT jx=0; (13) corresponds to the voltage gradient @xVinduced under the open-circuit condition. We thus get from Eq. (10): S= 31 JkBP 2eT Tc3=2 = (3)~Pm 2eA; (14) wherem= (2=32)JkBA(T=Tc)3=2=~is the magnonic contribution to the heat conductivity. Such magnon-drag thermopower has recently been observed in Fe and Co [9], with scaling/T3=2over a broad temperature range and opposite sign in the two metals. Note that the sign depends on = and the eective carrier charge e. Equations (10) and (14) constitute the main results of this paper. In the absence of Gilbert damping, !0, the magnon-drag thermopower Sis proportional to the heat conductivity. This contribution was studied in Ref. [3] and is understood as a viscous hydrodynamic drag. In simple model calculations [8], P > 0 and this hydrodynamic thermopower thus has the sign of the ef- fective carrier charge e. WhenP > 0, so that the ma- jority band is polarized along the spin order parameter3 n, the/contribution to the thermopower is opposite to the/contribution. (Note that is always>0, in order to yield the positive dissipation.) The underly- ing geometric meaning of this result is sketched in Fig. 1. Namely, the spin waves that are generated at the hot end and are propagating towards the cold end are associated with a decreasing solid angle, @x <0. The rst term in Eq. (1), which is rooted in the geometric Berry con- nection [10], is proportional to the gradient of this solid angle times the precession frequency, /!@i , resulting in a net force towards the hot side acting on the spins collinear with n. Note that we have neglected the Onsager-reciprocal backaction of the spin-polarized electron drift on the magnetic dynamics. This is justied as including the corresponding spin-transfer torque in the LLG equation would yield higher-order eects that are beyond our treatment [11]. The diusive contribution to the See- beck eect,/T=EF, whereEFis a characteristic Fermi energy, which has been omitted from our analysis, is ex- pected to dominate only at very low temperatures [9]. The conventional phonon-drag eects have likewise been disregarded. A systematic study of the relative impor- tance of the magnon and phonon drags is called upon in magnetic metals and semiconductors. This work is supported by the ARO under Contract No. 911NF-14-1-0016, FAME (an SRC STARnet center sponsored by MARCO and DARPA), the Stichting voor Fundamenteel Onderzoek der Materie (FOM), and the D-ITP consortium, a program of the Netherlands Orga- nization for Scientic Research (NWO) that is funded by the Dutch Ministry of Education, Culture, and Science (OCW).[1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008); K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa, H. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh, Nat. Mater. 9, 894 (2010). [2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and S. Maekawa, Phys. Rev. B 81, 214418 (2010). [3] M. E. Lucassen, C. H. Wong, R. A. Duine, and Y. Tserkovnyak, Appl. Phys. Lett. 99, 262506 (2011). [4] R. A. Duine, Phys. Rev. B 77, 014409 (2008); Y. Tserkovnyak and M. Mecklenburg, ibid.77, 134407 (2008). [5] S. Homan, K. Sato, and Y. Tserkovnyak, Phys. Rev. B 88, 064408 (2013). [6] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [7] W. F. Brown, Phys. Rev. 130, 1677 (1963). [8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J. Magn. Magn. Mater. 320, 1282 (2008). [9] S. J. Watzman, R. A. Duine, Y. Tserkovnyak, H. Jin, A. Prakash, Y. Zheng, and J. P. Heremans, \Magnon- drag thermopower and Nernst coecient in Fe and Co," arXiv:1603.03736. [10] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984); G. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83 (1987); S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 (2007); Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79, 014402 (2009). [11] The backaction by the spin-transfer torque would be ab- sent when the longitudinal spin current, ji=(PE i+ Fi=e)n, vanishes, where Eiis the electric eld and Fi is the spin-motive force (1). Understanding Eq. (10) as pertaining to the limit of the vanishing spin current ji rather than electric current ji=(Ei+PFi=e)nwould, however, result in higher-order (in T=T c) corrections to the Seebeck coecient (13). These are beyond the level of our approximations.

2016-05-21

Metallic ferromagnets subjected to a temperature gradient exhibit a magnonic drag of the electric current. We address this problem by solving a stochastic Landau-Lifshitz equation to calculate the magnon-drag thermopower. The long-wavelength magnetic dynamics result in two contributions to the electromotive force acting on electrons: (1) An adiabatic Berry-phase force related to the solid angle subtended by the magnetic precession and (2) a dissipative correction thereof, which is rooted microscopically in the spin-dephasing scattering. The first contribution results in a net force pushing the electrons towards the hot side, while the second contribution drags electrons towards the cold side, i.e., in the direction of the magnonic drift. The ratio between the two forces is proportional to the ratio between the Gilbert damping coefficient $\alpha$ and the coefficient $\beta$ parametrizing the dissipative contribution to the electromotive force.

Landau-Lifshitz theory of the magnon-drag thermopower

1605.06578v1

arXiv:2003.12967v1 [math.AP] 29 Mar 2020STABILITY RESULTS FOR AN ELASTIC-VISCOELASTIC WAVES INTER ACTION SYSTEMS WITH LOCALIZED KELVIN-VOIGT DAMPING AND WITH AN INT ERNAL OR BOUNDARY TIME DELAY MOUHAMMAD GHADER1, RAYAN NASSER1,2, AND ALI WEHBE1 Abstract. We investigate the stability of a one-dimensional wave equa tion with non smooth localized internal viscoelastic damping of Kelvin-Voigt type and with boundar y or localized internal delay feedback. The main novelty in this paper is that the Kelvin-Voigt and the delay d amping are both localized via non smooth coefficients. In the case that the Kelvin-Voigt damping is loc alized faraway from the tip and the wave is subjected to a locally distributed internal or boundary del ay feedback, we prove that the energy of the system decays polynomially of type t−4. However, an exponential decay of the energy of the system is established provided that the Kelvin-Voigt damping is localized near a p art of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin-Voigt an d the internal delay damping are both localized via non smooth coefficients near the tip, the energy of the syst em decays polynomially of type t−4. Frequency domain arguments combined with piecewise multiplier techn iques are employed. Contents 1. Introduction 1 2. Wave equation with local Kelvin-Voigt damping and with bo undary delay feedback 7 2.1. Wave equation with local Kelvin-Voigt damping far from the boundary and with boundary delay feedback 8 2.1.1. Well-posedness of the problem 9 2.1.2. Strong Stability 12 2.1.3. Polynomial Stability 16 2.2. Wave equation with local Kelvin-Voigt damping near the boundary and boundary delay feedback 21 3. Wave equation with local internal Kelvin-Voigt damping a nd local internal delay feedback 24 3.1. Well-posedness of the problem 25 3.2. Polynomial Stability 28 Appendix A. Notions of stability and theorems used 37 References 38 1Lebanese University, Faculty of sciences 1, Khawarizmi Lab oratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon. 2Université de Bretagne-Occidentale, France. E-mail addresses :mhammadghader@hotmail.com, rayan.nasser94@hotmail.co m, ali.wehbe@ul.edu.lb . 1991Mathematics Subject Classification. 35L05; 35B35; 93D15; 93D20. Key words and phrases. Wave equation; Kelvin-Voigt damping; Time delay; Semigrou p; Stability. iWAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY 1.Introduction Viscoelastic materials feature intermediate characteris tics between purely elastic and purely viscous behav- iors,i.e.they display both behaviors when undergoing deformation. I n wave equations, when the viscoelastic controlling parameter is null, the viscous property vanish es and the wave equation becomes a pure elastic wave equation. However, time delays arise in many applications a nd practical problems like physical, chemical, bi- ological, thermal and economic phenomena, where an arbitra ry small delay may destroy the well-posedness of the problem and destabilize it. Actually, it is well-known t hat simplest delay equations of parabolic type, ut(x,t) = ∆u(x,t−τ), or hyperbolic type utt(x,t) = ∆u(x,t−τ), with a delay parameter τ >0,are not well-posed. Their instability is due to the existenc e of a sequence of initial data remaining bounded, while the corresponding so lutions go to infinity in an exponential manner at a fixed time (see [ 16,23]). The stabilization of a wave equation with Kelvin-Voigt type damping and internal or boundary time delay has attracted the attention of many authors in the last five ye ars. Indeed, in 2016 Messaoudi et al. studied the stabilization of a wave equation with global Kelvin-Voi gt damping and internal time delay in the multidi- mensional case (see [ 30]), and they obtained an exponential stability result. In th e same year, Nicaise et al. in [33] considered the multidimensional wave equation with local ized Kelvin-Voigt damping and mixed boundary condition with time delay. They obtained an exponential dec ay of the energy regarding that the damping is acting on a neighborhood of part of the boundary via a smooth c oefficient. Also, in 2018, Anikushyn et al. in [15] considered the stabilization of a wave equation with globa l viscoelastic material subjected to an internal strong time delay where a global exponential decay rate was o btained. Thus, it seems to us that there are no previous results concerning the case of wave equations wi th internal localized Kelvin-Voigt type damping and boundary or internal time delay, especially in the absen ce of smoothness of the damping coefficient even in the one dimensional case. So, we are interested in studyin g the stability of elastic wave equation with local Kelvin–Voigt damping and with boundary or internal time del ay (see Systems ( 1.1) and ( 1.2)). This paper investigates the study of the stability of a strin g with Kelvin-Voigt type damping localized via non-smooth coefficient and subjected to a localized internal or boundary time delay. Indeed, in the first part of this paper, we study the stability of elastic wave equation w ith local Kelvin–Voigt damping, boundary feedback and time delay term at the boundary, i.e.we consider the following system (1.1) Utt(x,t)−/bracketleftbig κUx(x,t)+δ1χ(α,β)Uxt(x,t)/bracketrightbig x= 0,(x,t)∈(0,L)×(0,+∞), U(0,t) = 0, t∈(0,+∞), Ux(L,t) =−δ3Ut(L,t)−δ2Ut(L,t−τ), t ∈(0,+∞), (U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L), Ut(L,t) =f0(L,t), t ∈(−τ,0), whereL, τ, δ 1andδ3are strictly positive constant numbers, δ2is a non zero real number and the initial data (U0,U1,f0)belongs to a suitable space. Here 0≤α<β <L andU=uχ(0,α)+vχ(α,β)+wχ(β,L), withχ(a,b)is the characteristic function of the interval (a,b).We assume that there exist strictly positive constant numbe rs κ1, κ2, κ3, such that κ=κ1χ(0,α)+κ2χ(α,β)+κ3χ(β,L).In fact, here we will consider two cases. In the first case, we divide the bar into 3 pieces; the first piece is an elas tic part, the second piece is the viscoelastic part and in the third piece, the time delay feedback is effective at the ending point of the piece, i.e.we consider the caseα>0(see Figure 1). While, in the second case, we divide the bar into 2 pieces; t he first piece is the viscoelastic part and in the second piece the time delay feed back is effective at the ending point of the piece, i.e.we consider the case α= 0(see Figure 2). Remark, here, in both cases, the Kelvin–Voigt damping is effective on a part of the piece and the time delay is effective a tL. 1WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY • •α βv(x) u(x) w(x) 0 LElastic part Viscoelastic part Boundary delay feedback Figure 1. K-V damping is acting localized in the internal of the body an d time delay feedback is effective at L • 0 βv(x) w(x) LViscoelastic part Boundary delay feedback Figure 2. K-V damping is acting localized near the boundary of the body and time delay feedback is effective at L In the second part of this paper, we study the stability of ela stic wave equation with local Kelvin–Voigt damping and local internal time delay. This system takes the followi ng form (1.2) Utt(x,t)−/bracketleftbig κUx(x,t)+χ(α,β)(δ1Uxt(x,t)+δ2Uxt(x,t−τ))/bracketrightbig x= 0,(x,t)∈(0,L)×(0,+∞), U(0,t) =U(L,t) = 0, t∈(0,+∞), (U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L), Ut(x,t) =f0(x,t), (x,t)∈(0,L)×(−τ,0), whereL, τandδ1are strictly positive constant numbers, δ2is a non zero real number and the initial data (U0,U1,f0)belongs to a suitable space. Here 0<α<β <L andU=uχ(0,α)+vχ(α,β)+wχ(β,L), withχ(a,b)is the characteristic function of the interval (a,b).We assume that there exist strictly positive constant numbe rs κ1, κ2, κ3, such that κ=κ1χ(0,α)+κ2χ(α,β)+κ3χ(β,L).In fact, here we will divide the bar into 3 pieces; the first piece is an elastic part, in the second piece the Kelvin– Voigt damping and the time delay are effective and the third piece is an elastic part (see Figure 3). So, the Kelvin–Voigt damping and the time delay are effecti ve on(α,β). • •α βv(x) u(x) w(x) 0 LElastic part Viscoelastic part &delay feedback Elastic part Figure 3. Local internal K-V damping and Local internal delay feedbac k In the literature, Datko et al. studied in 1985 the one dimensional wave equation which mode ls the vibrations 2WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY of a string fixed at one end and free at the other one (see [ 14]). The system is given by the following: (1.3) utt(x,t)−uxx(x,t)+2aut(x,t)+a2u(x,t) = 0,(x,t)∈(0,1)×(0,+∞), u(0,t) = 0, t∈(0,+∞), ux(1,t) =−κut(1,t−τ), t ∈(0,+∞), where the delay parameter τis strictly positive, a>0andκ>0. So, the above system models a string having a boundary feedback with delay at the free end. They showed th at ifκ/parenleftbig e2a+1/parenrightbig <e2a−1,then System ( 1.3) is strongly stable for all small enough delays. However, if κ/parenleftbig e2a+1/parenrightbig >e2a−1,then there exists an open set D dense in (0,+∞), such that for all τinD, System ( 1.3) admits exponentially unstable solutions. Moreover, in the absence of delay in System ( 1.3) (i.eτ= 0) anda≥0,κ≥0, its energy decays exponentially to zero under the condition a2+κ2>0(see [12]). In 1990, Datko in [ 13] considered the boundary feedback stabilization of a one-dimensional wave equation with time delay (see Example 3.5 in [ 13]). The system is given by the following: (1.4) utt(x,t)−uxx(x,t)−δuxxt(x,t) = 0,(x,t)∈(0,1)×(0,+∞), u(0,t) = 0, t ∈(0,+∞), ux(1,t) =−κut(1,t−τ), t ∈(0,+∞), whereτ >0, κ >0andδ >0. He proved that System ( 1.4) is unstable for an arbitrary small value of τ. In 2006, Xu et al. in [44] investigated the following closed loop system with homoge neous Dirichlet boundary condition at x= 0and delayed Neumann boundary feedback at x= 1: (1.5) utt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞), u(0,t) = 0, t ∈(0,+∞), ux(1,t) =−κut(1,t)−κ(1−µ)ut(1,t−τ), t∈(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈(0,1), ut(1,t) =f0(1,t), t ∈(−τ,0). The above system represents a wave equation that is fixed at on e end and subjected to a boundary control input possessing a partial time delay of weight (1−µ)at the other end. They proved the following stability results: 1. Ifµ>2−1,then System ( 1.5) is uniformly stable. 2. Ifµ= 2−1andτ∈Q∩(0,1), then System ( 1.5) is unstable. 3. Ifµ= 2−1andτ∈(R\Q)∩(0,1), then System ( 1.5) is asymptotically stable. 4. Ifµ<2−1,then System ( 1.5) is always unstable. Later on, in 2008, Guo and Xu in [ 18] studied the stabilization of a wave equation in the 1-D case where it is effected by a boundary control and output observation sufferi ng from time delay. The system is given by the following: utt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞), u(0,t) = 0, t ∈(0,+∞), ux(1,t) =w(t), t ∈(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1), y(t) =ut(1,t−τ), t ∈(0,+∞), wherewis the control and yis the output observation. Using the separation principle, the authors proved that the above delayed system is exponentially stable. In 20 10, Gugat in [ 17] studied the wave equation which models a string of length Lthat is rigidly fixed at one end and stabilized with a boundary feedback and constant 3WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY delay at the other end. The problem is described by the follow ing system utt(x,t)−c2uxx(x,t) = 0, (x,t)∈(0,L)×(0,+∞), u(0,t) = 0, t ∈(0,+∞), ux(L,t) = 0, t ∈(0,2Lc−1), ux(L,t) =c−1λut/parenleftbig L,t−2Lc−1/parenrightbig , t ∈(2Lc−1,+∞), (u(x,0),ut(x,0),u(0,0)) = (u0(x),u1(x),0), x∈(0,L), whereλis a real number and c >0. Gugat proved that the above system is exponentially stable . In 2011, J. Wang et al. in [41] studied the stabilization of a wave equation under boundar y control and collocated observation with time delay. The system is given by the follo wing: utt(x,t)−uxx(x,t) = 0, (x,t)∈(0,1)×(0,+∞), u(0,t) = 0, t ∈(0,+∞), ux(1,t) =κut(1,t−τ), t ∈(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1). They showed that if the delay is equal to even multiples of the wave propagation time, then the above closed loop system is exponentially stable under sufficient and nece ssary conditions for κ. Else, if the delay is an odd multiple of the wave propagation time, thus the closed loop s ystem is unstable. In 2013, H. Wang et al. in [43], studied System ( 1.5) under the feedback control law ut(1,t) =w(t)provided that the weight of the feedback with delay is a real βand that of the feedback without delay is a real α. They found a feedback control law that stabilizes exponentially the system for any |α| /\e}atio\slash=|β|, by modifying the velocity feedback into the form u(t) =βwt(1,t)+αf(w(.,t),wt(.,t)), wherefis a linear functional. Finally, in 2017, Xu et al. in [42], studied the stability problem of a one dimensional wave equation wit h internal control and boundary delay term utt(x,t)−uxx(x,t)+2αut(x,t) = 0,(x,t)∈(0,1)×(0,+∞), u(0,t) = 0, t ∈(0,+∞), ux(1,t) =κut(1,t−τ), t ∈(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,1), ut(1,t) =f0(1,t), t ∈(−τ,0), whereτ >0,α>0andκis real. Based on the idea of Lyapunov functional, they prove d exponential stability of the above system under a certain relationship between αandκ. Going to the multidimensional case, the stability of wave eq uation with time delay has been studied in [32,6,37,30,33,15,3,4]. In 2006, Nicaise and Pignotti in [ 32] studied the multidimensional wave equation considering two cases. The first case concerns a wave equatio n with boundary feedback and a delay term at the boundary (1.6) utt(x,t)−∆u(x,t) = 0, (x,t)∈Ω×(0,+∞), u(x,t) = 0, (x,t)∈ΓD×(0,+∞), ∂u ∂ν(x,t) =−µ1ut(x,t)−µ2ut(x,t−τ),(x,t)∈ΓN×(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω, ut(x,t) =f0(x,t), (x,t)∈ΓN×(−τ,0). 4WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY The second case concerns a wave equation with an internal fee dback and a delayed velocity term ( i.ean internal delay) and a mixed Dirichlet-Neumann boundary condition (1.7) utt(x,t)−∆u(x,t)+µ1ut(x,t)+µ2ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞), u(x,t) = 0, (x,t)∈ΓD×(0,+∞), ∂u ∂ν(x,t) = 0, (x,t)∈ΓN×(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω, ut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0). In both systems, τ, µ1, µ2are strictly positive constants, ∂u/∂ν is the partial derivative, Ωis an open bounded domain of RNwith a boundary Γof classC2andΓ = ΓD∪ΓN, such that ΓD∩ΓN=∅. Under the assumption that the weight of the feedback with delay is smaller than tha t without delay (µ2< µ1), they obtained an exponential decay of the energy of both Systems ( 1.6) and ( 1.7). On the contrary, if the previous assumption does not hold (i.eµ2≥µ1), they found a sequence of delays for which the energy of some s olutions does not tend to zero (see [ 10] for the treatment of Problem ( 1.7) in more general abstract form). In 2009, Nicaise et al.in [34] studied System ( 1.6) in the one dimensional case where the delay time τis a function depending on time and they established an exponential stability result u nder the condition that the derivative of the decay function is upper bounded by a constant d<1and assuming that µ2<√ 1−d µ1. In 2010, Ammari et al. in [6] studied the wave equation with interior delay damping and d issipative undelayed boundary condition in an open domain ΩofRN, N≥2.The system is given by the following: (1.8) utt(x,t)−∆u(x,t)+aut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞), u(x,t) = 0, (x,t)∈Γ0×(0,+∞), ∂u ∂ν(x,t) =−κut(x,t), (x,t)∈Γ1×(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω, ut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0), whereτ >0,a>0andκ >0. Under the condition that Γ1satisfies the Γ-condition introduced in [ 25], they proved that System ( 1.8) is uniformly asymptotically stable whenever the delay coe fficient is sufficiently small. In 2012, Pignotti in [ 37] considered the wave equation with internal distributed ti me delay and local damping in a bounded and smooth domain Ω⊂RN,N≥1. The system is given by the following: (1.9) utt(x,t)−∆u(x,t)+aχωut(x,t)+κut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞), u(x,t) = 0, (x,t)∈Γ×(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω, ut(x,t) =f(x,t), (x,t)∈Ω×(−τ,0), whereκreal,τ >0anda >0. System ( 1.9) shows that the damping is localized, indeed, it acts on a neighborhood of a part of the boundary of Ω. Under the assumption that |κ|<κ0<a,the author established an exponential decay rate. Later, in 2016, Messaoudi et al. in [30] considered the stabilization of the following wave equation with strong time delay utt(x,t)−∆u(x,t)−µ1∆ut(x,t)−µ2∆ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞), u(x,t) = 0, (x,t)∈Γ×(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω, ut(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0), whereµ1>0andµ2is a non zero real number. Under the assumption that |µ2|< µ1, they obtained an exponential stability result. In addition, in the same year , Nicaise et al. in [33] studied the multidimensional 5WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY wave equation with localized Kelvin-Voigt damping and mixe d boundary condition with time delay (1.10) utt(x,t)−∆u(x,t)−div(a(x)∇ut) = 0,(x,t)∈Ω×(0,+∞), u(x,t) = 0, (x,t)∈Γ0×(0,+∞), ∂u ∂ν(x,t) =−a(x)∂ut ∂ν(x,t)−κut(x,t−τ),(x,t)∈Γ1×(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x ∈Ω, ut(x,t) =f0(x,t), (x,t)∈Γ1×(−τ,0), whereτ >0,κis real,a(x)∈L∞(Ω)anda(x)≥a0>0onωsuch thatω⊂Ωis an open neighborhood of Γ1. Under an appropriate geometric condition on Γ1and assuming that a∈C1,1(Ω),∆a∈L∞(Ω), they proved an exponential decay of the energy of System ( 1.10). Finally, in 2018, Anikushyn et al. in [15] considered an initial boundary value problem for a viscoelastic wave equa tion subjected to a strong time localized delay in a Kelvin-Voigt type. The system is given by the following: utt(x,t)−c1∆u(x,t)−c2∆u(x,t−τ)−d1∆ut(x,t)−d2∆ut(x,t−τ) = 0,(x,t)∈Ω×(0,+∞), u(x,t) = 0, (x,t)∈Γ0×(0,+∞), ∂u ∂ν(x,t) = 0, (x,t)∈Γ1×(0,+∞), (u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈Ω, u(x,t) =f0(x,t), (x,t)∈Ω×(−τ,0). Under appropriate conditions on the coefficients, a global ex ponential decay rate is obtained. We can also mention that Ammari et al. in [7] considered the stabilization problem for an abstract equa tion with delay and a Kelvin-Voigt damping in 2015. The system is given by the fol lowing: utt(t)+aBB∗ut(t)+BB∗u(t−τ), t∈(0,+∞), (u(0),ut(0)) = (u0,u1), B∗u(t) =f0(t), t ∈(−τ,0), for an appropriate class of operator Banda >0.Using the frequency domain approach, they obtained an exponential stability result. Finally, the transmission p roblem of a wave equation with global or local Kelvin- Voigt damping and without any time delay was studied by many a uthors in the one dimensional case (see [26,2,21,1,20,19,35,39,28]) and in the multidimensional case (see [ 31,45,40,27]) and polynomial and exponential stability results were obtained. In addition, the stability of wave equations on tree with local Kelvin-Voigt damping has been studied in [ 5]. Thus, as we confirmed in the beginning, the case of wave equati ons with localized Kelvin-Voigt type damping and boundary or internal time delay; as in our Systems ( 1.1) and ( 1.2), where the damping is acting in a non- smooth region is still an open problem. The aim of the present paper consists in studying the stability of the Systems ( 1.1) and ( 1.2). For System ( 1.1), we consider two cases. Case one, if α >0(see Figure 1), then using the semigroup theory of linear operators and a result o btained by Borichev and Tomilov, we show that the energy of the System ( 1.1) has a polynomial decay rate of type t−4. Case two, if α= 0(see Figure 2), then using the semigroup theory of linear operators and a res ult obtained by Huang and Prüss, we prove an exponential decay of the energy of System ( 1.1). For System ( 1.2), by using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov, we s how that the energy of the System ( 1.2) has a polynomial decay rate of type t−4. This paper is organized as follows: In Section 2, we study the stability of System ( 1.1). Indeed, in Subsection 2.1, we consider the case α>0. First, we prove the well-posedness of System ( 1.1). Next, we prove the strong stability of the system in the lack of the compactness of the r esolvent of the generator. Then, we establish a polynomial energy decay rate of type t−4(see Theorem 2.7). In addition, in Subsection 2.2, we consider the caseα= 0and we prove the exponential stability of system ( 1.1) (see Theorem 2.14). In Section 3, we study the stability of System ( 1.2). First, we prove the well-posedness of System ( 1.2). Next, we establish a polynomial energy decay rate of type t−4(see Theorem 3.2). 6WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY 2.Wave equation with local Kelvin-Voigt damping and with boun dary delay feedback This section is devoted to our first aim, that is to study the st ability of a wave equation with localized Kelvin- Voigt damping and boundary delay feedback (see System ( 1.1)). For this aim, let us introduce the auxiliary unknown η(L,ρ,t) =Ut(L,t−ρτ), ρ∈(0,1), t>0. Thus, Problem ( 1.1) is equivalent to (2.1) Utt(x,t)−/bracketleftbig κUx(x,t)+δ1χ(α,β)Uxt(x,t)/bracketrightbig x= 0,(x,t)∈(0,L)×(0,+∞), τηt(L,ρ,t)+ηρ(L,ρ,t) = 0, (ρ,t)∈(0,1)×(0,+∞), U(0,t) = 0, t∈(0,+∞), Ux(L,t) =−δ3Ut(L,t)−δ2η(L,1,t), t ∈(0,+∞), (U(x,0),Ut(x,0)) = (U0(x),U1(x)), x ∈(0,L), η(L,ρ,0) =f0(L,−ρτ), ρ ∈(0,1). LetUbe a smooth solution of System ( 2.1), we associate its energy defined by (2.2) E(t) =1 2/integraldisplayL 0/parenleftbig |Ut|2+κ|Ux|2/parenrightbig dx+τ 2/integraldisplay1 0|η|2dρ. Multiplying the first equation of ( 2.1) byUt, integrating over (0,L)with respect to x, then using by parts integration and the boundary conditions in ( 2.1) atx= 0and atx=L, we get (2.3)1 2d dt/integraldisplayL 0/parenleftbig |Ut|2+κ|Ux|2/parenrightbig dx=−δ1/integraldisplayβ α|Uxt|2dx−κ3δ3|Ut(L,t)|2−κ3δ2η(L,1,t)Ut(L,t). Multiplying the second equation of ( 2.1) byη, integrating over (0,1)with respect to ρ, then using the fact that η(L,0,t) =Ut(L,t), we get (2.4)τ 2d dt/integraldisplay1 0|η|2dρ=−1 2|η(L,1,t)|2+1 2|Ut(L,t)|2. Adding ( 2.3) and ( 2.4), we get (2.5)dE(t) dt=−δ1/integraldisplayβ α|Uxt|2dx−/parenleftbigg κ3δ3−1 2/parenrightbigg |Ut(L,t)|2−κ3δ2η(L,1,t)Ut(L,t)−1 2|η(L,1,t)|2. For allp>0, we have (2.6) −κ3δ2η(L,1,t)Ut(L,t)≤κ3|δ2||η(L,1,t)|2 2p+κ3|δ2|p|Ut(L,t)|2 2. Inserting ( 2.6) in (2.5), we get (2.7)dE(t) dt≤ −δ1/integraldisplayβ α|Uxt|2dx−/parenleftbigg1 2−κ3|δ2| 2p/parenrightbigg |η(L,1,t)|2−/parenleftbigg κ3δ3−1 2−κ3|δ2|p 2/parenrightbigg |Ut(L,t)|2. In the sequel, the assumption on κ3, δ1, δ2andδ3will ensure that (H) κ3>0, δ1>0, δ3>0, δ2∈R∗, δ3>1 2κ3,|δ2|<1 κ3/radicalbig 2κ3δ3−1. In this case, we easily check that there exists a strictly pos itive number psatisfying (2.8) κ3|δ2|<p<2 κ3|δ2|/parenleftbigg κ3δ3−1 2/parenrightbigg , such that 1 2−κ3|δ2| 2p>0andκ3δ3−1 2−κ3|δ2|p 2>0, so that the energies of the strong solutions satisfy E′(t)≤0.Hence, System ( 2.1) is dissipative in the sense that its energy is non increasing with respect to the time t. For studying the stability of System ( 2.1), we consider two cases. In Subsection 2.1, we consider the first case, 7WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY when the Kelvin-Voigt damping is localized in the internal o f the body, i.e.α >0. While in Subsection 2.2, we consider the second, when the Kelvin-Voigt damping is loc alized near the boundary of the body, i.e.α= 0. 2.1.Wave equation with local Kelvin-Voigt damping far from the b oundary and with boundary delay feedback. In this subsection, we assume that there exist αandβsuch that 0< α < β < L , in this case, the Kelvin-Voigt damping is localized in the internal of the body (see Figure 1). For this aim, we denote the longitudinal displacement by Uand this displacement is divided into three parts U(x,t) = u(x,t),(x,t)∈(0,α)×(0,+∞), v(x,t),(x,t)∈(α,β)×(0,+∞), w(x,t),(x,t)∈(β,L)×(0,+∞). In this case, System ( 2.1) is equivalent to the following system utt−κ1uxx= 0,(x,t)∈(0,α)×(0,+∞), (2.9) vtt−(κ2vx+δ1vxt)x= 0,(x,t)∈(α,β)×(0,+∞), (2.10) wtt−κ3wxx= 0,(x,t)∈(β,L)×(0,+∞), (2.11) τηt(L,ρ,t)+ηρ(L,ρ,t) = 0,(ρ,t)∈(0,1)×(0,+∞), (2.12) with the following boundary and transmission conditions u(0,t) = 0, t∈(0,+∞), (2.13) wx(L,t) =−δ3wt(L,t)−δ2η(L,1,t), t∈(0,+∞), (2.14) u(α,t) =v(α,t), t∈(0,+∞), (2.15) v(β,t) =w(β,t), t∈(0,+∞), (2.16) κ2vx(α,t)+δ1vxt(α,t) =κ1ux(α,t), t∈(0,+∞), (2.17) κ2vx(β,t)+δ1vxt(β,t) =κ3wx(β,t), t∈(0,+∞), (2.18) and with the following initial conditions (u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,α), (2.19) (v(x,0),vt(x,0)) = (v0(x),v1(x)), x∈(α,β), (2.20) (w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L), (2.21) η(L,ρ,0) =f0(L,−ρτ), ρ∈(0,1), (2.22) where the initial data (u0,u1,v0,v1,w0,w1,f0)belongs to a suitable Hilbert space. So, using ( 2.2), the energy of System ( 2.9)-(2.22) is given by E(t) =1 2/integraldisplayα 0/parenleftbig |ut|2+κ1|ux|2/parenrightbig dx+1 2/integraldisplayβ α/parenleftbig |vt|2+κ2|vx|2/parenrightbig dx+1 2/integraldisplayL β/parenleftbig |wt|2+κ3|wx|2/parenrightbig dx+τ 2/integraldisplay1 0|η|2dρ. Similar to ( 2.5) and ( 2.7), we get dE(t) dt=−δ1/integraldisplayβ α|vxt|2dx−1 2|η(L,1,t)|2−κ3δ2η(L,1,t)wt(L,t)−/parenleftbigg κ3δ3−1 2/parenrightbigg |wt(L,t)|2, ≤ −δ1/integraldisplayβ α|vxt|2dx−/parenleftbigg1 2−κ3|δ2| 2p/parenrightbigg |η(L,1,t)|2−/parenleftbigg κ3δ3−1 2−κ3|δ2|p 2/parenrightbigg |wt(L,t)|2, wherepis defined in ( 2.8). Thus, under hypothesis (H), the System ( 2.9)-(2.22) is dissipative in the sense that its energy is non increasing with respect to the time t.Now, we are in position to prove the existence and uniqueness of the solution of our system. 8WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY 2.1.1. Well-posedness of the problem. We start this part by formulating System ( 2.9)-(2.22) as an abstract Cauchy problem. For this aim, let us define Hm=Hm(0,α)×Hm(α,β)×Hm(β,L), m= 1,2, L2=L2(0,α)×L2(α,β)×L2(β,L), H1 L={(u,v,w)∈H1|u(0) = 0, u(α) =v(α), v(β) =w(β)}. Remark 2.1. The Hilbert space L2is equipped with the norm: /ba∇dbl(u,v,w)/ba∇dbl2 L2=/integraldisplayα 0|u|2dx+/integraldisplayβ α|v|2dx+/integraldisplayL β|w|2dx. Also, it is easy to check that the space H1 Lis Hilbert space over Cequipped with the norm: /ba∇dbl(u,v,w)/ba∇dbl2 H1 L=κ1/integraldisplayα 0|ux|2dx+κ2/integraldisplayβ α|vx|2dx+κ3/integraldisplayL β|wx|2dx. Moreover, by Poincaré inequality we can easily verify that the re existsC >0, such that /ba∇dbl(u,v,w)/ba∇dblL2≤C/ba∇dbl(u,v,w)/ba∇dblH1 L. /square We now define the Hilbert energy space Hby H=H1 L×L2×L2(0,1) equipped with the following inner product /a\}b∇acketle{tU,˜U/a\}b∇acket∇i}htH=κ1/integraldisplayα 0ux˜uxdx+κ2/integraldisplayβ αvx˜vxdx+κ3/integraldisplayL βwx˜wxdx+/integraldisplayα 0y˜ydx+/integraldisplayβ αz˜zdx+/integraldisplayL βφ˜φdx+τ/integraldisplay1 0η(L,ρ)˜η(L,ρ)dρ, whereU= (u,v,w,y,z,φ,η (L,·))∈ Hand˜U= (˜u,˜v,˜w,˜y,˜z,˜φ,˜η(L,·))∈ H. We use /ba∇dblU/ba∇dblHto denote the corresponding norm. We define the linear unbounded operator A:D(A)⊂ H −→ H by: D(A) =/braceleftbigg (u,v,w,y,z,φ,η (L,·))∈H1 L×H1 L×H1(0,1)|(u,κ2v+δ1z,w)∈H2, κ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0)/bracerightbigg and for all U= (u,v,w,y,z,φ,η (L,·))∈D(A) AU=/parenleftbig y,z,φ,κ 1uxx,(κ2vx+δ1zx)x,κ3wxx,−τ−1ηρ(L,·)/parenrightbig . IfU= (u,v,w,u t,vt,wt,η(L,·))is a regular solution of System ( 2.9)-(2.22), then we transform this system into the following initial value problem (2.23)/braceleftBigg Ut=AU, U(0) =U0, whereU0= (u0,v0,w0,u1,v1,w1,f0(L,−·τ))∈ H.We now use semigroup approach to establish well-posedness result for the System ( 2.9)-(2.22). According to Lumer-Phillips theorem (see [ 36]), we need to prove that the operator Ais m-dissipative in H. Therefore, we prove the following proposition. Proposition 2.2. Under hypothesis (H), the unbounded linear operator Ais m-dissipative in the energy space H. Proof. For allU= (u,v,w,y,z,φ,η (L,·))∈D(A),we have Re/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=κ1Re/integraldisplayα 0(yxux+uxxy)dx+Re/integraldisplayβ α(κ2zxvx+(κ2vx+δ1zx)xz)dx +κ3Re/integraldisplayL β/parenleftbig φxwx+wxxφ/parenrightbig dx−Re/integraldisplay1 0ηρ(L,ρ)η(L,ρ)dρ. 9WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Here Re is used to denote the real part of a complex number. Usi ng by parts integration in the above equation, we get (2.24)Re/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=−δ1/integraldisplayβ α|zx|2dx−1 2|η(L,1)|2+1 2|η(L,0)|2+κ3Re/parenleftbig wx(L)φ(L)/parenrightbig −κ1Re(ux(0)y(0))+ Re(κ1ux(α)y(α)−κ2vx(α)z(α)−δ1zx(α)z(α)) +Re/parenleftbig κ2vx(β)z(β)+δ1zx(β)z(β)−κ3wx(β)φ(β)/parenrightbig . On the other hand, since U∈D(A), we have (2.25) y(0) = 0, y(α) =z(α), z(β) =φ(β), κ1ux(α)−κ2vx(α)−δ1zx(α) = 0, κ2vx(β)+δ1zx(β)−κ3wx(β) = 0, wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0). Inserting ( 2.25) in (2.24), we get (2.26) Re /a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=−δ1/integraldisplayβ α|zx|2dx−1 2|η(L,1)|2−/parenleftbigg κ3δ3−1 2/parenrightbigg |η(L,0)|2−κ3δ2Re(η(L,0)η(L,1)). Under hypothesis (H), we easily check that there exists p>0such that κ3|δ2|<p<2 κ3|δ2|/parenleftbigg κ3δ3−1 2/parenrightbigg . By Young’s inequality, we get −κ3δ2Re(η(L,0)η(L,1))≤κ3|δ2||η(L,1)|2 2p+κ3|δ2|p|η(L,0)|2 2. Inserting the above inequality in ( 2.26), we get (2.27) Re /a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH≤ −δ1/integraldisplayβ α|zx|2dx−/parenleftbigg1 2−κ3|δ2| 2p/parenrightbigg |η(L,1)|2−/parenleftbigg κ3δ3−1 2−κ3|δ2|p 2/parenrightbigg |η(L,0)|2. From the construction of p, we have 1 2−κ3|δ2| 2p>0andκ3δ3−1 2−κ3|δ2|p 2>0. Therefore, from ( 2.27), we get Re/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH≤0, which implies that Ais dissipative. Now, let us go on with maximality. Let F= (f1,f2,f3,f4,f5,f6,f7(L,·))∈ Hwe look for U= (u,v,w,y,z,φ,η (L,·))∈D(A)solution of the equation (2.28) −AU=F. Equivalently, we consider the following system −y=f1, (2.29) −z=f2, (2.30) −φ=f3, (2.31) −κ1uxx=f4, (2.32) −(κ2vx+δ1zx)x=f5, (2.33) −κ3wxx=f6, (2.34) ηρ(L,ρ) =τf7(ρ). (2.35) In addition, we consider the following boundary conditions u(0) = 0, u(α) =v(α), v(β) =w(β), (2.36) κ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.37) wx(L) =−δ3η(L,0)−δ2η(L,1), (2.38) η(L,0) =φ(L). (2.39) 10WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY From ( 2.29)-(2.31) and the fact that F∈ H, it is clear that (y,z,φ)∈H1 L. Next, from ( 2.31), (2.39) and the fact thatf3∈H1(β,L), we get η(L,0) =φ(L) =−f3(L). From the above equation and Equation ( 2.35), we can determine η(L,ρ) =τ/integraldisplayρ 0f7(ξ)dξ−f3(L). It is clear that η(L,·)∈H1(0,1)andη(L,0) =φ(L) =−f3(L). Inserting the above equation in ( 2.38), then System ( 2.29)-(2.39) is equivalent to y=−f1, z=−f2, φ=−f3, η(L,ρ) =τ/integraldisplayρ 0f7(ξ)dξ−f3(L), (2.40) −κ1uxx=f4, (2.41) −(κ2vx+δ1zx)x=f5, (2.42) −κ3wxx=f6, (2.43) u(0) = 0, u(α) =v(α), v(β) =w(β), (2.44) κ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.45) wx(L) = (δ3+δ2)f3(L)−τδ2/integraldisplay1 0f7(ξ)dξ. (2.46) Let(ϕ,ψ,θ)∈H1 L. Multiplying Equations ( 2.41), (2.42), (2.43) byϕ,ψ,θ, integrating over (0,α),(α,β)and (β,L)respectively, taking the sum, then using by parts integrati on, we get (2.47)κ1/integraldisplayα 0uxϕxdx+/integraldisplayβ α(κ2vx+δ1zx)ψxdx+κ3/integraldisplayL βwxθxdx+κ1ux(0)ϕ(0) −κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α))ψ(α)−(κ2vx(β)+δ1zx(β))ψ(β)+κ3wx(β)θ(β) =/integraldisplayα 0f4ϕdx+/integraldisplayβ αf5ψdx+/integraldisplayL βf6θdx+κ3wx(L)θ(L). From the fact that (ϕ,ψ,θ)∈H1 L,we have ϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β). Inserting the above equation in ( 2.47), then using ( 2.40) and ( 2.44)-(2.46), we get (2.48)κ1/integraldisplayα 0uxϕxdx+κ2/integraldisplayβ αvxψxdx+κ3/integraldisplayL βwxθxdx=/integraldisplayα 0f4ϕdx +/integraldisplayβ α/parenleftbig δ1(f2)xψx+f5ψ/parenrightbig dx+/integraldisplayL βf6θdx+κ3/parenleftbigg (δ3+δ2)f3(L)−τδ2/integraldisplay1 0f7(ξ)dξ/parenrightbigg θ(L). We can easily verify that the left hand side of ( 2.48) is a bilinear continuous coercive form on H1 L×H1 L, and the right hand side of ( 2.48) is a linear continuous form on H1 L. Then, using Lax-Milgram theorem, we deduce that there exists (u,v,w)∈H1 Lunique solution of the variational Problem ( 2.48). Using stan- dard arguments, we can show that (u,κ2v+δ1z,w)∈H2. Finally, by seting y=−f1, z=−f2, φ=−f3 andη(L,ρ) =τ/integraldisplayρ 0f7(ξ)dξ−f3(L)and by applying the classical elliptic regularity we deduce thatU= (u,v,w,y,z,φ,η (L,·))∈D(A)is solution of Equation ( 2.28). To conclude, we need to show the uniqueness of U. So, letU= (u,v,w,y,z,φ,η (L,·))∈D(A)be a solution of ( 2.28) withF= 0, then we directly deduce that y=z=φ=η(L,ρ) = 0 and that (u,v,w)∈H1 Lsatisfies Problem ( 2.48) with zero in the right hand side. This implies that u=v=w= 0, in other words, ker A={0}and0belongs to the resolvent set ρ(A)ofA. Then, by contraction principale, we easily deduce that R(λI− A) =Hfor sufficiently small λ >0. This, together with the dissipativeness of A, imply that D(A)is dense in Hand that Ais m-dissipative in H(see Theorems 4.5, 4.6 in [ 36]). The proof is thus complete. /square Thanks to Lumer-Philips theorem (see [ 36]), we deduce that Agenerates a C0−semigroup of contractions etA inHand therefore Problem ( 2.9)-(2.22) is well-posed. Then we have the following result: 11WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Theorem 2.3. Under hypothesis (H), for any U0∈ H,Problem (2.23)admits a unique weak solution, U(x,ρ,t) =etAU0(x,ρ), such that U∈C0(R+,H). Moreover, if U0∈D(A),then U∈C1(R+,H)∩C0(R+,D(A)). 2.1.2. Strong Stability. Our main result in this part is the following theorem. Theorem 2.4. Under hypothesis (H), theC0−semigroup of contractions etAis strongly stable on the Hilbert spaceHin the sense that lim t→+∞||etAU0||H= 0, ∀U0∈ H. For the proof of Theorem 2.4, according to Theorem A.2, we need to prove that the operator Ahas no pure imaginary eigenvalues and σ(A)∩iRcontains only a countable number of continuous spectrum of A. The argument for Theorem 2.4relies on the subsequent lemmas. Lemma 2.5. Under hypothesis (H), forλ∈R,we haveiλI−Ais injective, i.e. ker(iλI−A) ={0},∀λ∈R. Proof. From Proposition 2.2, we have 0∈ρ(A).We still need to show the result for λ∈R∗.Suppose that there exists a real number λ/\e}atio\slash= 0andU= (u,v,w,y,z,φ,η (L,·))∈D(A)such that (2.49) AU=iλU. First, similar to Equation ( 2.27), we have 0 =Re/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH≤ −δ1/integraldisplayβ α|zx|2dx−/parenleftbigg1 2−κ3|δ2| 2p/parenrightbigg |η(L,1)|2−/parenleftbigg κ3δ3−1 2−κ3|δ2|p 2/parenrightbigg |η(L,0)|2≤0. Thus, (2.50) zx= 0 in(α,β)andη(L,1) =η(L,0) = 0. Next, writing ( 2.49) in a detailed form gives y=iλu, x∈(0,α), (2.51) z=iλv, x∈(α,β), (2.52) w=iλφ, x∈(β,L), (2.53) κ1uxx=iλy, x∈(0,α), (2.54) (κ2vx+δ1zx)x=iλz, x∈(α,β), (2.55) κ3wxx=iλφ, x∈(β,L), (2.56) ηρ(L,ρ) =−iλτη(L,ρ), ρ∈(0,1). (2.57) From ( 2.57) and ( 2.50), we get (2.58) η(L,·) =η(L,0)e−iλτ·= 0 in(0,1). Combining ( 2.50) with ( 2.52), we get that (2.59) vx=zx= 0 in(α,β). Thus, vxx=zxx= 0 in(α,β). Inserting the above result in ( 2.55), then taking into consideration ( 2.52), we obtain (2.60) v=z= 0 in(α,β). From the definition of D(A)and using ( 2.58)-(2.60), we get u(α) =v(α) = 0, w(β) =v(β) = 0, y(α) =z(α) = 0, φ(β) =z(β) = 0, κ1ux(α) =κ2vx(α)+δ1zx(α) = 0, κ3wx(β) =k2vx(β)+δ1zx(β) = 0, w(L) =iλφ(L) =iλη(L,0) = 0, wx(L) =−δ3η(L,0)−δ2η(L,1) = 0. 12WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Combining ( 2.51) with ( 2.54) and ( 2.53) with ( 2.56) and using the above equation as boundary conditions, we get uxx+λ2 κ1u= 0, x∈(0,α), u(0) =u(α) =ux(α) = 0, wxx+λ2 κ3w= 0, x∈(β,L), w(β) =w(L) =wx(β) =wx(L) = 0. Thus, (2.61) u(x) = 0∀x∈(0,α)andw(x) = 0∀x∈(β,L). Combining ( 2.61) with ( 2.51) and ( 2.53), we obtain y(x) = 0∀x∈(0,α)andφ(x) = 0∀x∈(β,L). Finally, from the above result, ( 2.58), (2.60) and ( 2.61), we get that U= 0.The proof is thus complete. /square Lemma 2.6. Under hypothesis (H), forλ∈R,we haveiλI−Ais surjective, i.e. R(iλI−A) =H,∀λ∈R. Proof. Since0∈ρ(A),we still need to show the result for λ∈R∗.For anyF= (f1,f2,f3,f4,f5,f6,f7(L,·))∈ H andλ∈R∗,we prove the existence of U= (u,v,w,y,z,φ,η (L,·))∈D(A)solution for the following equation (iλI−A)U=F. Equivalently, we consider the following problem y=iλu−f1inH1(0,α), (2.62) z=iλv−f2inH1(α,β), (2.63) φ=iλw−f3inH1(β,L), (2.64) iλy−κ1uxx=f4inL2(0,α), (2.65) iλz−(κ2vx+δ1zx)x=f5inL2(α,β), (2.66) iλφ−κ3wxx=f6inL2(β,L), (2.67) ηρ(L,·)+iτλη(L,·) =τf7(L,·)inL2(0,1), (2.68) with the following boundary conditions u(0) = 0, u(α) =v(α), v(β) =w(β), (2.69) κ2vx(α)+δ1zx(α) =κ1ux(α), κ2vx(β)+δ1zx(β) =κ3wx(β), (2.70) wx(L) =−δ3η(L,0)−δ2η(L,1), (2.71) η(L,0) =φ(L). (2.72) It follows from ( 2.68), (2.72) and ( 2.64) that (2.73) η(L,ρ) = (iλw(L)−f3(L))e−iτλρ+τ/integraldisplayρ 0eiτλ(ξ−ρ)f7(L,ξ)dξ. Inserting ( 2.62)-(2.64) and ( 2.73) in (2.65)-(2.72) and deriving ( 2.63) with respect to x, we get −λ2u−κ1uxx=iλf1+f4, (2.74) −λ2v−(κ2vx+δ1zx)x=iλf2+f5, (2.75) −λ2w−κ3wxx=iλf3+f6, (2.76) zx=iλvx−(f2)x, (2.77) u(0) = 0, u(α) =v(α), κ1ux(α) =κ2vx(α)+δ1zx(α), (2.78) w(β) =v(β), κ3wx(β) =κ2vx(β)+δ1zx(β), (2.79) wx(L) =−iλ/parenleftbig δ3+δ2e−iτλ/parenrightbig w(L)+/parenleftbig δ3+δ2e−iτλ/parenrightbig f3(L)−τδ2/integraldisplay1 0eiτλ(ξ−1)f7(L,ξ)dξ. (2.80) 13WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Let(ϕ,ψ,θ)∈H1 L. Multiplying Equations ( 2.74), (2.75), (2.76) byϕ,ψ,θ, integrating over (0,α),(α,β)and (β,L)respectively, taking the sum, then using by parts integrati on, we get (2.81)κ1/integraldisplayα 0uxϕxdx+/integraldisplayβ α(κ2vx+δ1zx)ψxdx+κ3/integraldisplayL βwxθxdx+κ1ux(0)ϕ(0) −κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α))ψ(α)−(κ2vx(β)+δ1zx(β))ψ(β)+κ3wx(β)θ(β) −λ2/integraldisplayα 0uϕdx−λ2/integraldisplayβ αvψdx−λ2/integraldisplayL βwθdx−κ3wx(L)θ(L) =/integraldisplayα 0(iλf1+f4)ϕdx+/integraldisplayβ α(iλf2+f5)ψdx+/integraldisplayL β(iλf3+f6)θdx. From the fact that (ϕ,ψ,θ)∈H1 L,we have ϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β). Inserting the above equation in ( 2.81), then using ( 2.77)-(2.80), we get (2.82) a((u,v,w),(ϕ,ψ,θ)) = F(ϕ,ψ,θ),∀(ϕ,ψ,θ)∈H1 L, where F(ϕ,ψ,θ) =/integraldisplayα 0(iλf1+f4)ϕdx+/integraldisplayβ α(iλf2+f5)ψdx+δ1/integraldisplayβ α(f2)xψxdx +/integraldisplayL β(iλf3+f6)θdx+κ3/parenleftbigg/parenleftbig δ3+δ2e−iτλ/parenrightbig f3(L)−τδ2/integraldisplay1 0eiτλ(ξ−1)f7(L,ξ)dξ/parenrightbigg θ(L) and a((u,v,w),(ϕ,ψ,θ)) =a1((u,v,w),(ϕ,ψ,θ))+a2((u,v,w),(ϕ,ψ,θ)), such that a1((u,v,w),(ϕ,ψ,θ)) =κ1/integraldisplayα 0uxϕxdx+(κ2+iδ1λ)/integraldisplayβ αvxψxdx+κ3/integraldisplayL βwxθxdx, a2((u,v,w),(ϕ,ψ,θ)) =−λ2/integraldisplayα 0uϕdx−λ2/integraldisplayβ αvψdx−λ2/integraldisplayL βwθdx+iκ3λ/parenleftbig δ3+δ2e−iτλ/parenrightbig w(L)θ(L). Let/parenleftbig H1 L/parenrightbig′be the dual space of H1 L. We define the operators A,A1andA2by /braceleftBigg A :H1 L→/parenleftbig H1 L/parenrightbig′ (u,v,w)→A(u,v,w)/braceleftBigg A1:H1 L→/parenleftbig H1 L/parenrightbig′ (u,v,w)→A1(u,v,w)/braceleftBigg A2:H1 L→/parenleftbig H1 L/parenrightbig′ (u,v,w)→A2(u,v,w) such that (2.83) (A(u,v,w))(ϕ,ψ,θ) =a((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1 L, (A1(u,v,w))(ϕ,ψ,θ) =a1((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1 L, (A2(u,v,w))(ϕ,ψ,θ) =a2((u,v,w),(ϕ,ψ,θ)),∀(ϕ,ψ,θ)∈H1 L. Our aim is to prove that the operator Ais an isomorphism. For this aim, we proceed the proof in three steps. Step 1. In this step we proof that the operator A1is an isomorphism. For this aim, according to ( 2.83), we have a1((u,v,w),(ϕ,ψ,θ)) =κ1/integraldisplayα 0uxϕxdx+(κ2+iδ1λ)/integraldisplayβ αvxψxdx+κ3/integraldisplayL βwxθxdx We can easily verify that a1is a bilinear continuous coercive form on H1 L×H1 L. Then, by Lax-Milgram lemma, the operator A1is an isomorphism. Step 2. In this step we proof that the operator A2is compact. First, for1 2<r<1, we introduce the Hilbert spaceHr Lby Hr L={(ϕ,ψ,θ)∈Hr(0,α)×Hr(α,β)×Hr(β,L)|ϕ(0) = 0, ϕ(α) =ψ(α), ψ(β) =θ(β)}. 14WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Thus by trace theorem, there exists C >0, such that (2.84) |θ(L)| ≤C/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr L. Now, according to ( 2.83), we have a2((u,v,w),(ϕ,ψ,θ)) =−λ2/integraldisplayα 0uϕdx−λ2/integraldisplayβ αvψdx−λ2/integraldisplayL βwθdx+iκ3λ/parenleftbig δ3+δ2e−iτλ/parenrightbig w(L)θ(L). Then, by using ( 2.84), we get |a2((u,v,w),(ϕ,ψ,θ))| ≤C1/ba∇dbl(u,v,w)/ba∇dblH1 L/ba∇dbl(ϕ,ψ,θ)/ba∇dblL2+C1/ba∇dbl(u,v,w)/ba∇dblH1 L/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr L, whereC1>0. Therefore, for all r∈(1 2,1)there exists C2>0, such that |a2((u,v,w),(ϕ,ψ,θ))| ≤C2/ba∇dbl(u,v,w)/ba∇dblH1 L/ba∇dbl(ϕ,ψ,θ)/ba∇dblHr L, which implies that A2∈ L/parenleftbig H1 L,(Hr L)′/parenrightbig . Finally, using the compactness embedding from (Hr L)′into/parenleftbig H1 L/parenrightbig′we deduce that A2is compact. From steps 1 and 2, we get that the operator A = A 1+A2is a Fredholm operator of index zero 0. Consequently, by Fredholm alternative, proving the operator Ais an isomorphism reduces to proving ker(A) = {0}. Step 3. In this step we proof that the ker(A) = {0}. For this aim, let (˜u,˜v,˜w)∈ker(A) ,i.e. a((˜u,˜v,˜w),(ϕ,ψ,θ)) = 0,∀(ϕ,ψ,θ)∈H1 L. Equivalently, /integraldisplayα 0/parenleftbig κ1˜uxϕx−λ2˜uϕ/parenrightbig dx+/integraldisplayβ α/parenleftbig (κ2+iδ1λ)˜vxψx−λ2˜vψ/parenrightbig dx+/integraldisplayL β/parenleftbig κ3˜wxθx−λ2˜wθ/parenrightbig dx +iκ3λ/parenleftbig δ3+δ2e−iτλ/parenrightbig ˜w(L)θ(L) = 0,∀(ϕ,ψ,θ)∈H1 L. Then, we find that −λ2˜u−κ1˜uxx= 0, −λ2˜v−(κ2+iδ1λ)˜vxx= 0, −λ2˜w−κ3˜wxx= 0, ˜u(0) = 0,˜u(α) = ˜v(α), κ1˜ux(α) = (κ2+iδ1λ)˜vx(α), ˜w(β) = ˜v(β), κ3˜wx(β) = (κ2+iδ1λ)˜vx(β), ˜wx(L) =−iλ/parenleftbig δ3+δ2e−iτλ/parenrightbig ˜w(L). Therefore, the vector ˜Vdefine by ˜V=/parenleftbig ˜u,˜v,˜w,iλ˜u,iλ˜v,iλ˜w,iλ˜w(L)e−iτλ·/parenrightbig belongs toD(A)and we have iλ˜V−A˜V= 0. Thus,˜V∈ker(iλI−A), therefore by Lemma 2.5, we get ˜V= 0, this implies that ˜u= 0,˜v= 0and˜w= 0, so ker(A) = {0}. Therefore, from step 3 and Fredholm alternative, we get that the operator Ais an isomorphism. It easy to see that the operator Fis continuous form on H1 L. Consequently, Equation ( 2.82) admits a unique solution (u,v,w)∈H1 L. Thus, using ( 2.62)-(2.64), (2.73) and a classical regularity arguments, we conclude that (iλI− A)U=Fadmits a unique solution U∈D(A). The proof is thus complete. /square Proof of Theorem 2.4.Form Lemma 2.5, we have that the operator Ahas no pure imaginary eigenvalues and by Lemma 2.6,R(iλI−A) =Hfor allλ∈R.Therefore, the closed graph theorem implies that σ(A)∩iR=∅. Thus, we get the conclusion by applying Theorem A.2of Arendt and Batty. The proof is thus complete. /square 15WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY 2.1.3. Polynomial Stability. In this part, we will prove the polynomial stability of Syste m (2.9)-(2.22). Our main result in this part is the following theorem. Theorem 2.7. Under hypothesis (H), for all initial data U0∈D(A),there exists a constant C >0independent ofU0such that the energy of System (2.9)-(2.22)satisfies the following estimation (2.85) E(t)≤C t4/ba∇dblU0/ba∇dbl2 D(A),∀t>0. From Lemma 2.5and Lemma 2.6, we have seen that iR⊂ρ(A),then for the proof of Theorem 2.7, according to Theorem A.5(part (ii)), we need to prove that (2.86) sup λ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble L(H)=O/parenleftBig |λ|1 2/parenrightBig . We will argue by contradiction. Indeed, suppose there exist s {(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(L,·)))}n≥1⊂R∗ +×D(A), such that (2.87) λn→+∞,/ba∇dblUn/ba∇dblH= 1 and there exists sequence Fn:= (f1,n,f2,n,f3,n,f4,n,f5,n,f6,n,f7,n(L,·))∈ H, such that (2.88) λℓ n(iλnI−A)Un=Fn→0inH. In case that ℓ=1 2, we will check condition ( 2.86) by finding a contradiction with /ba∇dblUn/ba∇dblH= 1such as/ba∇dblUn/ba∇dblH= o(1).From now on, for simplicity, we drop the index n. By detailing Equation ( 2.88), we get the following system iλu−y=λ−ℓf1inH1(0,α), (2.89) iλv−z=λ−ℓf2inH1(α,β), (2.90) iλw−φ=λ−ℓf3inH1(β,L), (2.91) iλy−κ1uxx=λ−ℓf4inL2(0,α), (2.92) iλz−(κ2vx+δ1zx)x=λ−ℓf5inL2(α,β), (2.93) iλφ−κ3wxx=λ−ℓf6inL2(β,L), (2.94) ηρ(L,·)+iτλη(L,·) =τλ−ℓf7(L,·)inL2(0,1). (2.95) Remark that, since U= (u,v,w,y,z,φ,η (L,·))∈D(A), we have the following boundary conditions (2.96)/braceleftBigg |ux(α)|=κ−1 1|κ2vx(α)+δ1zx(α)|,|y(α)|=|z(α)|, |wx(β)|=κ−1 3|κ2vx(β)+δ1zx(β)|,|z(β)|=|φ(β)| and (2.97) wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0). The proof of Theorem 2.7is divided into several lemmas. Lemma 2.8. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations (2.89)-(2.95)satisfies the following asymptotic behavior estimations /integraldisplayβ α|zx|2=o/parenleftbig λ−ℓ/parenrightbig , (2.98) |φ(L)|2=|η(L,0)|2=o/parenleftbig λ−ℓ/parenrightbig ,|η(L,1)|2=o/parenleftbig λ−ℓ/parenrightbig , (2.99) /integraldisplayβ α|vx|2dx=o/parenleftbig λ−ℓ−2/parenrightbig , (2.100) |wx(L)|2=o/parenleftbig λ−ℓ/parenrightbig . (2.101) 16WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Proof. Taking the inner product of ( 2.88) withUinH, then using the fact that Uis uniformly bounded in H, we get −Re/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=Re/a\}b∇acketle{t(iλI−A)U,U/a\}b∇acket∇i}htH=o/parenleftbig λ−ℓ/parenrightbig , Now, under hypothesis (H), similar to Equation ( 2.27), we get (2.102) 0≤δ1/integraldisplayβ α|zx|2dx+C1|η(L,1)|2+C2|η(L,0)|2≤ −Re/a\}b∇acketle{tAU,U/a\}b∇acket∇i}htH=o/parenleftbig λ−ℓ/parenrightbig , where C1=1 2−κ3|δ2| 2p>0andC2=κ3δ3−1 2−κ3|δ2|p 2>0. Therefore, from ( 2.102), we get ( 2.98) and ( 2.99). Next, from ( 2.90), (2.98) and the fact that (f2)x→0in L2(α,β), we get ( 2.100). Finally, from ( 2.97) and ( 2.99), we obtain ( 2.101). Thus, the proof of the lemma is complete. /square Lemma 2.9. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations (2.89)-(2.95)satisfies the following asymptotic behavior estimation (2.103)/integraldisplay1 0|η(L,ρ)|2dρ=o/parenleftbig λ−ℓ/parenrightbig . Proof. It follows from ( 2.95) that η(L,ρ) =η(L,0)e−iτλρ+τλ−ℓ/integraldisplayρ 0eiτλ(ξ−ρ)f7(L,ξ)dξ∀ρ∈(0,1). By using Cauchy Schwarz inequality, we get |η(L,ρ)|2≤2|η(L,0)|2+2τ2λ−2ℓ/parenleftbigg/integraldisplay1 0|f7(L,ξ)|dξ/parenrightbigg2 ≤2|η(L,0)|2+2τ2λ−2ℓ/integraldisplay1 0|f7(L,ξ)|2dξ∀ρ∈(0,1). Integrating over (0,1)with respect to ρ, then using ( 2.99) and the fact that f7(L,·)→0inL2(0,1), we get /integraldisplay1 0|η(L,ρ)|2dρ≤2|η(L,0)|2+2τ2λ−2ℓ/integraldisplay1 0|f7(L,ξ)|2dξ=o/parenleftbig λ−ℓ/parenrightbig , hence, we get ( 2.103). Thus, the proof of the lemma is complete. /square Lemma 2.10. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations (2.89)-(2.95)satisfies the following asymptotic behavior estimations /integraldisplayL β|φ|2dx=o/parenleftbig λ−ℓ/parenrightbig ,/integraldisplayL β|wx|2dx=o/parenleftbig λ−ℓ/parenrightbig , (2.104) |wx(β)|2=o/parenleftbig λ−ℓ/parenrightbig ,|φ(β)|2=o/parenleftbig λ−ℓ/parenrightbig , (2.105) |κ2vx(β)+δ1zx(β)|2=o/parenleftbig λ−ℓ/parenrightbig ,|z(β)|2=o/parenleftbig λ−ℓ/parenrightbig . (2.106) Proof. Multiplying Equation ( 2.94) byxwxand integrating over (β,L),we get (2.107) iλ/integraldisplayL βxφwxdx−κ3/integraldisplayL βxwxxwxdx=λ−ℓ/integraldisplayL βxf6wxdx. From ( 2.91), we deduce that iλwx=−φx−λ−ℓ(f3)x. Inserting the above result in ( 2.107), then using the fact that φ, wxare uniformly bounded in L2(β,L)and (f3)x, f6converge to zero in L2(β,L)gives −/integraldisplayL βxφφxdx−κ3/integraldisplayL βxwxxwxdx=o/parenleftbig λ−ℓ/parenrightbig . Taking the real part in the above equation, then using by part s integration, we get 1 2/integraldisplayL β|φ|2dx+κ3 2/integraldisplayL β|wx|2dx+β 2/parenleftbig κ3|wx(β)|2+|φ(β)|2/parenrightbig =L 2/parenleftbig κ3|wx(L)|2+|φ(L)|2/parenrightbig +o/parenleftbig λ−ℓ/parenrightbig . 17WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Inserting ( 2.99) and ( 2.101) in the above equation, we get 1 2/integraldisplayL β|φ|2dx+κ3 2/integraldisplayL β|wx|2dx+β 2/parenleftbig κ3|wx(β)|2+|φ(β)|2/parenrightbig =o/parenleftbig λ−ℓ/parenrightbig , hence, we get ( 2.104) and ( 2.105). Finally, from ( 2.96) and ( 2.105), we obtain ( 2.106). The proof is thus complete. /square Lemma 2.11. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations (2.89)-(2.95)satisfies the following asymptotic behavior estimations /integraldisplayβ α|z|2dx=o/parenleftBig λ−min(2ℓ+1 2,ℓ+1)/parenrightBig , (2.108) |z(α)|2=o/parenleftBig λ−min(2ℓ,ℓ+1 2)/parenrightBig ,|z(β)|2=o/parenleftBig λ−min(2ℓ,ℓ+1 2)/parenrightBig , (2.109) |κ2vx(α)+δ1zx(α)|2=o/parenleftBig λ−min(2ℓ−1,ℓ−1 2)/parenrightBig . (2.110) Proof. Letg∈C1([α,β])such that g(β) =−g(α) = 1,max x∈[α,β]|g(x)|=cgandmax x∈[α,β]|g′(x)|=cg′, wherecgandcg′are strictly positive constant numbers independent from λ. The proof is divided into three steps. Step 1. In this step, we prove the following asymptotic behavior est imate (2.111) |z(β)|2+|z(α)|2≤/parenleftBigg λ1 2 2+2cg′/parenrightBigg/integraldisplayβ α|z|2dx+o/parenleftBig λ−min(2ℓ,ℓ+1 2)/parenrightBig . First, from ( 2.90), we have (2.112) zx=iλvx−λ−ℓ(f2)xinL2(α,β). Multiplying ( 2.112) by2gzand integrating over (α,β),then taking the real part, we get /integraldisplayβ αg(x)(|z|2)xdx=Re/braceleftBigg 2iλ/integraldisplayβ αg(x)vxzdx/bracerightBigg −Re/braceleftBigg 2λ−ℓ/integraldisplayβ αg(x)(f2)xzdx/bracerightBigg , using by parts integration in the left hand side of above equa tion, we get /bracketleftbig g(x)|z|2/bracketrightbigβ α=/integraldisplayβ αg′(x)|z|2dx+Re/braceleftBigg 2iλ/integraldisplayβ αg(x)vxzdx/bracerightBigg −Re/braceleftBigg 2λ−ℓ/integraldisplayβ αg(x)(f2)xzdx/bracerightBigg , consequently, (2.113) |z(β)|2+|z(α)|2≤cg′/integraldisplayβ α|z|2dx+2λcg/integraldisplayβ α|vx||z|dx+2λ−ℓcg/integraldisplayβ α|(f2)x||z|dx. On the other hand, we have 2λcg|vx||z| ≤λ1 2|z|2 2+2λ3 2c2 g|vx|2and2λ−ℓcg|(f2)x||z| ≤cg′|z|2+c2 gλ−2ℓ cg′|(f2)x|2. Inserting the above equation in ( 2.113), then using ( 2.100) and the fact that (f2)x→0inL2(α,β), we get |z(β)|2+|z(α)|2≤/parenleftBigg λ1 2 2+2cg′/parenrightBigg/integraldisplayβ α|z|2dx+o/parenleftBig λ−min(2ℓ,ℓ+1 2)/parenrightBig , hence, we get ( 2.111). Step 2. In this step, we prove the following asymptotic behavior est imate (2.114) |κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2≤λ3 2 2/integraldisplayβ α|z|2dx+o/parenleftBig λ−ℓ+1 2/parenrightBig . 18WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY First, multiplying ( 2.93) by−2g(κ2vx+δ1zx)and integrating over (α,β),then taking the real part, we get /integraldisplayβ αg(x)/parenleftBig |κ2vx+δ1zx|2/parenrightBig xdx= 2Re/braceleftBigg iλ/integraldisplayβ αg(x)z(κ2vx+δ1zx)dx/bracerightBigg −2λ−ℓRe/braceleftBigg/integraldisplayβ αg(x)f5(κ2vx+δ1zx)dx/bracerightBigg , using by parts integration in the left hand side of above equa tion, we get /bracketleftBig g(x)|κ2vx+δ1zx|2/bracketrightBigβ α=/integraldisplayβ αg′(x)|κ2vx+δ1zx|2dx+2Re/braceleftBigg iλ/integraldisplayβ αg(x)z(κ2vx+δ1zx)dx/bracerightBigg −2λ−ℓRe/braceleftBigg/integraldisplayβ αg(x)f5(κ2vx+δ1zx)dx/bracerightBigg , consequently, |κ2vx(β)+δ1zx(β)|2+|κ2vx(α)+δ1zx(α)|2≤cg′/integraldisplayβ α|κ2vx+δ1zx|2dx+2λcg/integraldisplayβ α|z||κ2vx+δ1zx|dx +2λ−ℓcg/integraldisplayβ α|f5||κ2vx+δ1zx|dx. Now, using Cauchy Schwarz inequality, Equations ( 2.98), (2.100) and the fact that f5→0inL2(α,β)in the right hand side of above equation, we get (2.115) |κ2vx(β)+δ1zx(β)|2+|κ2vx(α)+δ1zx(α)|2≤2λcg/integraldisplayβ α|z||κ2vx+δ1zx|dx+o/parenleftbig λ−ℓ/parenrightbig . On the other hand, we have 2λcg|z||κ2vx+δ1zx| ≤λ3 2 2|z|2+2λ1 2c2 g|κ2vx+δ1zx|2. Inserting the above equation in ( 2.115), then using Equations ( 2.98) and ( 2.100), we get |κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2≤λ3 2 2/integraldisplayβ α|z|2dx+o/parenleftBig λ−ℓ+1 2/parenrightBig , hence, we get ( 2.114). Step 3. In this step, we prove the asymptotic behavior estimations o f (2.108)-(2.110). First, multiplying ( 2.93) by−iλ−1zand integrating over (α,β),then taking the real part, we get /integraldisplayβ α|z|2dx=−Re/braceleftBigg iλ−1/integraldisplayβ α(κ2vx+δ1zx)xzdx/bracerightBigg −Re/braceleftBigg iλ−ℓ−1/integraldisplayβ αf5zdx/bracerightBigg , consequently, (2.116)/integraldisplayβ α|z|2dx≤λ−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ α(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+λ−ℓ−1/integraldisplayβ α|f5||z|dx. From the fact that zis uniformly bounded in L2(α,β)andf5→0inL2(α,β), we get (2.117) λ−ℓ−1/integraldisplayβ α|f5||z|dx=o/parenleftbig λ−ℓ−1/parenrightbig . 19WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY On the other hand, using by parts integration and ( 2.98), (2.100), we get (2.118)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ α(κ2v+δ1z)xxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle[(κ2vx+δ1zx)z]β α−/integraldisplayβ α(κ2vx+δ1zx)zxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤ |κ2vx(β)+δ1zx(β)||z(β)|+|κ2vx(α)+δ1zx(α)||z(α)|+/integraldisplayβ α|κ2vx+δ1zx||zx|dx ≤ |κ2vx(β)+δ1zx(β)||z(β)|+|κ2vx(α)+δ1z(α)||z(α)|+o/parenleftbig λ−ℓ/parenrightbig . Inserting ( 2.117) and ( 2.118) in (2.116), we get (2.119)/integraldisplayβ α|z|2dx≤λ−1|κ2vx(β)+δ1zx(β)||z(β)|+λ−1|κ2vx(α)+δ1zx(α)||z(α)|+o/parenleftbig λ−ℓ−1/parenrightbig . Now, forζ=βorζ=α, we have λ−1|κ2vx(ζ)+δ1zx(ζ)||z(ζ)| ≤λ−1 2 2|z(ζ)|2+λ−3 2 2|κ2vx(ζ)+δ1zx(ζ)|2. Inserting the above equation in ( 2.119), we get /integraldisplayβ α|z|2dx≤λ−1 2 2/parenleftbig |z(α)|2+|z(β)|2/parenrightbig +λ−3 2 2/parenleftBig |κ2vx(α)+δ1zx(α)|2+|κ2vx(β)+δ1zx(β)|2/parenrightBig +o/parenleftbig λ−ℓ−1/parenrightbig . Next, inserting Equations ( 2.111) and ( 2.114) in the above inequality, we obtain /integraldisplayβ α|z|2dx≤/parenleftbigg1 2+cg′ λ1 2/parenrightbigg/integraldisplayβ α|z|2dx+o/parenleftBig λ−min(2ℓ+1 2,ℓ+1)/parenrightBig , consequently,/parenleftbigg1 2−cg′ λ1 2/parenrightbigg/integraldisplayβ α|z|2dx≤o/parenleftBig λ−min(2ℓ+1 2,ℓ+1)/parenrightBig . Sinceλ→+∞, by choosing λ>4c2 g′, we get 0</parenleftbigg1 2−cg′ λ1 2/parenrightbigg/integraldisplayβ α|z|2dx≤o/parenleftBig λ−min(2ℓ+1 2,ℓ+1)/parenrightBig , hence, we get ( 2.108). Finally, inserting ( 2.108) in (2.111) and ( 2.114) and using the first asymptotic estimates of (2.106), we get ( 2.109) and ( 2.110). Thus, the proof of the lemma is complete. /square Remark 2.12. An example about g, we can take g(x) = cos/parenleftbigg(β−x)π β−α/parenrightbigg to get g(β) =−g(α) = 1, g∈C1([α,β]),max x∈[α,β]|g(x)|= 1,max x∈[α,β]|g′(x)|=π β−α. Also, we can take g(x) =/parenleftbiggβ−x β−α/parenrightbigg2 −3/parenleftbiggβ−x β−α/parenrightbigg +1. /square Lemma 2.13. Under hypothesis (H), for allℓ≥0, the solution (u,v,w,y,z,φ,η (L,·))∈D(A)of Equations (2.89)-(2.95)satisfies the following asymptotic behavior estimations (2.120)/integraldisplayα 0|y|2dx=o/parenleftBig λ−min(2ℓ−1,ℓ−1 2)/parenrightBig and/integraldisplayα 0|ux|2dx=o/parenleftBig λ−min(2ℓ−1,ℓ−1 2)/parenrightBig . Proof. Multiply Equation ( 2.92) byxuxand integrating over (0,α),we get (2.121) iλ/integraldisplayα 0xyuxdx−κ1/integraldisplayα 0xuxxuxdx=λ−ℓ/integraldisplayα 0xf4uxdx. From ( 2.89), we deduce that iλux=−yx−λ−ℓ(f1)x. 20WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Inserting the above result in ( 2.121), then using the fact that ux, yare uniformly bounded in L2(0,α)and (f1)x, f4converge to zero in L2(0,α)gives −/integraldisplayα 0xyyxdx−κ1/integraldisplayα 0xuxxuxdx=o/parenleftbig λ−ℓ/parenrightbig . Taking the real part in the above equation, then using by part s integration, we get (2.122)1 2/integraldisplayα 0|y|2dx+κ1 2/integraldisplayα 0|ux|2dx−α 2/parenleftbig κ1|ux(α)|2+|y(α)|2/parenrightbig =o/parenleftbig λ−ℓ/parenrightbig . Inserting the boundary conditions ( 2.96) atx=αin (2.122) gives 1 2/integraldisplayα 0|y|2dx+κ1 2/integraldisplayα 0|ux|2dx=α 2/parenleftbig κ−1 1|κ2vx(α)+δ1zx(α)|2+|z(α)|2/parenrightbig +o/parenleftbig λ−ℓ/parenrightbig . Inserting ( 2.109)-(2.110) in the above equation, we obtain the first and the second asym ptotic estimates of (2.120). The proof is thus complete. /square Proof of Theorem 2.7.From Lemma 2.8, Lemma 2.9, Lemma 2.10, Lemma 2.11and Lemma 2.13, we get /ba∇dblU/ba∇dbl2 H=κ1/integraldisplayα 0|ux|2dx+κ2/integraldisplayβ α|vx|2dx+κ3/integraldisplayL β|wx|2dx+/integraldisplayα 0|y|2dx +/integraldisplayβ α|z|2dx+/integraldisplayL β|φ|2dx+τ/integraldisplay1 0|η(L,ρ)|2dρ=o/parenleftBig λ−min(2ℓ−1,ℓ−1 2)/parenrightBig . To obtain /ba∇dblU/ba∇dblH=o(1), we need min/parenleftbigg 2ℓ−1,ℓ−1 2/parenrightbigg ≥0, so we choose ℓ=1 2as the optimal value. Hence, we obtain that /ba∇dblU/ba∇dblH=o(1)which contradicts ( 2.87). Therefore, the energy of System ( 2.9)-(2.22) satisfies estimation ( 2.85) for all initial data U0∈D(A). /square 2.2.Wave equation with local Kelvin-Voigt damping near the boun dary and boundary delay feedback. In this subsection, we study the stability of System ( 2.1), but in the case that the Kelvin-Voigt damping is near the boundary, i.e.α= 0and0<β <L (see Figure 2). For this aim, we denote the longitudinal displacement by Uand this displacement is divided into two parts U(x,t) =/braceleftBigg v(x,t),(x,t)∈(α,β)×(0,+∞), w(x,t),(x,t)∈(β,L)×(0,+∞). In this case, System ( 2.1) is equivalent to the following system (2.123) vtt−(κ2vx+δ1vxt)x= 0, (x,t)∈(0,β)×(0,+∞), wtt−κ3wxx= 0, (x,t)∈(β,L)×(0,+∞), τηt(L,ρ,t)+ηρ(L,ρ,t) = 0, (ρ,t)∈(0,1)×(0,+∞), v(0,t) = 0, t ∈(0,+∞), wx(L,t) =−δ3wt(L,t)−δ2η(L,1,t), t∈(0,+∞), v(β,t) =w(β,t), t ∈(0,+∞), κ2vx(β,t)+δ1vxt(β,t) =κ3wx(β,t), t∈(0,+∞), (v(x,0),vt(x,0)) = (v0(x),v1(x)), x ∈(α,β), (w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L), η(L,ρ,0) =f0(L,−ρτ), ρ ∈(0,1), where the initial data (v0,v1,w0,w1,f0)belongs to a suitable space. Similar to Section 2.1, we define Xm=Hm(0,β)×Hm(β,L), m= 1,2, X0=L2(0,β)×L2(β,L),X1 L={(v,w)∈X1|v(0) = 0, v(β) =w(β)}, 21WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY where the Hilbert space X0is equipped with the norm: /ba∇dbl(v,w)/ba∇dbl2 X0=/integraldisplayβ 0|v|2dx+/integraldisplayL β|w|2dx. Moreover, it is easy to check that the space X1 Lis Hilbert space over Cequipped with the norm: /ba∇dbl(v,w)/ba∇dbl2 X1 L=κ2/integraldisplayβ 0|vx|2dx+κ3/integraldisplayL β|wx|2dx. In addition, by Poincaré inequality we can easily verify tha t there exists C >0, such that /ba∇dbl(v,w)/ba∇dblX0≤C/ba∇dbl(v,w)/ba∇dblX1 L. We now define the Hilbert energy space by H1=X1 L×X0×L2(0,1) equipped with the following inner product /a\}b∇acketle{tU,˜U/a\}b∇acket∇i}htH1=κ2/integraldisplayβ 0vx˜vxdx+κ3/integraldisplayL βwx˜wxdx+/integraldisplayβ 0z˜zdx+/integraldisplayL βφ˜φdx+τ/integraldisplay1 0η(L,ρ)˜η(L,ρ)dρ, whereU= (v,w,z,φ,η (L,·))∈ H1and˜U= (˜v,˜w,˜z,˜φ,˜η(L,·))∈ H1. We use /ba∇dblU/ba∇dblH1to denote the corresponding norm. We define the linear unbounded operator A1:D(A1)⊂ H1−→ H 1by: D(A1) =/braceleftbigg U= (v,w,z,φ,η (L,·))∈X1 L×X1 L×H1(0,1)|(κ2v+δ1z,w)∈X2, κ2vx(β)+δ1zx(β) =κ3wx(β), wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0)/bracerightbigg and for all U= (v,w,z,φ,η (L,·))∈D(A1) A1U=/parenleftbig z,φ,(κ2vx+δ1zx)x,κ3wxx,−τ−1ηρ(L,·)/parenrightbig . IfU= (v,w,v t,wt,η(L,·))is a regular solution of System ( 2.123), then we transform this system into the following initial value problem (2.124)/braceleftBigg Ut=A1U, U(0) =U0, whereU0= (v0,w0,v1,w1,f0(L,−·τ))∈ H1.Note thatD(A1)is dense in H1and that for all U∈D(A1), we have (2.125) Re /a\}b∇acketle{tA1U,U/a\}b∇acket∇i}htH1≤ −δ1/integraldisplayβ 0|zx|2dx−/parenleftbigg1 2−κ3|δ2| 2p/parenrightbigg |η(L,1)|2−/parenleftbigg κ3δ3−1 2−κ3|δ2|p 2/parenrightbigg |η(L,0)|2, wherepis defined in ( 2.8). Consequently, under hypothesis (H), the system becomes d issipative. We can easily adapt the proof in Subsection 2.1.1to prove the well-posedness of System ( 2.124). Theorem 2.14. Under hypothesis (H), for all initial data U0∈ H1,the System (2.123)is exponentially stable. According to Theorem A.5(part (i)), we have to check if the following conditions hold , (2.126) iR⊆ρ(A1) and (2.127) sup λ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A1)−1/vextenddouble/vextenddouble/vextenddouble L(H1)=O(1). Proof. First, we can easily adapt the proof in Subsection 2.1.2to prove the strong stability (condition ( 2.126)) of System ( 2.123). Next, we will prove condition ( 2.127) by a contradiction argument. Indeed, suppose there exists {(λn,Un:= (vn,wn,zn,φn,ηn(L,·)))}n≥1⊂R∗ +×D(A1), such that (2.128) λn→+∞,/ba∇dblUn/ba∇dblH1= 1 22WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY and there exists sequence Gn:= (g1,n,g2,n,g3,n,g4,n,g5,n(L,·))∈ H1, such that (2.129) (iλnI−A1)Un=Gn→0inH1. We will check condition ( 2.127) by finding a contradiction with /ba∇dblUn/ba∇dblH1= 1such as/ba∇dblUn/ba∇dblH1=o(1).From now on, for simplicity, we drop the index n. By detailing Equation ( 2.129), we get the following system iλv−z=g1inH1(0,β), (2.130) iλw−φ=g2inH1(β,L), (2.131) iλz−(κ2vx+δ1zx)x=g3inL2(0,β), (2.132) iλφ−κ3wxx=g4inL2(β,L), (2.133) ηρ(L,·)+iτλη(L,·) =τg5(L,·)inL2(0,1). (2.134) Remark that, since U= (v,w,z,φ,η (L,·))∈D(A1), we have the following boundary conditions |wx(β)|=κ−1 3|κ2vx(β)+δ1zx(β)|,|z(β)|=|φ(β)|, (2.135) wx(L) =−δ3η(L,0)−δ2η(L,1), φ(L) =η(L,0). (2.136) Taking the inner product of ( 2.129) withUinH1, then using ( 2.125), hypothesis (H) and the fact that Uis uniformly bounded in H1, we obtain (2.137)/integraldisplayβ 0|zx|2=o(1),|φ(L)|2=|η(L,0)|2=o(1),|η(L,1)|2=o(1). From ( 2.130), then using the first asymptotic estimate of ( 2.137) and the fact that (g1)x→0inL2(0,β), we get (2.138)/integraldisplayβ 0|vx|2dx=o/parenleftbig λ−2/parenrightbig . From the first asymptotic estimate of ( 2.136), then using the second and the third asymptotic estimates o f (2.137), we obtain (2.139) |wx(L)|2=o(1). Similar to Lemma 2.9, withℓ= 0, from ( 2.134), then using the second and the third asymptotic estimates o f (2.137), we obtain (2.140)/integraldisplay1 0|η(L,ρ)|2dρ=o(1). Similar to Lemma 2.10, withℓ= 0, multiplying Equation ( 2.133) byxwxand integrating over (β,L),after that using the fact that iλwx=−φx−(g2)x, then using the fact that φ, wxare uniformly bounded in L2(β,L)and (g2)x, g4converge to zero in L2(β,L)gives −/integraldisplayL βxφφxdx−κ3/integraldisplayL βxwxxwxdx=o(1). Taking the real part in the above equation, then using by part s integration, Equation ( 2.139) and the second asymptotic estimate of ( 2.137), we obtain 1 2/integraldisplayL β|φ|2dx+κ3 2/integraldisplayL β|wx|2dx+β 2/parenleftbig κ3|wx(β)|2+|φ(β)|2/parenrightbig =o(1), hence, we get (2.141)/integraldisplayL β|φ|2dx=o(1),/integraldisplayL β|wx|2dx=o(1),|wx(β)|2=o(1),|φ(β)|2=o(1). Inserting the third and the fourth asymptotic estimates of ( 2.141) in (2.135), we get (2.142) |κ2vx(β)+δ1zx(β)|=o(1),|z(β)|=o(1). 23WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Similar to step 3 of Lemma 2.11, withα= 0andℓ= 0, multiplying ( 2.132) by−iλ−1zand integrating over (0,β),taking the real part, then using the fact that zis uniformly bounded in L2(0,β)andg3→0inL2(0,β), we get (2.143)/integraldisplayβ 0|z|2dx≤λ−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ 0(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+o/parenleftbig λ−1/parenrightbig . On the other hand, using by parts integration, the fact that z(0) = 0 , and Equations ( 2.137)-(2.138), (2.142), we get (2.144)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ 0(κ2vx+δ1zx)xzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle[(κ2vx+δ1zx)z]β 0−/integraldisplayβ 0(κ2vx+δ1zx)zxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤ |κ2vx(β)+δ1zx(β)||z(β)|+/integraldisplayβ 0|κ2vx+δ1zx||zx|dx=o(1). Inserting ( 2.144) in (2.143), we get (2.145)/integraldisplayβ 0|z|2dx=o/parenleftbig λ−1/parenrightbig . Finally, from ( 2.138), (2.140), (2.141) and ( 2.145), we get /ba∇dblU/ba∇dblH1=o(1), which contradicts ( 2.128). Therefore, ( 2.127) holds and the result follows from Theorem A.5(part (i)). /square 3.Wave equation with local internal Kelvin-Voigt damping and local internal delay feedback In this section, we study the stability of System ( 1.2). We assume that there exists αandβsuch that 0< α < β < L , in this case, the Kelvin-Voigt damping and the time delay fe edback are locally internal (see Figure 3). For this aim, we denote the longitudinal displacement by Uand this displacement is divided into three parts U(x,t) = u(x,t),(x,t)∈(0,α)×(0,+∞), v(x,t),(x,t)∈(α,β)×(0,+∞), w(x,t),(x,t)∈(β,L)×(0,+∞). Furthermore, let us introduce the auxiliary unknown η(x,ρ,t) =vt(x,t−ρτ), x∈(α,β), ρ∈(0,1), t>0. In this case, System ( 1.2) is equivalent to the following system utt−κ1uxx= 0,(x,t)∈(0,α)×(0,+∞), (3.1) vtt−(κ2vx+δ1vxt(x,t)+δ2ηx(x,1,t))x= 0,(x,t)∈(α,β)×(0,+∞), (3.2) wtt−κ3wxx= 0,(x,t)∈(β,L)×(0,+∞), (3.3) τηt(x,ρ,t)+ηρ(x,ρ,t) = 0,(x,ρ,t)∈(α,β)×(0,1)×(0,+∞), (3.4) with the Dirichlet boundary conditions (3.5) u(0,t) =w(L,t) = 0, t∈(0,+∞), with the following transmission conditions (3.6) u(α,t) =v(α,t), v(β,t) =w(β,t), t ∈(0,+∞), κ1ux(α,t) =κ2vx(α,t)+δ1vxt(α,t)+δ2ηx(α,1,t), t∈(0,+∞), κ3wx(β,t) =κ2vx(β,t)+δ1vxt(β,t)+δ2ηx(β,1,t), t∈(0,+∞), 24WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY and with the following initial conditions (3.7) (u(x,0),ut(x,0)) = (u0(x),u1(x)), x∈(0,α), (v(x,0),vt(x,0)) = (v0(x),v1(x)), x ∈(α,β), (w(x,0),wt(x,0)) = (w0(x),w1(x)), x∈(β,L), η(x,ρ,0) =f0(x,−ρτ), (x,ρ)∈(α,β)×(0,1), where the initial data (u0,u1,v0,v1,w0,w1,f0)belongs to a suitable space. To a strong solution of System (3.1)-(3.7), we associate the energy defined by E(t) =1 2/integraldisplayα 0/parenleftbig |ut(x,t)|2+κ1|ux(x,t)|2/parenrightbig dx+1 2/integraldisplayβ α/parenleftbig |vt(x,t)|2+κ2|vx(x,t)|2/parenrightbig dx +1 2/integraldisplayL β/parenleftbig |wt(x,t)|2+κ3|wx(x,t)|2/parenrightbig dx+τ|δ2| 2/integraldisplay1 0/integraldisplayβ α|ηx(x,ρ,t)|2dρdx. Multiplying ( 3.1), (3.2), (3.3) and ( 3.4)xbyut,yt,wtand|δ2|ηx, integrating over (0,α),(α,β),(β,L)and (α,β)×(0,1)respectively, taking the sum, then using by parts integrati on and the boundary conditions in (3.5)-(3.6), we get E′(t) =/parenleftbigg −δ1+|δ2| 2/parenrightbigg/integraldisplayβ α|vxt(x,t)|2dx−|δ2| 2/integraldisplayβ α|ηx(x,1,t)|2dx−δ2/integraldisplayβ αvxt(x,t)ηx(x,1,t)dx. Using Young’s inequality for the third term in the right, we g et E′(t)≤(−δ1+|δ2|)/integraldisplayβ α|vxt(x,t)|2dx. In the sequel, the assumption on δ1andδ2will ensure that (H1) δ1>0, δ2∈R∗,|δ2|<δ1. In this case, the energies of the strong solutions satisfy E′(t)≤0.Hence, the System ( 3.1)-(3.7) is dissipative in the sense that its energy is non increasing with respect to the timet. 3.1.Well-posedness of the problem. We start this part by formulating System ( 3.1)-(3.7) as an abstract Cauchy problem. For this aim, let us define L2 ∗=L2(0,α)×L2 ∗(α,β)×L2(β,L), H1 ∗={(u,v,w)∈H1(0,α)×H1 ∗(α,β)×H1(β,L)|u(0) = 0, u(α) =v(α), v(β) =w(β), w(L) = 0}, H2=H2(0,α)×H2(α,β)×H2(β,L). Here we consider L2 ∗(α,β) =/braceleftBigg z∈L2(α,β)|/integraldisplayβ αzdx= 0/bracerightBigg andH1 ∗(α,β) =H1(α,β)∩L2 ∗(α,β). The spaces L2 ∗andH1 ∗are obviously a Hilbert spaces equipped respectively with t he norms /ba∇dbl(u,v,w)/ba∇dbl2 L2∗=/integraldisplayα 0|u|2dx+/integraldisplayβ α|v|2dx+/integraldisplayL β|w|2dx and /ba∇dbl(u,v,w)/ba∇dbl2 H1∗=κ1/integraldisplayα 0|ux|2dx+κ2/integraldisplayβ α|vx|2dx+κ3/integraldisplayL β|wx|2dx. In addition by Poincaré inequality we can easily verify that there exists C >0, such that /ba∇dbl(u,v,w)/ba∇dblL2∗≤C/ba∇dbl(u,v,w)/ba∇dblH1∗. Let us define the energy Hilbert space H2by H2=H1 ∗×L2 ∗×L2/parenleftbig (0,1),H1 ∗(α,β)/parenrightbig 25WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY equipped with the following inner product /a\}b∇acketle{tU,˜U/a\}b∇acket∇i}htH2=κ1/integraldisplayα 0ux˜uxdx+κ2/integraldisplayβ αvx˜vxdx+κ3/integraldisplayL βwx˜wxdx +/integraldisplayα 0y˜ydx+/integraldisplayβ αz˜zdx+/integraldisplayL βφ˜φdx+τ|δ2|/integraldisplay1 0/integraldisplayβ αηx(x,ρ)˜ηx(x,ρ)dxdρ, whereU= (u,v,w,y,z,φ,η (·,·))∈ H2and˜U= (˜u,˜v,˜w,˜y,˜z,˜φ,˜η(·,·))∈ H2. We use /ba∇dblU/ba∇dblH2to denote the corresponding norm. We define the linear unbounded operator A2:D(A2)⊂ H2−→ H 2by: D(A2) =/braceleftbigg (u,v,w,y,z,φ,η (·,·))∈ H2|(y,z,φ)H1 ∗ (u,κ2v+δ1z+δ2η(·,1),w)∈H2, κ2vx(α)+δ1zx(α)+δ2ηx(α,1) =κ1ux(α), κ2vx(β)+δ1zx(β)+δ2ηx(β,1) =κ3wx(β), η, ηρ∈L2/parenleftbig (0,1),H1 ∗(α,β)/parenrightbig , z(·) =η(·,0)/bracerightbigg and for all U= (u,v,w,y,z,φ,η (·,·))∈D(A2) A2U=/parenleftbig y,z,φ,κ 1uxx,(κ2vx+δ1zx+δ2ηx(·,1))x,κ3wxx,−τ−1ηρ(·,·)/parenrightbig . IfU= (u,v,w,u t,vt,wt,η(·,·))is a regular solution of System ( 3.1)-(3.7), then we transform this system into the following initial value problem (3.8)/braceleftBigg Ut=A2U, U(0) =U0, whereU0= (u0,v0,w0,u1,v1,w1,f0(·,−·τ))∈ H2.We now use semigroup approach to establish well-posedness result for the System ( 3.1)-(3.7). We prove the following proposition. Proposition 3.1. Under hypothesis (H1), the unbounded linear operator A2is m-dissipative in the energy spaceH2. Proof. For allU= (u,v,w,y,z,φ,η (·,·))∈D(A2),we have Re/a\}b∇acketle{tA2U,U/a\}b∇acket∇i}htH2=κ1Re/integraldisplayα 0(yxux+uxxy)dx+Re/integraldisplayβ α(κ2zxvx+(κ2vx+δ1zx+δ2ηx(·,1))xz)dx +κ3Re/integraldisplayL β/parenleftbig φxwx+wxxφ/parenrightbig dx−|δ2|Re/integraldisplayβ α/integraldisplay1 0ηxρ(x,ρ)ηx(x,ρ)dxdρ. Using by parts integration in the above equation, we get (3.9)Re/a\}b∇acketle{tA2U,U2/a\}b∇acket∇i}htH2=−δ1/integraldisplayβ α|zx|2dx−δ2Re/integraldisplayβ αηx(·,1)zxdx+|δ2| 2/integraldisplayβ α|ηx(x,0)|2dx −|δ2| 2/integraldisplayβ α|ηx(x,1)|2dx−κ1Re(ux(0)y(0))+κ3Re/parenleftbig wx(L)φ(L)/parenrightbig +Re(κ1ux(α)y(α)−κ2vx(α)z(α)−δ1zx(α)z(α)−δ2ηx(α,1)z(α)) +Re/parenleftbig κ2vx(β)z(β)+δ1zx(β)z(β)+δ2ηx(β,1)z(β)−κ3wx(β)φ(β)/parenrightbig . SinceU∈D(A2), we have /braceleftBigg y(0) =φ(0) = 0, y(α) =z(α), z(β) =φ(β), z(x) =η(x,0), κ1ux(α)−κ2vx(α)−δ1zx(α)−δ2ηx(α,1) = 0, κ2vx(β)+δ1zx(β)+δ2ηx(β,1)−κ3wx(β) = 0. Substituting the above boundary conditions in ( 3.9), then using Young’s inequality, we get (3.10)Re/a\}b∇acketle{tA2U,U/a\}b∇acket∇i}htH2=/parenleftbigg −δ1+|δ2| 2/parenrightbigg/integraldisplayβ α|zx|2dx−|δ2| 2/integraldisplayβ α|ηx(x,1)|2dx−δ2Re/integraldisplayβ αηx(·,1)zxdx ≤(−δ1+|δ2|)/integraldisplayβ α|zx|2dx, 26WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY hence under hypothesis (H1), we get Re/a\}b∇acketle{tA2U,U/a\}b∇acket∇i}htH2≤0, which implies that A2is dissipative. To prove that A2is m-dissipative, it is enough to prove that 0∈ρ(A2) sinceA2is a closed operator and D(A2) =H2. LetF= (f1,f2,f3,f4,f5,f6,f7(·,·))∈ H2.We should prove that there exists a unique solution U= (u,v,w,y,z,φ,η (·,·))∈D(A2)of the equation −A2U=F. Equivalently, we consider the following system −y=f1, (3.11) −z=f2, (3.12) −φ=f3, (3.13) −κ1uxx=f4, (3.14) −(κ2vx+δ1zx+δ2ηx(·,1))x=f5, (3.15) −κ3wxx=f6, (3.16) ηρ(x,ρ) =τf7(x,ρ). (3.17) In addition, we consider the following boundary conditions u(0) = 0, u(α) =v(α), v(β) =w(β), w(L) = 0, (3.18) κ2vx(α)+δ1zx(α)+δ2ηx(α,1) =κ1ux(α), κ2vx(β)+δ1zx(β)+δ2ηx(β,1) =κ3wx(β), (3.19) η(·,0) =z(·). (3.20) From ( 3.11)-(3.13) and the fact that F∈ H, we obtain (y,z,φ)∈H1 ∗. Next, from ( 3.12), (3.20) and the fact thatf2∈H1 ∗(α,β), we get η(·,0) =z(·) =−f2(·)∈H1 ∗(α,β). From the above equation and Equation ( 3.17), we can determine (3.21) η(x,ρ) =τ/integraldisplayρ 0f7(x,ξ)dξ−f2(x). Sincef2∈H1 ∗(α,β)andf7∈L2/parenleftbig (0,1),H1 ∗(α,β)/parenrightbig , then it is clear that η, ηρ∈L2((0,1),H1 ∗(0,1)). Now, let (ϕ,ψ,θ)∈H1 ∗. Multiplying Equations ( 3.14), (3.15), (3.16) byϕ,ψ,θ, integrating over (0,α),(α,β)and(β,L) respectively, taking the sum, then using by parts integrati on, we get (3.22)κ1/integraldisplayα 0uxϕxdx+/integraldisplayβ α(κ2vx+δ1zx+δ2ηx(·,1))ψxdx+κ3/integraldisplayL βwxθxdx+κ1ux(0)ϕ(0)−κ3wx(L)θ(L) −κ1ux(α)ϕ(α)+(κ2vx(α)+δ1zx(α)+δ2ηx(α,1))ψ(α)−(κ2vx(β)+δ1zx(β)+δ2ηx(β,1))ψ(β) +κ3wx(β)θ(β) =/integraldisplayα 0f4ϕdx+/integraldisplayβ αf5ψdx+/integraldisplayL βf6θdx. From the fact that (ϕ,ψ,θ)∈H1 ∗,we have ϕ(0) = 0, ϕ(α) =ψ(α), θ(β) =ψ(β), θ(L) = 0. Inserting the above equation in ( 3.22), then using ( 3.12), (3.19) and ( 3.21), we get (3.23)κ1/integraldisplayα 0uxϕxdx+κ2/integraldisplayβ αvxψxdx+κ3/integraldisplayL βwxθxdx =/integraldisplayα 0f4ϕdx+/integraldisplayβ αf5ψdx+/integraldisplayL βf6θdx+/integraldisplayβ α/parenleftbigg (δ1+δ2)(f2)x−δ2τ/integraldisplay1 0(f7(·,ξ))xdξ/parenrightbigg ψxdx. We can easily verify that the left hand side of ( 3.23) is a bilinear continuous coercive form on H1 ∗×H1 ∗, and the right hand side of ( 3.23) is a linear continuous form on H1 ∗. Then, using Lax-Milgram theorem, we deduce that there exists (u,v,w)∈H1 ∗unique solution of the variational Problem ( 3.23). Using standard arguments, we can show that (u,κ2v+δ1z+δ2η(·,1),w)∈H2. Thus, from ( 3.11)-(3.13), (3.21) and applying the classical elliptic regularity we deduce that U= (u,v,w,y,z,φ,η (·,·))∈D(A2). The proof is thus complete. /square 27WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Thanks to Lumer-Philips theorem (see [ 36]), we deduce that A2generates a C0−semigroup of contractions etA2 inH2and therefore Problem ( 3.1)-(3.7) is well-posed. 3.2.Polynomial Stability. The main result in this subsection is the following theorem. Theorem 3.2. Under hypothesis (H1), for all initial data U0∈D(A2),there exists a constant C >0indepen- dent ofU0such that the energy of System (3.1)-(3.7)satisfies the following estimation E(t)≤C t4/ba∇dblU0/ba∇dbl2 D(A2),∀t>0. According to Theorem A.5(part (ii)), we have to check if the following conditions hol d, (3.24) iR⊆ρ(A2) and (3.25) sup λ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−A2)−1/vextenddouble/vextenddouble/vextenddouble L(H2)=O/parenleftBig |λ|1 2/parenrightBig . The next proposition is a technical result to be used in the pr oof of Theorem 3.2given below. Proposition 3.3. Under hypothesis (H1), let(λ,U:= (u,v,w,y,z,φ,η (·,·)))∈R∗×D(A2),such that (3.26) (iλI−A2)U=F:= (f1,f2,f3,f4,f5,f6,g(·,·))∈ H2, i.e. iλu−y=f1 inH1(0,α), (3.27) iλv−z=f2 inH1 ∗(α,β), (3.28) iλw−φ=f3 inH1(β,L), (3.29) iλy−κ1uxx=f4 inL2(0,α), (3.30) iλz−(κ2vx+δ1zx+δ2ηx(·,1))x=f5 inL2 ∗(α,β), (3.31) iλφ−κ3wxx=f6 inL2(β,L), (3.32) ηρ(·,·)+iτλη(·,·) =τg(·,·)inL2/parenleftbig (0,1),H1 ∗(α,β)/parenrightbig . (3.33) Then, we have the following inequality (3.34) /ba∇dblU/ba∇dbl2 H2≤K1λ−4(|λ|+1)6/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . In addition, if |λ| ≥M >0,then we have (3.35) /ba∇dblU/ba∇dbl2 H2≤K2/parenleftbigg√ M+1√ M/parenrightbigg2 |λ|1 2/parenleftBig 1+|λ|−1 2/parenrightBig8/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . Here and below we denote by Kja positive constant number independent of λ. Before stating the proof of Proposition 3.3, leth∈C1([α,β])such that h(α) =−h(β) = 1,max x∈[α,β]|h(x)|=Chandmax x∈[α,β]|h′(x)|=Ch′, whereChandCh′are strictly positive constant numbers independent of λ. An example about h, we can take h(x) =−2(x−α) β−α+1to get h(α) =−h(β) = 1, h∈C1([α,β]), Ch= 1, Ch′=2 β−α. For the proof of Proposition 3.3, we need the following lemmas. 28WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Lemma 3.4. Under hypothesis (H1), the solution (u,v,w,y,z,φ,η (·,·))∈D(A2)of Equation (3.26)satisfies the following estimations /integraldisplayβ α|zx|2dx≤K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2, (3.36) /integraldisplayβ α|vx|2dx≤K4λ−2/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , (3.37) /integraldisplayβ α/integraldisplay1 0|ηx(x,ρ)|2dxdρ≤K5/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , (3.38) /integraldisplayβ α|κ2vx+δ1zx+δ2ηx(·,1)|2dx≤K6/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , (3.39) where /braceleftBigg K3= (δ1−|δ2|)−1, K4= 2max/parenleftbig K3,κ−1 2/parenrightbig , K5= 2max/parenleftbig K3,τ|δ2|−1/parenrightbig , K6= 3max/parenleftbig κ2 2K4,δ2 1K3+δ2 2K5/parenrightbig . Proof. First, taking the inner product of ( 3.26) withUinH2, then using hypothesis (H1), arguing in the same way as ( 3.10), we obtain /integraldisplayβ α|zx|2dx≤ −1 δ1−|δ2|Re/a\}b∇acketle{tA2U,U/a\}b∇acket∇i}htH2=1 δ1−|δ2|Re/a\}b∇acketle{tF,U/a\}b∇acket∇i}htH2≤1 δ1−|δ2|/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2, hence we get ( 3.36). Next, from ( 3.28), (3.36) and the fact that κ2/integraldisplayβ α|(f2)x|2dx≤ /ba∇dblF/ba∇dbl2 H2, we obtain /integraldisplayβ α|vx|2dx≤2λ−2/integraldisplayβ α|zx|2dx+2λ−2/integraldisplayβ α|(f2)x|2dx ≤2λ−2/parenleftBig K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+κ−1 2/ba∇dblF/ba∇dbl2 H2/parenrightBig ≤2λ−2max/parenleftbig K3,κ−1 2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . therefore we get ( 3.37). Now, from ( 3.33) and using the fact that U∈D(A2) (i.e.η(·,0) =z(·)), we obtain (3.40) η(x,ρ) =z(x)e−iτλρ+τ/integraldisplayρ 0eiτλ(ξ−ρ)g(x,ξ)dξ(x,ρ)∈(α,β)×(0,1), consequently, we obtain /integraldisplayβ α/integraldisplay1 0|ηx(x,ρ)|2dxdρ≤2/integraldisplayβ α|zx|2dx+2τ2/integraldisplayβ α/integraldisplay1 0|gx(x,ξ)|2dξdx. Inserting ( 3.36) in the above equation, then using the fact that τ|δ2|/integraldisplayβ α/integraldisplay1 0|gx(x,ξ)|2dξdx≤ /ba∇dblF/ba∇dbl2 H2, we obtain /integraldisplayβ α/integraldisplay1 0|ηx(x,ρ)|2dxdρ≤2K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+2τ|δ2|−1/ba∇dblF/ba∇dbl2 H2 ≤2max/parenleftbig K3,τ|δ2|−1/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get ( 3.38). On the other hand, from ( 3.40), we get ηx(x,1) =zx(x)e−iτλ+τ/integraldisplay1 0eiτλ(ξ−1)gx(x,ξ)dξ x∈(α,β). From the above equation and ( 3.36), we obtain /integraldisplayβ α|ηx(x,1)|2dx≤2/integraldisplayβ α|zx|2dx+2τ2/integraldisplayβ α/integraldisplay1 0|gx(x,ξ)|2dξdx ≤2K3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+2τ|δ2|−1/ba∇dblF/ba∇dbl2 H2 ≤2max/parenleftbig K3,τ|δ2|−1/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . 29WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Finally, from ( 3.36), (3.37) and the above inequality, we get /integraldisplayβ α|κ2vx+δ1zx+δ2ηx(·,1)|2dx≤3κ2 2/integraldisplayβ α|vx|2dx+3δ2 1/integraldisplayβ α|zx|2dx+3δ2 2/integraldisplayβ α|ηx(·,1)|2dx ≤3/parenleftbig κ2 2K4λ−2+δ2 1K3+δ2 2K5/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig ≤3max/parenleftbig κ2 2K4,δ2 1K3+δ2 2K5/parenrightbig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get ( 3.39). /square Lemma 3.5. Under hypothesis (H1), for alls1, s2∈Randr1, r2∈R∗ +, the solution (u,v,w,y,z,φ,η (·,·))∈ D(A2)of Equation (3.26)satisfies the following estimations (3.41) |z(β)|2+|z(α)|2≤/parenleftBigg Ch′+|λ|1 2−s1 r1/parenrightBigg/integraldisplayβ α|z|2dx+K7r1C2 h|λ|−1 2+s1/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig and (3.42)|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2 ≤|λ|3 2−s2 r2/integraldisplayβ α|z|2dx+K8/parenleftBig Ch′+Ch+r2C2 h|λ|1 2+s2/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , where K7= 2/parenleftbig K4+κ−1 2/parenrightbig , K8=K6+1. Proof. First, from Equation ( 3.28), we have −zx= (f2)x−iλvx. Multiplying the above equation by 2hz, integrating over (α,β)and taking the real parts, then using by parts integration and the fact that h(α) =−h(β) = 1, we get (3.43)|z(β)|2+|z(α)|2=−/integraldisplayβ αh′|z|2dx−2Re/braceleftBigg/integraldisplayβ αh(iλvx−(f2)x)zdx/bracerightBigg ≤Ch′/integraldisplayβ α|z|2dx+2Ch|λ|/integraldisplayβ α|vx||z|dx+2Ch/integraldisplayβ α|(f2)x||z|dx. On the other hand, for all s1∈Randr1∈R∗ +, we have 2Ch|λ||vx||z| ≤|λ|1 2−s1|z|2 2r1+2r1C2 h|λ|3 2+s1|vx|2and2Ch|(f2)x||z| ≤|λ|1 2−s1|z|2 2r1+2r1C2 h|λ|−1 2+s1|(f2)x|2. Inserting the above equation in ( 3.43), then using ( 3.37) and the fact that/integraldisplayβ α|(f2)x|2dx≤κ−1 2/ba∇dblF/ba∇dbl2 H2,we get |z(β)|2+|z(α)|2 ≤/parenleftBigg Ch′+|λ|1 2−s1 r1/parenrightBigg/integraldisplayβ α|z|2dx+2r1C2 h|λ|s1/parenleftBigg |λ|3 2/integraldisplayβ α|vx|2dx+|λ|−1 2/integraldisplayβ α|(f2)x|2dx/parenrightBigg ≤/parenleftBigg Ch′+|λ|1 2−s1 r1/parenrightBigg/integraldisplayβ α|z|2dx+2r1C2 h|λ|s1−1 2/parenleftBig K4/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig +κ−1 2/ba∇dblF/ba∇dbl2 H2/parenrightBig ≤/parenleftBigg Ch′+|λ|1 2−s1 r1/parenrightBigg/integraldisplayβ α|z|2dx+2r1C2 h|λ|s1−1 2/parenleftbig K4+κ−1 2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get ( 3.41). Next, multiplying Equation ( 3.31) by2h(κ2vx+δ1zx+δ2ηx(·,1)), integrating over (α,β) and taking the real parts, then using by parts integration, w e get |κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2 =−/integraldisplayβ αh′|κ2vx+δ1zx+δ2ηx(·,1)|2dx+2Re/integraldisplayβ αh(f5−iλz)(κ2vx+δ1zx+δ2ηx(·,1))dx, 30WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY consequently, we have (3.44)|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2 ≤Ch′/integraldisplayβ α|κ2vx+δ1zx+δ2ηx(·,1)|2dx+2Ch/integraldisplayβ α|f5||κ2vx+δ1zx+δ2ηx(·,1)|dx +2Ch|λ|/integraldisplayβ α|z||κ2vx+δ1zx+δ2ηx(·,1)|dx. On the other hand, for all s2∈Randr2∈R∗ +, we have 2Ch|f5|||κ2vx+δ1zx+δ2ηx(·,1)| ≤Ch|f5|2+Ch|κ2vx+δ1zx+δ2ηx(·,1)|2, 2Ch|λ||z|||κ2vx+δ1zx+δ2ηx(·,1)| ≤|λ|3 2−s2 r2|z|2+r2C2 h|λ|1 2+s2||κ2vx+δ1zx+δ2ηx(·,1)|2. Inserting the above equation in ( 3.44), we get |κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2≤|λ|3 2−s2 r2/integraldisplayβ α|z|2dx +/parenleftBig Ch′+Ch+r2C2 h|λ|1 2+s2/parenrightBig/bracketleftBigg/integraldisplayβ α||κ2vx+δ1zx+δ2ηx(·,1)|2dx+/integraldisplayβ α|f5|2dx/bracketrightBigg . Inserting ( 3.39) in the above equation, then using the fact that /integraldisplayβ α|f5|2dx≤ /ba∇dblF/ba∇dbl2 H2≤/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , we get |κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2 ≤|λ|3 2−s2 r2/integraldisplayβ α|z|2dx+(K6+1)/parenleftBig Ch′+Ch+r2C2 h|λ|1 2+s2/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get ( 3.42). /square Lemma 3.6. Under hypothesis (H1), for alls1, s2∈R, the solution (u,v,w,y,z,φ,η (·,·))∈D(A2)of Equation (3.26)satisfies the following estimation (3.45)/integraldisplayα 0|y|2dx+κ1/integraldisplayα 0|ux|2dx+/integraldisplayL β|φ|2dx+κ3/integraldisplayL β|wx|2dx ≤K9/parenleftBig 1+|λ|1 2−s1+|λ|3 2−s2/parenrightBig/integraldisplayβ α|z|2dx +K10/parenleftBig 1+|λ|−1 2+s1+|λ|1 2+s2/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , where K9= max/bracketleftbig max(α,L−β),max/parenleftbig κ−1 1α,κ−1 3(L−β)/parenrightbig/bracketrightbig max(Ch′,1) and K10= 2max/bracketleftBig K7C2 hmax(α,L−β), K8max/parenleftbig κ−1 1α,κ−1 3(L−β)/parenrightbig max/parenleftbig Ch′+Ch,C2 h/parenrightbig ,4/parenleftBig ακ−1 2 1+(L−β)κ−1 2 3/parenrightBig/bracketrightBig . Proof. First, multiplying Equation ( 3.30) by2xux, integrating over (0,α)and taking the real parts, then using by parts integration, we get (3.46) 2Re/braceleftbigg iλ/integraldisplayα 0xyuxdx/bracerightbigg +κ1/integraldisplayα 0|ux|2dx=κ1α|ux(α)|2+2Re/braceleftbigg/integraldisplayα 0xf4uxdx/bracerightbigg . From ( 3.27), we deduce that iλux=−yx−(f1)x. 31WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Inserting the above result in ( 3.46), then using by parts integration, we get /integraldisplayα 0|y|2dx+κ1/integraldisplayα 0|ux|2dx=κ1α|ux(α)|2+α|y(α)|2+2Re/braceleftbigg/integraldisplayα 0xf4uxdx/bracerightbigg +2Re/braceleftbigg/integraldisplayα 0xy(f1)xdx/bracerightbigg , consequently, we get (3.47)/integraldisplayα 0|y|2dx+κ1/integraldisplayα 0|ux|2dx≤κ1α|ux(α)|2+α|y(α)|2+2α/parenleftbigg/integraldisplayα 0|ux||f4|dx+/integraldisplayα 0|y||(f1)x|dx/parenrightbigg . Using Cauchy Schwarz inequality, we get (3.48)/integraldisplayα 0|ux||f4|dx+/integraldisplayα 0|y||(f1)x|dx≤2κ−1 2 1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2. On the other hand, since U∈D(A2), we have (3.49)/braceleftBigg κ1|ux(α)|=|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|,|y(α)|=|z(α)|, κ3|wx(β)|=|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|,|φ(β)|=|z(β)|. Substituting ( 3.48) and the boundary conditions ( 3.49) atx=αin (3.47), we obtain (3.50)/integraldisplayα 0|y|2dx+κ1/integraldisplayα 0|ux|2dx≤ακ−1 1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2+α|z(α)|2+4ακ−1 2 1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2. Next, by the same way, we multiply equation ( 3.32) by2(x−L)wxand integrate over (β,L),then we use ( 3.29). Arguing in the same way as ( 3.47), we get /integraldisplayL β|φ|2dx+κ3/integraldisplayL β|wx|2dx≤κ3(L−β)|wx(β)|2+(L−β)|φ(β)|2+4(L−β)κ−1 2 3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2. Substituting the boundary conditions ( 3.49) atx=βin the above equation, we obtain /integraldisplayL β|φ|2dx+κ3/integraldisplayL β|wx|2dx≤(L−β)/bracketleftBig κ−1 3|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|z(β)|2+4κ−1 2 3/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2/bracketrightBig . Now, adding the above equation and ( 3.50), we get /integraldisplayα 0|y|2dx+κ1/integraldisplayα 0|ux|2dx+/integraldisplayL β|φ|2dx+κ3/integraldisplayL β|wx|2dx≤max(α,L−β)/parenleftbig |z(α)|2+|z(β)|2/parenrightbig +max/parenleftbig κ−1 1α,κ−1 3(L−β)/parenrightbig/parenleftbig |κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2+|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2/parenrightbig +4/parenleftBig ακ−1 2 1+(L−β)κ−1 2 3/parenrightBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2. Inserting ( 3.41) and ( 3.42) withr1=r2= 1in the above estimation, we get /integraldisplayα 0|y|2dx+κ1/integraldisplayα 0|ux|2dx+/integraldisplayL β|φ|2dx+κ3/integraldisplayL β|wx|2dx ≤max/bracketleftbig max(α,L−β),max/parenleftbig κ−1 1α,κ−1 3(L−β)/parenrightbig/bracketrightbig/parenleftBig Ch′+|λ|1 2−s1+|λ|3 2−s2/parenrightBig/integraldisplayβ α|z|2dx +max(α,L−β)K7C2 h|λ|−1 2+s1/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig +max/parenleftbig κ−1 1α,κ−1 3(L−β)/parenrightbig K8/parenleftBig Ch′+Ch+C2 h|λ|1 2+s2/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig +4/parenleftBig ακ−1 2 1+(L−β)κ−1 2 3/parenrightBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2. In the above equation, using the fact that /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2≤/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2≤/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , 32WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY we get /integraldisplayα 0|y|2dx+κ1/integraldisplayα 0|ux|2dx+/integraldisplayL β|φ|2dx+κ3/integraldisplayL β|wx|2dx ≤max/bracketleftbig max(α,L−β),max/parenleftbig κ−1 1α,κ−1 3(L−β)/parenrightbig/bracketrightbig max(Ch′,1)/parenleftBig 1+|λ|1 2−s1+|λ|3 2−s2/parenrightBig/integraldisplayβ α|z|2dx +K7C2 hmax(α,L−β)|λ|−1 2+s1/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig +K8max/parenleftbig κ−1 1α,κ−1 3(L−β)/parenrightbig max/parenleftbig Ch′+Ch,C2 h/parenrightbig/parenleftBig 1+|λ|1 2+s2/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig +4/parenleftBig ακ−1 2 1+(L−β)κ−1 2 3/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence, we get ( 3.45). /square Lemma 3.7. Under hypothesis (H1), for alls1, s2, s3∈Randr1, r2,r3∈R∗ +, the solution (u,v,w,y,z,φ,η (·,·))∈ D(A2)of Equation (3.26)satisfies the following estimations (3.51)/ba∇dblU/ba∇dbl2 H2≤K11/parenleftBig 1+|λ|1 2−s1+|λ|3 2−s2/parenrightBig/integraldisplayβ α|z|2dx +K12/parenleftBig 1+|λ|1 2+s2+|λ|−1 2+s1/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig and (3.52) R1,λ/integraldisplayβ α|z|2dx≤K13R2,λ/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , such that R1,λ= 1−1 2/parenleftbigg|λ|−s3−s2 r2r3+r3|λ|−s1+s3 r1+r3Ch′|λ|s3−1 2/parenrightbigg , R2,λ=C2 h|λ|−1(r1r3|λ|s1+s3+r2r−1 3|λ|s2−s3)+r−1 3|λ|−s3−3 2(Ch′+Ch)+|λ|−1, where K11=K9+1, K12=K10+max(κ2K4,τ|δ2|K5), K13=max(K3+K6+2,max(K7,K8)) 2. Proof. First, from ( 3.37), (3.38) and ( 3.45), we get /ba∇dblU/ba∇dbl2 H2≤/parenleftBig K9+1+K9/parenleftBig |λ|1 2−s1+|λ|3 2−s2/parenrightBig/parenrightBig/integraldisplayβ α|z|2dx +K10/parenleftBig 1+|λ|1 2+s2+|λ|−1 2+s1/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig +max(κ2K4,τ|δ2|K5)/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get ( 3.51). Next, multiplying ( 3.31) by−iλ−1zand integrating over (α,β),then taking the real part, then using by parts integration, we get /integraldisplayβ α|z|2dx=−Re/braceleftBigg iλ−1/integraldisplayβ αf5zdx/bracerightBigg +Re/braceleftBigg iλ−1/integraldisplayβ α(κ2v+δ1z+δ2η(·,1))xzxdx/bracerightBigg −Re/braceleftbig iλ−1(κ2vx(β)+δ1zx(β)+δ2ηx(β,1))z(β)/bracerightbig +Re/braceleftbig iλ−1(κ2vx(α)+δ1zx(α)+δ2ηx(α,1))z(α)/bracerightbig , consequently, (3.53)/integraldisplayβ α|z|2dx≤ |λ|−1/integraldisplayβ α|f5||z|dx+|λ|−1/integraldisplayβ α|κ2vx+δ1zx+δ2ηx(·,1)||zx|dx +|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|. Using Cauchy Schwarz inequality, we have (3.54) |λ|−1/integraldisplayβ α|f5||z|dx≤ |λ|−1/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2≤ |λ|−1/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . 33WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY From ( 3.36) and ( 3.39), we get /integraldisplayβ α|κ2vx+δ1zx+δ2ηx(·,1)||zx|dx ≤1 2/integraldisplayβ α|zx|2dx+1 2/integraldisplayβ α|κ2vx+δ1zx+δ2ηx(·,1)|2dx ≤K3 2/ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+K6 2/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig ≤K3+K6 2/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . Inserting ( 3.54) and the above estimation in ( 3.53), we get (3.55)/integraldisplayβ α|z|2dx≤K3+K6+2 2|λ|−1/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig +|λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)|. Now, for all s3∈R,r3∈R∗ +and forζ=αorζ=β, we get |λ|−1|κ2vx(ζ)+δ1zx(ζ)+δ2ηx(ζ,1)||z(ζ)| ≤r3|λ|s3−1 2 2|z(ζ)|2+|λ|−s3−3 2 2r3|κ2vx(ζ)+δ1zx(ζ)+δ2ηx(ζ,1)|2. From the above inequality, we get |λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)| ≤|λ|−s3−3 2 2r3/parenleftBig |κ2vx(β)+δ1zx(β)+δ2ηx(β,1)|2+|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)|2/parenrightBig +r3|λ|s3−1 2 2/parenleftbig |z(α)|2+|z(β)|2/parenrightbig . Inserting ( 3.41) and ( 3.42) in the above estimation, we obtain |λ|−1|κ2vx(β)+δ1zx(β)+δ2ηx(β,1)||z(β)|+|λ|−1|κ2vx(α)+δ1zx(α)+δ2ηx(α,1)||z(α)| ≤1 2/parenleftbigg|λ|−s3−s2 r2r3+r3|λ|−s1+s3 r1+r3Ch′|λ|s3−1 2/parenrightbigg/integraldisplayβ α|z|2dx +max(K7,K8) 2R3,λ/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , where R3,λ=C2 h|λ|−1(r1r3|λ|s1+s3+r2r−1 3|λ|s2−s3)+r−1 3|λ|−s3−3 2(Ch′+Ch). Finally, inserting the above equation in ( 3.55), we get /bracketleftbigg 1−1 2/parenleftbigg|λ|−s3−s2 r2r3+r3|λ|−s1+s3 r1+r3Ch′|λ|s3−1 2/parenrightbigg/bracketrightbigg/integraldisplayβ α|z|2dx ≤max(K3+K6+2,max(K7,K8)) 2/parenleftbig R3,λ+|λ|−1/parenrightbig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get ( 3.52). /square Proof of Proposition 3.3.We now divide the proof in two steps: Step 1. In this step, we prove the asymptotic behavior estimate of ( 3.34). Takings3=s1=−s2=1 2, r1=1 Ch′, r2= 9Ch′andr3=1 3Ch′in Lemma 3.7, we get 1 2/integraldisplayβ α|z|2dx≤K13λ−4/parenleftbiggC2 h 3C2 h′λ2+|λ|+3Ch′/parenleftbig 9C2 hCh′+Ch+Ch′/parenrightbig/parenrightbigg/parenleftbig λ2+1/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , /ba∇dblU/ba∇dbl2 H2≤2K11/parenleftbig λ2+1/parenrightbig/integraldisplayβ α|z|2dx+3K12λ−2/parenleftbig λ2+1/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . 34WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY In the above equation, using the fact that C2 h 3C2 h′λ2+|λ|+3Ch′/parenleftbig 9C2 hCh′+Ch+Ch′/parenrightbig ≤max/parenleftbiggC2 h 3C2 h′,3Ch′/parenleftbig 9C2 hCh′+Ch+Ch′/parenrightbig ,1/parenrightbigg/parenleftbig λ2+|λ|+1/parenrightbig ≤max/parenleftbiggC2 h 3C2 h′,3Ch′/parenleftbig 9C2 hCh′+Ch+Ch′/parenrightbig ,1/parenrightbigg (|λ|+1)2 and λ2+1≤(|λ|+1)2, we get (3.56)/integraldisplayβ α|z|2dx≤K14λ−4(|λ|+1)4/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig and (3.57) /ba∇dblU/ba∇dbl2 H2≤2K11(|λ|+1)2/integraldisplayβ α|z|2dx+3K12λ−2(|λ|+1)2/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , where K14= 2K13max/parenleftbiggC2 h 3C2 h′,3Ch′/parenleftbig 9C2 hCh′+Ch+Ch′/parenrightbig ,1/parenrightbigg . Inserting ( 3.56) in (3.57), we get /ba∇dblU/ba∇dbl2 H2≤/parenleftBig 2K11K14(|λ|+1)4+3K12λ2/parenrightBig λ−4(|λ|+1)2/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , ≤(2K11K14+3K12)λ−4(|λ|+1)6/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get ( 3.34). Step 2. In this step, we prove the asymptotic behavior estimate of ( 3.35). LetM∈R∗such that |λ| ≥M >0. In this case, taking s1=s2=s3= 0,r1=3√ M 2Ch′,r2=3Ch′√ Mandr3=√ M 2Ch′in Lemma 3.7, we get (3.58)/ba∇dblU/ba∇dbl2 H2≤K11|λ|3 2/parenleftBig 1+|λ|−1+|λ|−3 2/parenrightBig/integraldisplayβ α|z|2dx +K12|λ|1 2/parenleftBig 1+|λ|−1 2+|λ|−1/parenrightBig/parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig and (3.59)1 2/parenleftBigg 1−√ M 2|λ|1 2/parenrightBigg/integraldisplayβ α|z|2dx ≤K13|λ|−1/bracketleftBigg 1+3C2 hM 4C2 h′+6C2 hC2 h′ M+2Ch′(Ch+Ch′)|λ|−1 2√ M/bracketrightBigg /parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . From the fact that |λ| ≥M, we get 1 2/parenleftBigg 1−√ M 2|λ|1 2/parenrightBigg ≥1 4>0. Therefore, from the above inequality and ( 3.59), we get (3.60)/integraldisplayβ α|z|2dx≤K15|λ|−1/parenleftBigg 1+M+1 M+|λ|−1 2√ M/parenrightBigg /parenleftbig 1+λ−2/parenrightbig/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . where K15= 4K13max/bracketleftbigg 1,3C2 h 4C2 h′,6C2 hC2 h′,2Ch′(Ch+Ch′)/bracketrightbigg . 35WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY In Estimation ( 3.59), using the fact that 1+M+1 M≤/parenleftbigg√ M+1√ M/parenrightbigg2 ,1√ M≤/parenleftbigg√ M+1√ M/parenrightbigg2 , 1+λ−2≤/parenleftBig 1+|λ|−1 2/parenrightBig4 . we get (3.61)/integraldisplayβ α|z|2dx≤K15|λ|−1/parenleftbigg√ M+1√ M/parenrightbigg2/parenleftBig 1+|λ|−1 2/parenrightBig5/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig . Inserting ( 3.61) in (3.58), then using the fact that 1+|λ|−1+|λ|−3 2≤/parenleftBig 1+|λ|−1 2/parenrightBig3 ,/parenleftBig 1+|λ|−1 2+|λ|−1/parenrightBig/parenleftbig 1+λ−2/parenrightbig ≤/parenleftBig 1+|λ|−1 2/parenrightBig6 ≤/parenleftBig 1+|λ|−1 2/parenrightBig8 , we get /ba∇dblU/ba∇dbl2 H2≤max(K11K15,K12)|λ|1 2/parenleftBig 1+|λ|−1 2/parenrightBig8/bracketleftBigg/parenleftbigg√ M+1√ M/parenrightbigg2 +1/bracketrightBigg/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig ≤2max(K11K15,K12)|λ|1 2/parenleftBig 1+|λ|−1 2/parenrightBig8/parenleftbigg√ M+1√ M/parenrightbigg2/parenleftBig /ba∇dblF/ba∇dblH2/ba∇dblU/ba∇dblH2+/ba∇dblF/ba∇dbl2 H2/parenrightBig , hence we get estimation of ( 3.35). The proof is thus complete. /square Proof of Theorem 3.2.First, we will prove condition ( 3.24). Remark that it has been proved in Proposition 3.1that0∈ρ(A2).Now, suppose ( 3.24) is not true, then there exists ω∈R∗such thatiω/\e}atio\slash∈ρ(A2). According to Lemma A.3and Remark A.4, there exists {(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(·,·)))}n≥1⊂R∗×D(A2), withλn→ωasn→ ∞,|λn|<|ω|and/ba∇dblUn/ba∇dblH2= 1, such that (iλnI−A2)Un=Fn:= (f1,n,f2,n,f3,n,f4,n,f5,6,f6,n,f7,n(·,·))→0inH2,asn→ ∞. We will check condition ( 3.24) by finding a contradiction with /ba∇dblUn/ba∇dblH2= 1such as/ba∇dblUn/ba∇dblH2→0.According to Equation ( 3.34) in Proposition 3.3withU=Un, F=Fnandλ=λn, we obtain 0≤ /ba∇dblUn/ba∇dbl2 H2≤K1|λn|−4(|λn|+1)6/parenleftBig /ba∇dblFn/ba∇dblH2/ba∇dblUn/ba∇dblH2+/ba∇dblFn/ba∇dbl2 H2/parenrightBig , asn→ ∞,we get/ba∇dblUn/ba∇dbl2 H2→0,which contradicts /ba∇dblUn/ba∇dblH2= 1. Thus, condition ( 3.24) is holds true. Next, we will prove condition ( 3.25) by a contradiction argument. Suppose there exists {(λn,Un:= (un,vn,wn,yn,zn,φn,ηn(·,·)))}n≥1⊂R∗×D(A2), with|λn| ≥1without affecting the result, such that |λn| →+∞,and/ba∇dblUn/ba∇dblH2= 1and there exists a sequence Gn:= (g1,n,g2,n,g3,n,g4,n,g5,6,g6,n,g7,n(·,·))∈ H2, such that (iλnI−A2)Un=λ−1 2nGn→0inH2. We will check condition ( 3.25) by finding a contradiction with /ba∇dblUn/ba∇dblH2= 1such as/ba∇dblUn/ba∇dblH2=o(1).According to Equation ( 3.35) in Proposition 3.3withU=Un, F=λ−1 2Gn, λ=λnandM= 1, we get /ba∇dblUn/ba∇dbl2 H2≤4K2/parenleftBig 1+|λn|−1 2/parenrightBig8/parenleftBig /ba∇dblGn/ba∇dblH2/ba∇dblUn/ba∇dblH2+|λn|−1 2/ba∇dblGn/ba∇dbl2 H2/parenrightBig , as|λn| → ∞,we get/ba∇dblUn/ba∇dbl2 H2=o(1),which contradicts /ba∇dblUn/ba∇dblH2= 1. Thus, condition ( 3.25) is holds true. The result follows from Theorem A.5(part (ii)). The proof is thus complete. /square Remark 3.8. In the case that α= 0andβ/\e}atio\slash=Lorβ=Landα/\e}atio\slash= 0, we can proceed similar to the proof of Theorem 3.2to check that the energy of System (3.1)-(3.7)decays polynomially of order t−4. /square 36WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Appendix A.Notions of stability and theorems used We introduce here the notions of stability that we encounter in this work. Definition A.1. Assume that Ais the generator of a C 0-semigroup of contractions/parenleftbig etA/parenrightbig t≥0on a Hilbert space H. TheC0-semigroup/parenleftbig etA/parenrightbig t≥0is said to be 1.strongly stable if lim t→+∞/ba∇dbletAx0/ba∇dblH= 0,∀x0∈H; 2.exponentially (or uniformly) stable if there exist two posit ive constants Mandǫsuch that /ba∇dbletAx0/ba∇dblH≤Me−ǫt/ba∇dblx0/ba∇dblH,∀t>0,∀x0∈H; 3.polynomially stable if there exists two positive constants Candαsuch that /ba∇dbletAx0/ba∇dblH≤Ct−α/ba∇dblAx0/ba∇dblH,∀t>0,∀x0∈D(A). /square For proving the strong stability of the C0-semigroup/parenleftbig etA/parenrightbig t≥0, we will recall two methods, the first result obtained by Arendt and Batty in [ 8]. Theorem A.2 (Arendt and Batty in [ 8]).Assume that Ais the generator of a C 0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0on a Hilbert space H. IfAhas no pure imaginary eigenvalues and σ(A)∩iRis countable, where σ(A)denotes the spectrum of A, then theC0-semigroup/parenleftbig etA/parenrightbig t≥0is strongly stable. /square The second one is a classical method based on Arendt and Batty theorem and the contradiction argument (see page 25 in [ 29]). Lemma A.3. Assume that Ais the generator of a C 0−semigroup of contractions/parenleftbig etA/parenrightbig t≥0on a Hilbert space H. Furthermore, Assume that 0∈ρ(A).If there exists ω∈R∗, such that iω/\e}atio\slash∈ρ(A), then (A.1)/braceleftBig iλsuch thatλ∈R∗and|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|/bracerightBig ⊂ρ(A)and sup |λ|</bardblA−1/bardbl−1≤|ω|/vextenddouble/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble/vextenddouble=∞. Proof. Since0∈ρ(A), for any real number λwith|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1, we deduce from the contraction mapping theorem that operator iλI−A=−A−1(I−iλA)is invertible. Therefore, we get /braceleftBig iλsuch thatλ∈R∗and|λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1/bracerightBig ⊂ρ(A). In addition, if there exists ω∈R∗, such that iω/\e}atio\slash∈ρ(A), then/vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |w|, hence we get the first estimation of (A.1). Next, since iω/\e}atio\slash∈ρ(A), then for all |λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|the operator iωI−A= (iλI−A)/parenleftBig I+i(ω−λ)(iλI−A)−1/parenrightBig is not invertible, hence from the contraction mapping theor em we deduce /vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble≥1 |ω−λ|,∀ |λ|</vextenddouble/vextenddoubleA−1/vextenddouble/vextenddouble−1≤ |ω|, therefore sup |λ|</bardblA−1/bardbl−1≤|ω|/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble≥sup |λ|</bardblA−1/bardbl−1≤|ω|1 |ω−λ|=∞, hence we get second estimation of ( A.1). /square Remark A.4. Condition (A.1)turns out that there exists {(λn,Un)}n≥1⊂R∗×D(A),withλn→ωas n→ ∞,|λn|<|ω|and/ba∇dblUn/ba∇dblH2= 1, such that (iλnI−A)Un=Fn→0inH, asn→ ∞. Then, we will check condition iR⊂ρ(A)by finding a contradiction with /ba∇dblUn/ba∇dblH= 1such as/ba∇dblUn/ba∇dblH=o(1)./square We now recall the following standard result which is stated i n a comparable way (see [ 22,38] for part (i) and [9,11,9] for part (ii)). 37WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY Theorem A.5. Assume that Ais the generator of a strongly continuous semigroup of contr actions/parenleftbig etA/parenrightbig t≥0 onH. Assume that iR⊂ρ(A).Then; (i)The semigroup etAis exponentially stable if and only if lim λ→∞/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble<∞. (ii)The semigroup etAis polynomially stable of order α>0if and only if lim λ→∞|λ|−1 α/vextenddouble/vextenddouble/vextenddouble(iλ−A)−1/vextenddouble/vextenddouble/vextenddouble<∞. /square References [1] M. Alves, J. M. Rivera, M. Sepúlveda, and O. V. Villagrán. The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin–Voigt Dissipation .SIAM Journal on Applied Mathematics , 74(2):345–365, Jan. 2014. 6 [2] M. Alves, J. M. Rivera, M. Sepúlveda, O. V. Villagrán, and M. Z. Garay. The asymptotic behavior of the linear transmission problem in viscoelasticity .Mathematische Nachrichten , 287(5-6):483–497, Oct. 2013. 6 [3] K. Ammari and B. Chentouf. Asymptotic behavior of a delayed wave equation without disp lacement term .Zeitschrift für angewandte Mathematik und Physik , 68(5), Sept. 2017. 4 [4] K. Ammari and B. Chentouf. On the exponential and polynomial convergence for a delayed wave equation without displace- ment.Applied Mathematics Letters , 86:126–133, Dec. 2018. 4 [5] K. Ammari, Z. Liu, and F. Shel. Stability of the wave equations on a tree with local Kelvin–V oigt damping .Semigroup Forum , Sept. 2019. 6 [6] K. Ammari, S. Nicaise, and C. Pignotti. Feedback boundary stabilization of wave equations with int erior delay .Systems & Control Letters , 59(10):623–628, Oct. 2010. 4,5 [7] K. Ammari, S. Nicaise, and C. Pignotti. Stability of an abstract-wave equation with delay and a Kelv in–Voigt damping . Asymptotic Analysis , 95(1-2):21–38, Oct. 2015. 6 [8] W. Arendt and C. J. K. Batty. Tauberian theorems and stability of one-parameter semigro ups.Trans. Amer. Math. Soc. , 306(2):837–852, 1988. 37 [9] C. J. K. Batty and T. Duyckaerts. Non-uniform stability for bounded semi-groups on Banach sp aces.J. Evol. Equ. , 8(4):765– 780, 2008. 37 [10] E. M. A. Benhassi, K. Ammari, S. Boulite, and L. Maniar. Feedback stabilization of a class of evolution equations wi th delay . Journal of Evolution Equations , 9(1):103–121, Feb. 2009. 5 [11] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigro ups.Math. Ann. , 347(2):455–478, 2010. 37 [12] G. Chen. Energie decay estimates and exact boundary val ue controllability for the wave equation in a bounded domain .J. Math. Pures Appl. (9) , 58:249–273, 1979. 3 [13] R. Datko. Two questions concerning the boundary control of certain el astic systems .Journal of Differential Equations , 92(1):27–44, July 1991. 3 [14] R. Datko, J. Lagnese, and M. Polis. An example of the effect of time delays in boundary feedback st abilization of wave equations . In1985 24th IEEE Conference on Decision and Control . IEEE, Dec. 1985. 3 [15] H. Demchenko, A. Anikushyn, and M. Pokojovy. On a Kelvin–Voigt Viscoelastic Wave Equation with Strong De lay.SIAM Journal on Mathematical Analysis , 51(6):4382–4412, Jan. 2019. 1,4,6 [16] M. Dreher, R. Quintanilla, and R. Racke. Ill-posed problems in thermomechanics .Applied Mathematics Letters , 22(9):1374– 1379, Sept. 2009. 1 [17] M. Gugat. Boundary feedback stabilization by time delay for one-dime nsional wave equations .IMA Journal of Mathematical Control and Information , 27(2):189–203, Apr. 2010. 3 [18] B.-Z. GUO and C.-Z. XU. Boundary Output Feedback Stabilization of A One-Dimension al Wave Equation System With Time Delay.IFAC Proceedings Volumes , 41(2):8755–8760, 2008. 3 [19] F. Hassine. Stability of elastic transmission systems with a local Kelv in–Voigt damping .European Journal of Control , 23:84– 93, May 2015. 6 [20] F. Hassine. Energy decay estimates of elastic transmission wave/beam s ystems with a local Kelvin–Voigt damping .Interna- tional Journal of Control , 89(10):1933–1950, June 2016. 6 [21] F. Huang. On the Mathematical Model for Linear Elastic Systems with An alytic Damping .SIAM Journal on Control and Optimization , 26(3):714–724, may 1988. 6 [22] F. L. Huang. Characteristic conditions for exponentia l stability of linear dynamical systems in Hilbert spaces. Ann. Differential Equations , 1(1):43–56, 1985. 37 [23] P. Jordan, W. Dai, and R. Mickens. 35(6):414–420, Sept. 2008. 1 [24] T. Kato. Perturbation Theory for Linear Operators . Springer Berlin Heidelberg, 1995. [25] J.-L. Lions. Contrôlabilité exacte, perturbations et stabilisation de s ystèmes distribués. Tome 1 , volume 8 of Recherches en Mathématiques Appliquées . Masson, Paris, 1988. 5 [26] K. Liu, S. Chen, and Z. Liu. Spectrum and Stability for Elastic Systems with Global or Lo cal Kelvin–Voigt Damping .SIAM Journal on Applied Mathematics , 59(2):651–668, jan 1998. 6 38WAVE EQUATION WITH KELVIN-VOIGT DAMPING AND TIME DELAY [27] K. Liu and B. Rao. Exponential stability for the wave equations with local Kel vin–Voigt damping .Zeitschrift für angewandte Mathematik und Physik , 57(3):419–432, Jan. 2006. 6 [28] Z. Liu and Q. Zhang. Stability of a String with Local Kelvin–Voigt Damping and No nsmooth Coefficient at Interface .SIAM Journal on Control and Optimization , 54(4):1859–1871, Jan. 2016. 6 [29] Z. Liu and S. Zheng. Semigroups associated with dissipative systems , volume 398 of Chapman &Hall/CRC Research Notes in Mathematics . Chapman &Hall/CRC, Boca Raton, FL, 1999. 37 [30] S. A. Messaoudi, A. Fareh, and N. Doudi. Well posedness and exponential stability in a wave equation with a strong damping and a strong delay .Journal of Mathematical Physics , 57(11):111501, Nov. 2016. 1,4,5 [31] R. Nasser, N. Noun, and A. Wehbe. Stabilization of the wave equations with localized Kelvin– Voigt type damping under optimal geometric conditions .Comptes Rendus Mathematique , 357(3):272–277, Mar. 2019. 6 [32] S. Nicaise and C. Pignotti. Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks .SIAM Journal on Control and Optimization , 45(5):1561–1585, Jan. 2006. 4 [33] S. Nicaise and C. Pignotti. Stability of the wave equation with localized Kelvin–Voigt damping and boundary delay feedback . Discrete and Continuous Dynamical Systems - Series S , 9(3):791–813, Apr. 2016. 1,4,5 [34] S. Nicaise, J. Valein, and E. Fridman. Stability of the heat and of the wave equations with boundary time-varying delays . Discrete and Continuous Dynamical Systems - Series S , 2(3):559–581, June 2009. 5 [35] H. P. Oquendo. Frictional versus Kelvin–Voigt damping in a transmission p roblem .Mathematical Methods in the Applied Sciences , 40(18):7026–7032, July 2017. 6 [36] A. Pazy. Semigroups of linear operators and applications to partial differential equations , volume 44 of Applied Mathematical Sciences . Springer-Verlag, New York, 1983. 9,11,28 [37] C. Pignotti. A note on stabilization of locally damped wave equations wit h time delay .Systems &Control Letters , 61(1):92–97, Jan. 2012. 4,5 [38] J. Prüss. On the spectrum of C0-semigroups .Trans. Amer. Math. Soc. , 284(2):847–857, 1984. 37 [39] J. E. M. Rivera, O. V. Villagran, and M. Sepulveda. Stability to localized viscoelastic transmission problem .Communications in Partial Differential Equations , 43(5):821–838, May 2018. 6 [40] L. Tebou. A constructive method for the stabilization of the wave equa tion with localized Kelvin–Voigt damping .Comptes Rendus Mathematique , 350(11-12):603–608, June 2012. 6 [41] J.-M. Wang, B.-Z. Guo, and M. Krstic. Wave Equation Stabilization by Delays Equal to Even Multipl es of the Wave Propa- gation Time .SIAM Journal on Control and Optimization , 49(2):517–554, Jan. 2011. 4 [42] Y. Xie and G. Xu. 10(3):557–579, 2017. 4 [43] Y. Xie and G. Xu. Exponential stability of 1-d wave equation with the boundar y time delay based on the interior control . 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2020-03-29

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin-Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin-Voigt and the delay damping are both localized via non smooth coefficients. In the case that the Kelvin-Voigt damping is localized faraway from the tip and the wave is subjected to a locally distributed internal or boundary delay feedback, we prove that the energy of the system decays polynomially of type t^{-4}. However, an exponential decay of the energy of the system is established provided that the Kelvin-Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin-Voigt and the internal delay damping are both localized via non smooth coefficients near the tip, the energy of the system decays polynomially of type t^{-4}. Frequency domain arguments combined with piecewise multiplier techniques are employed.

Stability results for an elastic-viscoelastic waves interaction systems with localized Kelvin-Voigt damping and with an internal or boundary time delay

2003.12967v1

Magnetisation dynamics of the compensated ferrimagnet Mn 2RuxGa G. Bonglio Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands K. Rode, K. Siewerska, J. Besbas, G. Y. P. Atcheson, P. Stamenov, and J.M.D. Coey CRANN, AMBER and School of Physics, Trinity College Dublin, Ireland A.V. Kimel, Th. Rasing, and A. Kirilyuk Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands and FELIX Laboratory, Radboud University, Toernooiveld 7c, 6525 ED Nijmegen, The Netherlands Here we study both static and time-resolved dynamic magnetic properties of the compensated ferrimagnet Mn 2RuxGa from room temperature down to 10 K, thus crossing the magnetic compen- sation temperature TM. The behaviour is analysed with a model of a simple collinear ferrimagnet with uniaxial anisotropy and site-specic gyromagnetic ratios. We nd a maximum zero-applied- eld resonance frequency of 160 GHz and a low intrinsic Gilbert damping 0:02, making it a very attractive candidate for various spintronic applications. I. INTRODUCTION Antiferromagnets (AFM) and compensated ferrimag- nets (FiM) have attracted a lot of attention over the last decade due to their potential use in spin electronics1,2. Due to their lack of a net magnetic moment, they are insensitive to external elds and create no demagnetis- ing elds of their own. In addition, their spin dynamics reach much higher frequencies than those of their ferro- magnetic (FM) counterparts due to the contribution of the exchange energy in the magnetic free energy3. Despite these clear advantages, AFMs are scarcely used apart from uni-directional exchange biasing rela- tively in spin electronic applications. This is because the lack of net moment also implies that there is no direct way to manipulate their magnetic state. Fur- thermore, detecting their magnetic state is also compli- cated and is usually possible only by neutron diraction measurements4, or through interaction with an adjacent FM layer5. Compensated, metallic FiMs provide an interesting al- ternative as they combine the high-speed advantages of AFMs with those of FMs, namely, the ease to manipu- late their magnetic state. Furthermore, it has been shown that such materials are good candidates for the emerging eld of All-Optical Switching (AOS) in which the mag- netic state is solely controlled by a fast laser pulse6{8. A compensated, half-metallic ferrimagnet was rst en- visaged by van Leuken and de Groot9. In their model two magnetic ions in crystallographically dierent po- sitions couple antiferromagnetically and perfectly com- pensate each-other, but only one of the two contributes to the states at the Fermi energy responsible for elec- tronic transport. The rst experimental realisation of this, Mn 2RuxGa (MRG), was provided by Kurt et al.10. MRG crystallises in the XAHeusler structure, space groupF43m, with Mn on the 4 aand 4csites11. Substrate-induced bi-axial strain imposes a slight tetrag- onal distortion, which leads to perpendicular magneticanisotropy. Due to the dierent local environment of the two sublattices, the temperature dependence of their magnetic moments dier, and perfect compensation is therefore obtained at a specic temperature TMthat depends on the Ru concentration xand the degree of biaxial strain. It was previously shown that MRG ex- hibits properties usually associated with FMs: a large anomalous Hall angle12, that depends only on the mag- netisation of the 4 cmagnetic sublattice13; tunnel magne- toresistance (TMR) of 40 %, a signature of its high spin polarisation14, was observed in magnetic tunnel junc- tions (MTJs) based on MRG15; and a clear magneto- optical Kerr eect and domain structure, even in the ab- sence of a net moment16,17. Strong exchange bias of a CoFeB layer by exchange coupling with MRG through a Hf spacer layer18, as well as single-layer spin-orbit torque19,20showed that MRG combined the qualities of FMs and AFMs in spin electronic devices. The spin dynamics in materials where two distinct sublattices are subject to diering internal elds (ex- change, anisotropy, . . . ) is much richer than that of a simple FM, as previously demonstrated by the obers- vation of single-pulse all-optical switching in amorphous GdFeCo21,22and very recently in MRG8. Given that the magnetisation of MRG is small, escpecially close to the compensation point, and the related frequency is high, normal ferromagnetic resonance (FMR) spectroscopy is unsuited to study their properties. Therefore, we used the all-optical pump-probe technique to characterize the resonance frequencies at dierent temperatures in vicin- ity of the magnetic compensation point. This, together with the simulation of FMR, make it possible to deter- mine the eective g-factors, the anisotropy constants and their evolution across the compensation point. We found, in particular, that our ferrimagnetic half-metallic Heusler alloy has resonance frequency up to 160 GHz at zero-eld and a relatively low Gilbert damping.arXiv:1909.09085v1 [cond-mat.mtrl-sci] 19 Sep 20192 FIG. 1. Net moment measured by magnetometry and coercive eld measured by static Faraday eect. The upturn of the net moment below T50 K is due to paramagnetic impurities in the MgO substrate. TMis indicated by the vertical dotted line. As expected the maximum available applied eld 0H= 7 T is insucient to switch the magnetisation close to TM. II. EXPERIMENTAL DETAILS Thin lm samples of MRG were grown in a `Sham- rock' sputter deposition cluster with a base pressure of 2108Torr on MgO (001) substrates. Further infor- mation on sample deposition can be found elsewhere23. The substrates were kept at 250C, and a protective 3 nm layer of aluminium oxide was added at room tem- perature. Here we focus on a 53 nm thick sample with x= 0:55, leading to TM80 K as determined by SQUID magnetometry using a Quantum Design 5 T MPMS sys- tem (see FIG. 1). We are able to study the magneto- optical properties both above and below TM. The magnetisation dynamics was investigated using an all-optical two-colour pump-probe scheme in a Faraday geometry inside a 0Hmax= 7 T superconducting coil- cryostat assembly. Both pump and probe were produced by a Ti:sapphire femtosecond pulsed laser with a cen- tral wavelength of 800 nm, a pulse width of 40 fs and a repetition rate of 1 kHz. After splitting the beam in two, the high-intensity one was doubled in frequency by a BBO crystal (giving = 400 nm) and then used as the pump while the lower intensity 800 nm beam acted as the probe pulse. The time delay between the two was adjusted by a mechanical delay stage. The pump was then modulated by a synchronised mechanical chopper at 500 Hz to improve the signal to noise ratio by lock-in detection. Both pump and probe beams were linearly polarized, and with spot sizes on the sample of 150 µm and 70 µm, respectively. The pump pulse hit the sample at an incidence angle of 10. After interaction with the sample, we split the probe beam in two orthogonally polarized parts using a Wollaston prism and detect the changes in transmission and rotation by calculating the FIG. 2. Comparison of hysteresis loops obtained by Faraday, AHE, and magnetometry recorded at room temperature. The two former were recorded with the applied eld perpendicular to the sample surface, while for the latter we show results for both eld applied parallel and perpendicular to the sample. sum and the dierence in intensity of the two signals. The external eld was applied at 75to the easy axis of magnetization thus tilting the magnetisation away from the axis. Upon interaction with the pump beam the mag- netisation is momentarily drastically changed24and we monitor its return to the initial conguration via remag- netisation and then precession through the time depen- dent Faraday eect on the probe pulse. The static magneto-optical properties were examined in the same cryostat/magnet assembly. III. RESULTS & DISCUSSION A. Static magnetic properties We rst focus on the static magnetic properties as observed by the Faraday eect, and compare them to what is inferred from magnetometry and the anomalous Hall eect. In FIG. 2 we present magnetic hysteresis loops as recorded using the three techniques. Due to the half metallic nature of the sample, the magnetotrans- port properties depend only on the 4 csublattice. As the main contribution to the MRG dielectric tensor in the visible and near infrared arises from the Drude tail16, both AHE and Faraday eect probe essentially the same properties (mainly the spin polarised conduction band of MRG), hence we observe overlapping loops for the two techniques. Magnetometry, on the other hand, measures the net moment, or to be precise the small dierence between two large sublattice moments. The 4 asublat- tice, which is insignicant for AHE and Faraday here contributes on equal footing. FIG. 2 shows a clear dier- ence in shape between the magnetometry loop and the3 FIG. 3. Time resolved Faraday eect recorded at T= 290 K in applied elds ranging from 1 T to 7 T. After the initial demagnetisation seen as a sharp increase in the signal at t 0 ps, magnetisation is recovered and followed by precession around the eective eld until fully damped. The lines are ts to the data. The inset shows the experimental geometry further detailed in the main text. AHE or Faraday loops. We highlight here that the ap- parent `soft' contribution that shows switching close to zero applied eld, is not a secondary magnetic phase, but a signature of the small dierences in the eld-behaviour of the two sublattices. We also note that this behaviour is a result of the non-collinear magnetic order of MRG. A complete analysis of the dynamic properties therefore requires knowledge of the anisotropy constants on both sublattices as well as the (at least) three intra and in- ter sublattice exchange constants. Such an analysis is beyond the scope of this article, and we limit our anal- ysis to the simplest model of a single, eective uniaxial anisotropy constant Kuin the exchange approximation of the ferrimagnet. B. Dynamic properties We now turn to the time-resolved Faraday eect and spin dynamics. Time-resolved Faraday eect data were recorded at ve dierent temperatures 10 K, 50 K, 100 K, 200 K and 290 K, with applied elds ranging from 1 T to 7 T. FIG. 3 shows the eld-dependence of the Faraday ef- fect as a function of the delay between the pump and the probe pulses, recorded at T= 290 K. Negative de- lay indicates the probe is hitting the sample before the pump. After the initial demagnetisation, the magneti- sation recovers and starts precessing around the eec- tive eld which is determined by the anisotropy and the applied eld. The solid lines in FIG. 3 are ts to the data to extract the period and the damping of the pre-cession in each case. The tting model was an expo- nentially damped sinusoid with a phase oset. We note that the apparent evolution of the amplitude and phase with changing applied magnetic eld is due to the quasi- resonance of the spectrum of the precessional motion with the low-frequency components of the convolution between the envelope of the probe pulse and the phys- ical relaxation of the system. The latter include both electron-electron and electron-lattice eects. A rudimen- tary model based on a classical oscillator successfully re- produces the main features of the amplitude and phase observed. In two-sublattice FiMs, the gyromagnetic ratios of the two sublattices are not necessarily the same. This is par- ticularly obvious in rare-earth/transition metal alloys, and is also the case for MRG despite the two sublat- tices being chemically similar; they are both Mn. Due to the dierent local environment however, the degree of charge transfer for the two diers. This leads to two characteristic temperatures, a rst TMwhere the mag- netic moments compensate, and a second TAwhere the angular momenta compensate. It can be shown that for the ferromagnetic mode, the eective gyromagnetic ratio ecan then be written25 e=M4c(T)M4a(T) M4c(T)= 4cM4a(T)= 4a(1) subscripti= 4a;4cdenotes sublattice i,Mi(T) the temperature-dependent magnetisation, and ithe sublattice-specic gyromagnetic ratio. eis related to the eective g-factor ge= eh B(2) wherehis the Planck constant and Bthe Bohr magne- ton. The frequency of the precession is determined by the eective eld, which can be inferred from the derivative of the magnetic free energy density with respect to M. For an external eld applied at a given xed angle with respect to the easy axis this leads to the Smit-Beljers formula26 !FMR = evuut1 M2ssin2" 2E 22E 22E 2# (3) whereandare the polar and azimuthal angles of the magnetisation vector, and Ethe magnetic free energy density E=0H
M+Kusin2+0M2 scos2=2 (4) where the terms correspond to the Zeeman, anisotropy and demagnetising energies, respectively, and Msis the net saturation magnetisation. It should be mentioned that the magnetic anisotropy constant Kuis related to M, which is being considered constant in magnitude, via Ku=0M2 s=2,a dimensionless parameter.4 FIG. 4. Observed precession frequency as a function of the applied eld for various temperatures. The solid lines are ts to the data as described in the main text. Based on Eqs. (1) through (4) we t our entire data set with eandKuas the only free parameters. The exper- imental data and the associated ts are shown as points and solid lines in FIG. 4. At all temperatures our simple model with one eective gyromagnetic ratio eand a single uniaxial anisotropy parameter Kureproduces the experimental data reasonably well. The model systemat- ically underestimates the resonance frequency for inter- mediate elds, with the point of maximum disagreement increasing with decreasing temperature. We speculate this is due to the use of a simple uniaxial anisotropy in the free energy (see Eq. 4), while the real situation is more likely to be better represented as a sperimagnet. In particular, the non-collinear nature of MRG that leads to a deviation from 180of the angle between the two sublattice magnetisations, depending on the applied eld and temperature. From the ts in FIG. 4 we infer the values of geand the anisotropy eld 0Ha=2Ku=Ms. The result is shown in FIG. 5. The anisotropy eld is monotonically increas- ing with decreasing temperature as the magnetisation of the 4csublattice increases in the same temperature range. We highlight here the advantage of determining this eld through time-resolved magneto-optics as op- posed to static magnetometry and optics. Indeed the anisotropy eld as seen by static methods is sensitive to the combination of anisotropy and the netmagnetic mo- ment, as illustrated in FIG. 1, where the coercive eld diverges as T!TM. In statics one would expect a di- vergence of the anisotropy eld at the same temperature. The time-resolved methods however distinguish between the net and the sublattice moments, hence better re ect- ing the evolution of the intrinsic material properties of the ferrimagnet. The temperature dependence of the anisotropy con- stants was a matter for discussion for many years27,28. FIG. 5. Eective g-factor,ge, and the anisotropy eld as determined by time-resolved Faraday eect. ge, orange squares, increases from near the free electron value of 2 to 4 just belowTM, while the anisotropy eld, blue triangles, in- creases near-linearly with decreasing temperature. A M3t, red dashes line, of the anisotropy behaviour shows the almost- metallic origin of it, indicating the dominant character of the 4c sublattice. Written in spherical harmonics the 3 danisotropy can be expressed as, k2Y0 2() +k4Y0 4()29wherek2/ M(T)3andk4/M(T)10. The experimental measured anisotropy is then, K2(T) =ak2(T)+bk4(T), withaand bthe contributions of the respective spherical harmonics. FIG. 5 shows that a reasonable t of our data is ob- tained with M(T)3which means, rst, that the contri- bution of the 4thorder harmonic can be neglected, and second, that the contribution of the TMand 2ndsublat- tice is negligible, indicating the dominant character of the 4c sublattice. In addition, we should note here that the high fre- quency exchange mode was never observed on our exper- iments. While far from TMits frequency might be too high to be observable, in the vicinity of TM, in contrast, its frequency is expected to be in the detection range. Moreover, given the dierent electronic structure of the two sublattices, it is expected that the laser pulse should selectively excite the sublattice 4c, and therefore lead to the eective excitation of the exchange mode. We argue that it is the non-collinearity of the sublattices (see sec- tion III A) that smears out the coherent precession at high frequencies. The eective gyromagnetic ratio, ge, shows a non- monotonic behaviour. It increases with decreasing Tto- wardsTM, reaching a maximum at about 50 K before decreasing again at T= 10 K. We alluded above to the dierence between the magnetic and the angular mo- menta compensation temperatures. We expect that ge reaches a maximum when T=TA30, here between the measurement at T= 50 K and the magnetic compensa- tion temperature TM80 K.5 FIG. 6. Intrinsic and anisotropic broadening in MRG across theTM. The inset shows the evaluation process of the two damping parameters. A linear t is used to evaluate intercept (anisotropic broadening) and slope (intrinsic damping) of the frequencies versus the inverse of the decay time. The data point are obtained from the t of time-resolved Faraday eect measurements (an example is shown in Fig.4). From XMCD data11, we could estimate spin and or- bital moment components of the magnetic moments of the two sublattices, what allowed us to derive the ef- fective g-factors for the sublattices as g4a= 2:05 and g4c= 2:00. In this case we expect the angular momentum compensation temperature TAto be below TM, opposite to what is observed for GdFeCo21. Given this small dif- ference however, TAandTMare expected to be rather close to each other, consistent with the limited increase ofgeacross the compensation points. We turn nally to the damping of the precessional mo- tion of Maround the eective eld 0He. Damping is usually described via the dimensionless parameter in the Landau-Lifshiz-Gilbert equation, and it is a measure of the dissipation of magnetic energy in the system. In this model, is a scalar constant and the observed broad- ening in the time domain is therefore a linear function of the frequency of precession31{33. We infer 0, the total damping, from our ts of the time-resolved Faraday eect as0= (d)1, wheredis the decay time of the ts. We then, for each temperature, plot 0as a function of the observed frequency and regress the data using a straight line t. The intrinsic is the slope of this line, while the intercept represents the anisotropic broadening. FIG. 6 shows the intrinsic damping and the anisotropic broadening as a function of temperature. Anisotropic broadening is usually attributed to a vari- ation of the anisotropy eld in the region probed by the probe pulse34. For MRG this is due to slight lateral vari- ations in the Ru content xin the thin lm sample. Such a variation leads to a variation in eective TMandTAand can therefore have a large in uence on the broadening asa function of temperature. Despite this, the anisotropic broadening is reasonably low in the entire temperature range above TM, and a more likely explanation for its rapid increase below TMis that the applied magnetic eld is insucient to completely remagnetize the sam- ple between two pump pulses. As observed in Fig.5, the anisotropy eld reaches almost 4 T at low temperature, comparable to our maximum applied eld of 7 T. The intrinsic damping is less than 0.02 far from TM, but increases sharply at T= 100 K. We tentatively attribute this to an increasing portion of the available power be- ing transferred into the high-energy exchange mode, al- though we underline that we have not seen any direct evidence of such a mode in any of the experimental data. IV. CONCLUSION We have shown that the time-resolved Faraday eect is a powerful tool to determine the spin dynamic proper- ties in compensated, metallic ferrimagnets. The high spin polarisation of MRG enables meaningful Faraday data to be recorded even near TMwhere the net magnetisation is vanishingly small, and the dependence of the dynamics on the sublattice as opposed to the net magnetic prop- erties provides a more physical understanding of the ma- terial. Furthermore, we nd that the ferromagnetic-like mode of MRG reaches resonance frequencies as high as 160 GHz in zero applied eld, together with a small in- trinsic damping. This value is remarkable if compared to well-known materials such as GdFeCo which, at zero eld, resonates at tens of GHz21or [Co/Pt] nmultilay- ers at 80 GHz35but with higher damping. We should however stress that, in the presence of strong anisotropy elds, higher frequencies can be reached. Example of that can be found for ferromagnetic Fe/Pt with 280 GHz (Ha= 10T)36, and for Heusler-like ferrimagnet (Mn 3Ge and Mn 3Ga) with500 GHz (Ha= 20T)37,38. Never- theless, the examples cited above show a considerably higher intrinsic damping compared to MRG. In addi- tion, it was recently shown that MRG exhibits unusu- ally strong intrinsic spin-orbit torque20. Thus, taking into account the material parameters we have determined here, it seems likely it will be possible to convert a DC driven current into a sustained ferromagnetic resonance atf= 160 GHz, at least. These characteristics make MRG, as well as any future compensated half-metallic ferrimagnet, particularly promising materials for both spintronics and all-optical switching. ACKNOWLEDGMENTS This project has received funding from the NWO pro- gramme Exciting Exchange, the European Union's Hori- zon 2020 research and innovation programme under grant agreement No 737038 `TRANSPIRE', and from Science6 Foundation Ireland through contracts 12/RC/2278 AM- BER and 16/IA/4534 ZEMS.The authors would like to thank D. Betto for help ex- tractinghLiandhSi. 1A. B. Shick, S. Khmelevskyi, O. N. Mryasov, J. Wunder- lich, and T. Jungwirth, Phys. Rev. B 81, 212409 (2010). 2L. Caretta, M. Mann, F. B uttner, K. Ueda, B. Pfau, C. M. G unther, P. Hessing, A. Churikova, C. Klose, M. Schneider, D. Engel, C. Marcus, D. Bono, K. Bagschik, S. Eisebitt, and G. S. Beach, Nat. Nanotechnol. 13, 1154 (2018). 3E. V. Gomonay and V. M. Loktev, Low Temp. Phys. 40, 17 (2014). 4C. G. Shull and J. S. Smart, Phys. Rev. 76, 1256 (1949). 5T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. 11, 231 (2016). 6C. D. Stanciu, A. Tsukamoto, A. V. Kimel, F. Hansteen, A. Kirilyuk, A. 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2019-09-19

Here we study both static and time-resolved dynamic magnetic properties of the compensated ferrimagnet from room temperature down to 10K, thus crossing the magnetic compensation temperature $T_{M}$. The behaviour is analysed with a model of a simple collinear ferrimagnet with uniaxial anisotropy and site-specific gyromagnetic ratios. We find a maximum zero-applied-field resonance frequency of $\sim$160GHz and a low intrinsic Gilbert damping $\alpha$$\sim$0.02, making it a very attractive candidate for various spintronic applications.

Magnetization dynamics of the compensated ferrimagnet $Mn_{2}Ru_{x}Ga$

1909.09085v1

IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 1 Damping dependence of spin-torque effects in thermally assisted magnetization reversal Y.P. Kalmykov,1 D. Byrne,2 W.T. Coffey,3 W. J. Dowling,3 S.V.Titov,4 and J.E. Wegrowe5 1Univ. Perpignan Via Domitia, Laboratoire de Mathématiques et Physique, F -66860, Perpignan, France 2School of Physics, University College Dublin, Belfield, Dublin 4, Ireland 3Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland 4Kotel’nikov Institute of Radio Engineering and Electronics of the Russia n Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region, 141120, Russia 5Laboratoire des Solides Irradiés, Ecole Polytechnique, 91128 Palaiseau Cedex, France Thermal fluctuations of nanomagnets driven by spin -polarized currents are treated via the Landau -Lifshitz -Gilbert equation as generalized to include both the random thermal noise field and Slonczewski spin -transfer torque (STT) term s. The magnetization reversal time of such a nanomagnet is then evaluated for wide ranges of damping by using a method which generalizes the solution of the so -called Kramers turnover problem for mechanical Brownian particles thereby bridging the very low damping (VLD) and intermediate damping (ID) Kramers escape rates , to the analogous magnetic turnover problem. The reversal time is then evaluated for a nanomagnet with the free energy density given in the standard form of superimposed easy -plane and in -plane easy -axis anisotropies with the dc bias field along the easy axis. Index Terms — Escape rate, Nanomagnets, Reversal time of the magnetization, Spin -transfer t orque . I. INTRODUCTION ue to the spin-transfer torque (STT) effect [1 -6], the magnetization of a nanoscale ferromagnet may be altered by spin -polarized currents . This phenomenon occurs because an electri c current with spin polarization in a ferromagnet has an associated flow of angular momentum [3,7] ther eby exerting a macroscopic spin torque. The phenomenon is the origin of the novel subject of spintronics [7,8], i.e., current - induced control over magnet ic nanostructures . Common applications are very high-speed current -induced magnetization switching by (a) reversing the orien tation of magnetic bits [3,9 ] and (b) using spin polarized currents to control steady state microwave oscillations [9 ]. This is accomplished via the steady state magnetization precession due to STT representing the conversion of DC input into an AC output voltage [3]. Unfortunately , thermal fluctuations cannot now be ignored due to the nanometric size of STT devices, e.g., leading to mainly noise -induced switching at currents far less than the critical switching current without noise [10] as corroborated by experiments (e.g., [11]) demonstrating that STT near room temperature significantly alters thermally activated switching processes . These now exhibit a pronounced dependence on both material and geometrical parameters. Consequently, an accurate account of STT switching effects at finite temperatures is necessary in order to achieve further improvements in the design and interpretatio n of experiments, in view of the manifold practical applications in spintronics, random access memory technology, and so on. During the last decade, various analytical and numerical approaches to the study of STT effects in the thermally assisted magnetiza tion reversal (or switching) time in nanoscale ferromagnets have been developed [6,7,12 -26]. Their objective being to generalize methods originally developed for zero STT [12,27 -32] such as stochastic dynamics simulations (e. g., Refs. [21 -25]) and extensio ns to spin Hamiltonians of the mean first passage time (MFPT) method (e.g., Refs. [16] and [17] ) in the Kramers escape rate theory [33,34]. However, unlike zero STT substantial progress in escape rate theory including STT effects has so far been achieved o nly in the limit of very low damping (VLD), corresponding to vanishingly small values of the damping parameter in the Landau -Lifshitz -Gilbert -Slonczewski equation (see Eq. (5) below). Here the pronounced time separation between fast precessional and slow energy changes in lightly damped closed phase space trajectories (ca lled Stoner -Wohlfarth orbits) has been exploited in Refs. [7,14, 16,17] to formulate a one -dimensional Fokker -Planck equation for the energy distribution function which may be solved by quadratures. This equation is essentially similar to that derived by Kramers [ 33] in treating the VLD noise - activated escape rate of a point Brownian particle from a potential well although the Hamiltonian of the magnetic problem is no longer separable and additive and the barrier height is now STT depend ent. The Stoner -Wohlfarth orbits and steady precession along such an orbit of constan t energy occur if the spin -torque is strong enough to cancel out the dissipative torque. The origin of the orbits arises from the bistable (or, indeed, in general multistable) structure of the anisotropy potential. This structure allows one to define a nonconservative “effective” potential with damping - and D Manuscript received April 6, 2017; revised June 27, 2017; accepted July 24, 2017. Date of publication July 27, 2017; date of current ver -sion September 18, 2017. Correspondin g author: Y. P. Kalmykov (e -mail: kalmykov@univ -perp.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org . Digital Object Identifier: 10.1109/TMAG.2017. 2732944 IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 2 current -dependent potential barrier s between stationary self - oscillatory states of the magnetization, thereby permitting one to estimate the reversal (switching) time between these states . The magnetizat ion reversal time in the VLD limit is then evaluated [16,17,35 ] both for zero and nonzero STT. In particular, for nonzero STT , the VLD reversal time has been evaluated analytically in Refs. [16,17 ]. Here it has been shown that in the high barrier limit, an asymptotic equation for the VLD magnetization reversal time from a single well in the presence of the STT is given by VLD TST1 CES . (1) In Eq. (1), is the damping parameter arising from the surroundings , TST AE Efe is the escape rate render ed by transition state theory (TST) which ignores effects due to the loss of spins at the barrier [34], AEf is the well precession frequency, E is the damping and spin -polarized -current dependent effective en ergy barrier, and CES is the dimensionless action at the saddle point C (the action is given by Eq. (13) below). The most essential fea ture of the results obtained in Refs. [16,17,35 ] and how they pertain to this paper is that they apply at VLD only where the inequality 1 CES holds meaning that the energy loss per cycle of the almost periodic motion at the critical en ergy is much less than the thermal energy . Unfortunately for typical values of the material parameters CES may be very high ( 310 ), meaning that this inequality can be fulfilled only for 0.001 . In addition, both experimental and theoretical estimates suggest higher values of of the order of 0.001 -0.1 ( see, e.g., Refs. [6,36 -38]), implying that the VLD asymptotic results are no longer valid as they will now differ substantially from the true value of the reversal time . These considerations suggest that the asymptotic calculations for STT should be extended to include both the VLD and intermediate damping (ID) regions. This is our primar y objective here . Now like point Brownian particles which are governed by a separable and additive Hamiltonian , in the escape rate problem as it pertains to magnetic moments of nanoparticles, three regimes of damping appear [ 12,33,34]. These are (i) very low damping ( 1) CES , (ii) intermediate - to-high damping (IHD) ( 1) CES , and (iii) a more or less critically damped turnover regime ( ~ 1) CES . Also , Kramers [33] obtained his now -famous VLD and IHD escape rate formulas for point Brownian particles by assuming in both cases that the energy barrier is much greater than the thermal energy so that the concept of an escape rate applies. He mentioned, however, that he could not find a general method of attack in order to obtain an escape rate formula valid for any damp ing regime. This problem, namely the Kramers turnover, was initially solved by Mel’nikov and Meshkov [39]. They obtained an escape rate that is valid for all values of the damping by a semi heuristic argument, thus constituting a solution of the Kramers tu rnover problem for point particles. Later, Grabert [40] and Pollak et al . [41] have presented by using a coupled oscillator model of the thermal bath , a complete solution of the Kramers turnover problem and have shown that the turnover escape rate formula can be obtained without the ad hoc interp olation between the VLD and IHD regimes as used by Mel’nikov and Meshkov . Finally, Coffey et al. [42,43 ] have shown for classical spins that at zero STT , the magnetization reversal time for values of damping up to intermediate values, 1, can also be evaluated via the turnover formula for the escape rate bridging the VLD and ID escape rates, namely, TST1 () CE AS , (2) where ()Az is the so-called depopulation factor, namely [39- 42] 2 2 0ln 1 exp[ ( 1/4)]1 1/4()z d A z e . (3) Now the ID reversal time (or the lower bound of the reversal time) may always be evaluated via TST as [32,34] ID TST1 . (4) Therefore b ecause () CCEE A S S is the energy loss per cycle at the critical energy 0 CES [39] (i.e. , in the VLD limit) , Eq. (2) transparently reduces to the VLD Kramers result, Eq. (1). Moreover in the ID range, where ( ) 1 CE AS , Eq. (2) reduces to the TST Eq. (4). Nevertheless in the high barrier limit 1, CES given by Eq. (2) can substantially deviate in the damping range 0.001 1 both from ID , Eq. (4), and VLD , Eq. (1). Now, the approach of Coffey et al. [42,43 ] generalizing the Kramers turnover results to classical spins (nanomagnets) was developed for zero STT, nevertheless, it can also be used to account for STT effects. Here we shall extend th e zero STT results of Refs. [14,16,17,39 -42] treating the damping dependence of STT effects in the magnetization reversal of nano scaled ferro magnets via escape rate theory in the most important range of damping comprising the VLD and ID ranges , 1. II. MODEL The object of our study is the role played by STT effects in the thermally assist ed magnetization reversal using an adaptation of the theory of thermal fluctuations in nanomagnets developed in the seminal work s of Néel [27] and Brown [28,29]. The Néel -Brown theory i s effect ively an adaptation of the Kramers theory [ 33,34 ] originally given for point Brownian particles to magnetization relaxa tion governed by a gyromagnetic -like equation which is taken as the Langevin equation of the pro cess. Hence, the verification of that theory in the pure (i.e., without STT) nanomagnet context nicely illustrates the Kramers conception of a thermal relaxation process as escape over a potential barrier arising from the IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 3 shuttling action of the Brownian m otion. However, it should be recalled throughout that unlike nanomagnets at zero STT (where the giant spin escape rate theory may be effectively regarded as fully developed), devices based on STT , due to the injection of the spin -polarized current, invaria bly represent an open system in an out-of-equilibrium steady state. This is in marked contrast to the conventional steady state of nanostructures characterized by the Boltzmann equilibrium distribution that arises when STT is omitted . Hence both the gover ning Fokker -Planck and Langevin equations and the escape rate theory based on these must be modified . To facilitate our di scussion, we first describe a schematic model of the STT effect. The archetypal model (Fig. 1 (a)) of a STT device is a nanostructure compr ising two magnetic strata label ed the free and fixed layers and a nonmagnetic conducting spacer. The fixed layer is much more strongly pinned along its orientation than the free one. If an electric current is passsed through the fixed layer it become s spin - polarized . Thus , the current , as it encounters the free layer, induces a STT . Hence, the magnetization M of the free layer is altered . Both ferromagnetic layers are assumed to be uniformly magnetized [3,6]. Although th is gia nt coherent spin approximation cannot explain all observations of the magnetization dynamics in spin -torque systems, nevertheless many qualitative features needed to interpret experimental data are satisfactorily reproduced. Indeed, the current -induced magnetization dynamics in the free layer may be described by the Landau -Lifshitz -Gilbert -Slonczewski equation including thermal fluctuations , i.e., the usual Landau -Lifshitz -Gilbert equation [ 44] incl uding STT, however augmented by a random magnetic field ()tη which is regarded as white noise. Hence it now becomes a magnetic Langevin equation [3,6,7,12 ], viz., S u u H η u u u u I . (5) Here /SMuM is the unit vector directed along M , SM is the saturation magnetization, and is the gyromagnetic -type constant . The effective magnetic field H comprising the anisotropy and external applied fields is defined as 0SkT E vMHu . (6) Here E is the normalized free energy density of the free layer constituting a conservative potential, v is the free layer volume , 7 2 1 04 10 JA m in SI units, and kT is the thermal energy. For purposes of illustration , we sh all take ,)(E in the standard form of superimposed easy -plane and in-plane easy -axis anisotropies plus the Zeeman term due to the applied magnetic field 0H [45] (in our notation): 22 2, ) sin cos sin cos )( ( 2 cos h E . (7) In Eq. (7) and are the polar and azimuthal angles in the usual spherical polar coordinate system , 0S/ (2 ) h H M D and 2 0S / ( ) v M D kT are the external field and anisotropy parameters, /1DD is the biaxiality parameter characterized by D and D thereby encompassing both demagnetizing and magnet ocrystalline anisotropy effects (since and are determined by both the volume an d the thickness of the free layer, th eir numerical values may vary through a very large range, in particular, they can be very large , > 100 [45]). The form of Eq. (7) implies that both the applied field 0H and the unit vector Pe identifying the magnetization direction in the fixed layer are directed along the easy X-axis (see Fig. 1(a)) . In general, ,()E as rendered by Eq. (7) has two equivalent saddle points C and two nonequivalent wells at A and A (see Fig.1(b) ). Finally , the STT induced field SI is given by 0S SkT vMIu , (8) where is the normalized non conservative potential due to the spin -polarized current, which in its simplest form i s ( , )PJ eu . (9) In Eq. (9), ()P J b I e kT is the dimensionless STT parameter , I is the spin -polarized current regarded as positive if electrons flow from the free into the fixed layer, e is the electronic charge, is Planck’s reduced constant , and Pb is a parameter determined by the spin polarization factor P [1]. Accompanying the magnetic Langevin equation (5) (i.e., the stochastic differential equation of the random magnetization process) , one has the Fokker -Planck equation for th e evolution of the associated probability density function ( , , )Wt of orientations of M on the unit sphere, viz., [ 6,12,16 ] X e u Z Y M easy axis H0 fixed layer free layer I eP (a) (b) Fig. 1. (a) Geometry of the problem: A STT device consists of two ferromagnetic strata labelled the free and fixed layers, respectively, and a normal conducting spacer all sandwiched on a pillar between two ohmic contact s [3,6]. Here I is the spin -polarized current, M is the magnetization of the free layer, H0 is the dc bias magnetic field. The magnetization of the fixed layer is directed along the unit vector eP. (b) Free energy potential of the free layer presented in the standard form of superimposed easy -plane and in-plane easy -axis anisotropies, Eq. (7), at = 20 and h = 0.2 . IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 4 FPLWWt , (10) where FPL is the Fokker -Planck operator in phase space ( , ) defined via [6,12,26] 1 N 1 11FP1 ( )L sin2 sin 1 ( ) 1 sin sin ( ) ( )sinWEWW EW EEW (11) and 0 N1 S( ) / (2 ) v M kT is the free diffus ion time of the magnetic moment. If 0= (zero STT), Eq. (10) becomes the original Fokker -Planck equation derived by Brown [33] for magnetic nanoparticles . III. ESCAPE RATES AND REVE RSAL TIME IN THE DAM PING RANGE 1 The magnetization reversal tim e can be calculated exactly by evaluating the smallest nonvanishing eigenvalue 1 of the Fokker -Planck operator L FP in Eq. (10) [32,34 ,42]. Thus 1 is the inverse of the longest relaxation time of the magnetization 11/ , which is usually associated with the reversal time . In the manner of zero STT [42,43], the calculation of 1 can be approximately accomplished using the Mel’nikov -Meshkov formalism [39]. This relies on the fact that in the high barrier and underda mped limit s, one may rewrite the Fokker -Planck equation, Eq. (10), as an energy -action diffusion equation. This in turn is very similar to that for translating point Brownian particles moving along the x-axis in an external potential V(x) [7,17,42] . In the under damp ed case, which is the range of interest, for the escape of spins from a single potential well with a minimum at a point A of the magnetocristalline anisotropy over a single saddle point C, the energy distribution function ()WE for magnetic moments precessing in the potential well can then be found via an integral equation [42], which can be solved for ()WE by the Wiener –Hopf method. Then, the flux -over-population method [33,34] yields the decay ( escape ) rate as 1/CAJN . Here constCJ is the probability current density over the sadd le point and ()C AE AEN W E dE is the well population while the escape rate is rendered as the product of the depopulation factor ( ), CE AS Eq. (3), and the TST escape rate TST AE Efe . In the preceding equation E is the effective spin-polari zed current dependent energy barrier given by 1 AC CE E EAEVdE E E ES , (12) where AE is the energy at the bottom of the potential well, CE is the energy at the saddle point, and the dimensionless action ES and the dimensionless work EV done by the STT are defined as [7,17] EEd SE uuu , (13) EE d Vuuu , (14) respectively. T he contour integrals in Eqs. (13) and (14) are taken along the energy trajectory constE and are to be evaluated in the vanishing damping sense. For the bistable potential, Eq. (7), having two nonequivalent wells A and A with minima ( 1 2 ) Eh at 0A and A , respectively, and two equivalent saddle points C with 2 CEh at cosC h (see Fig. 1(b)) we see that two wells and two escape routes over two saddle points are involved in the relaxation process . Thus, a finite probability for the magnetic dipole to return to the initi al well having already visited the second one exists. This possibility cannot be ignored in the underdamped regime because then the magnetic dipole having entered the second well loses its energy so slowly that even after several precessions, thermal fluctuations may still reverse it back over the potential barrier. In such a situation, on applying the Mel’nikov -Meshkov formalism [39] to the free energy potential, Eq. (7), and the nonconservative potential, Eq. (9), the energy distribution function s ()WE and ()WE for magne tic moments precessing in the two potential well s can then be found by solving two coupled integral equations for ()WE and ()WE . These then yield the depopulation factor , () CCEE A S S via the Mel’nik ov-Meshkov formula for two wells, viz., [39] ( ) ( ) ((), )CC CC CCEE EE EEA S A S A S SA S S . Here ()Az is the depopulation factor for a single well introduced in accordance with Eq. (3) above while CES are the dimensionless action s at the energy saddle point s for two wells. These are to be calculated via Eq. (13) by integrating along the energy trajectories C EE between two saddle points and are explicitly given by 2 3/2 12 21 12(1 ) (1(1 2 a4 (1 rct) )1an )(1 ) )1 (1CCEEh hhhES hd h h uuu (15) (at zero dc bias field, h = 0, these simplify to CCEESS 4 ). Furthermore, the overall TST escape rate TST for a bistable potential, Eq. (7), is estimated via the individual escape rates TST from each of the two wells as TST TSTTST2.EEffee (16) In Eq. (16), the factor 2 occurs because two magnetization escape r outes from each well over the two saddle points exist, while E are the effective spin -polarized current dependent IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 5 barrier heights for two wells (explicit equations for E are derived in Appendix A). In addit ion 01(1 )(1 )2f h h (17) are the corresponding well precession frequencies, where 1 0S 2MD is a precession time constant . Thus, the decay rate 1 becomes 2 2(1 ) ( , )1 0 (1 ) ( , )(1 )(1 ) (( ) ( ) () , 1 )(1 )CC CCJh F hEE EE Jh F hA S A S Ah h e hhS eS (18) where both the functions ( , )Fh occurring in each exponential are given by the analytical formula: 22 2 1 12 2 1 1 2(1 ) (1( , ) 12 2 2 (1 (1 21 arctan (1)(1 ) ) )1 )(1 ) 1 (1 )hhFhhh h hh h h h h (19) and 0.38 is a numerical par ameter (see Eq. (A.6), etc. in Appendix A ). For zero STT, J = 0, Eq. (18) reduces to the known results of the Néel -Brown theory [32,43] for classical magnetic moments with superimposed easy -plane and in -plane easy-axis anisotropies plus the Zeeman term due to the applied magnetic field. In contrast to zero STT, for normalized spin currents J 0, depends on not only through the depopulation factors () CE AS but also through the spin- polarized current dependent effective barrier heights E . This i s so because part s of the arguments of the exponentials in Eq. (18) , namely Eq. (19), are markedly dependent on the ratio /J and the dc bias field parameter. The turnover Eq. (18) also yields a n asymptotic estimate for the inverse of the smallest nonvanishing eigenvalue of the Fokker -Planck operator FPL in Eq. (10). In additio n, one may estimate two individual reversal times, namely, from the deeper well around the energy minimum at 0A and from the shallow well around the energy minimum at A (see Fig. 1(b)) as 2(1 ) ( , ) 02 ( ) (1 )(1 ) CJh F h Ee A S h h . (20) The individual times are in general unequal, i.e., . In deriving Eqs. (18) and (20), all terms of order 22, , ,JJ etc. are neglected. This hypothesis is true only for the underdamped regime , α < 1, and weak spin-polarized currents, J<<1. ( Despite these restrictions as we will see below Eqs. (18) and (20) still yield accurate estimates for for much higher values of J). Now, can also be calculated numerically via the method of statistical moments developed in Ref. [26] whereby t he solution of the Fokker -Planck equation (10) in configuration space is reduced to the task of solving an infinite hierarchy of differential -recurrence equations for the averaged spherical harmonics ( , ) ( )lmYt governing the magnetization relaxation . (The ( , )lmY are the spherical harmonics [46 ], and the angular brackets denote the statistical aver aging ). Thus one can evaluate numerically via 1 of the Fokker -Planck operator L FP in Eq. (10) by using matrix continued fr actions as described in Ref. [47 ]. We remark that the r anges of applicability of the escape rate theory and the matrix continued -fraction method are in a sense complementary because escape rate theory cannot be used for low potential barrie rs, 3E , while the matrix continued - fraction method encounters substantial computation al difficulties for very high potential barriers 25E in the VLD range, 410 . Thus , in the foregoing se nse, numerical methods and escape rate theory are very useful for the determination of τ for low and very high potential barriers, respectively. Nevertheless , in certain (wide) ranges of model parameters both methods yield accurate results for the reversal time ( here these ranges are 5 30, 3, and 410 ). Then the numerically exact benchmark solution provided by the matrix continued fraction method allows one to test the accuracy of the analytical es cape rate equations given above. IV. RESULTS AND DISCUSSIO N Throughout the calculations, the anisotropy and spin - polarization parameters will be taken as 0.034 D , 20 , and 0.3P ( 0.3 0.4P are typical of ferromagnetic metals) just as in Ref. 6. Thus for 5 1 1mA s . 10 , 22 300T K , 24~10v 3m , and a current density of the order of 7~ 10 2A cm in a 3 nm thick layer of cobalt with 61 S 1 1. Am 04 M , we have the following estimates for the anisotropy (or inverse temperature ) parameter 20.2 , characteristic time 1 0S2()MD 0.48 ps, and spin - polarized current parameter ( ) ~1P J b I e kT . In Figs. 2 and 3, we compare from the asymptotic escape rate Eq . (18) with 1 1 of the Fokker –Planck operator as calculated numerically via matrix continued fraction s [26]. Apparently, as rendered by the turnover equation (18) and 1 1 both lie very close to each other in the high barrier limit, where the asymptotic Eq. (18) provides an accurate approximation to 1 1. In Fig. 2, is plotte d as a function of for various J. As far as STT effects are concerned they are governed by the ratio /J so that by altering /J the ensuing variation of may exceed several or ders of magnitude (Fig. 2) . Invariably for J << 1, which is a condition of applicability of the escape rate equations (1) and (18), STT effects on the magnetization relaxation are pronounced only at very low damping, << 1 . For 1 , i.e. high damping, STT influences the reversal process very weak ly because the STT term in Eq. (5) is then small compared to the damping and random field terms . Furthermore, may greatly exceed or, on the other hand, be very much less than the value for zero STT , i.e., J = 0 (see Fig. 2). For example, as J decreases from positive values, IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 6 exponentially increases attaining a maximum at a critical value of the spin -polarized current and then smoothly switches over to exponential decrease as J is further increased through negative values of J [26]. Now, t he temperature , external d.c. bias field, and dam ping dependence of can readily be understood in terms of the effective potential barriers E in Eq. (18). For example, for 5, the temperature dependence of has the customary Arrhenius behavior ~,Ee where E , Eq. (19), is markedly dependent on /J (see Fig. 3 a). Furthermore, the slope of 1()T significantly decreases as the dc bias field parameter h increases due to lowering of the barrier height E owing to the action of the external field (see Fig. 3b) . Now, although the range of applicability of Eqs. (18) and (20) is ostensibly confined to weak spin -polarized currents, J << 1, they can still yield accurate estimates for the reversal time for much higher values of J far exceeding this condition (see Fig. 3 a). Thus , the turnover formula for , Eqs. (18) and (20), bridgi ng the Kramers VLD and ID escape rates as a function of the damping parameter for point particles [35,39 -41] as extended by Coffey et al. [42,43] to the magnetization relaxation in nanoscale ferromagnets allows us (via the further extension to include STT embodied in Eq. (18)) to accurately evaluate STT effects in the magnetization reversal time of a nanomagnet driven by spin -polarized current in the highly relevant ID to VLD damping range. This (underdamped) range is characterized by 1 and the asymptotic escape rates are in complete agreement with independent numerical results [17]. Two particular merits of the escape rate equations for the reversal time are that (i) they are relatively simple ( i.e., expressed via elementary functions) and (ii) that they can be used in those parameter ranges, where numerical methods (such as matrix continued fractions [17]) may be no longer applicable , e.g., for very high barriers , 25E . Hence , one may conclude that the damping dependence of the magnetization reversal time is very marked in the underdamped regime 1 , a fact which may be very significant in int erpreting many STT experiments. V. APPENDIX A: CALCULATION OF ( , )Fh IN EQ. (19) For the bistable potential given by Eq. (7), and the nonconservative potential, Eq. (9), the spin -polarized current dependent effective barrier heights E for each of the two wells are given by (cf. Eq. (12)) 21(1 ) ( , )h J F E h , (A.1) where ( , )C AVFhSd , (A.2) with /E , / 1 2AAEh , 2/CCEh . The dimensionless action S and the dimensionless work done by the STT V for the deeper well can be calculated analytically via elliptic integrals as described in detail in Ref. [17] yielding 2 2 0 2 22(1 ) 1 2 ( )11 ( ) ( ) )(2 1 (1 )( ) () (142),(1 )( ( ( ) ) 1)p Ehd hpf Em hqq q m K m q h q mhpq q mS m Km uuu (A.3) 54321: J = 0.2 2: J = 0.1 3: J = 0 4: J = 0.1 5: J = 0.2/ h =0.15 =20 = 201 Fig. 2. Reversal time 0/ vs the damping parameter for various values of the spin-polarized current parameter J. Solid lines : numerical calculations of the inverse of t he smallest nonvanishing eigenvalue 1 01() of the Fokker –Planck operator , Eq. (11). Asterisks: the turnover formula, Eq. (18). 54 / 3211: J = 1 2: J = 3: J = 4: J = 5: J = h = 0.1 = 0.01 = 20 (a) (b) 4 / 321 1: h = 0.0 2: h = 0.1 3: h = 0.2 4: h = 0.3 = 0.01 = 20 J = Fig. 3 . Reversal time 0/ vs. the anisotropy (inverse temperature) parameter for various spin-polarized currents J (a) and dc bias field parameters h (b). Solid lines: numerical solution for the inverse of the smallest nonvanishing eigenvalue 1 01() of the Fokker –Planck operator , Eq. (11). Asterisks: Eq. (18). IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 7 0 2 23 2 2( 1) 1 ( 1) ( | )1 2 21 2 1() ( ) (1 (|)121 ( ) (,))phV m m qhdhf q hpK hp q E m m qqq q m K mm ueu (A.4) where 2 2 21( 1)ph , 1 1eqe , 1 (1 ) (1 ) 1eemee , 2 ( 1)hepph , ()Km , ()Em , and ( | )am are the complete elliptic integrals of the fir st, seco nd, and third kinds, respectively [48], and f is the precession frequency in the deeper well at a given energy, namely, 0( 1)(1 ())(1 ) 8p e efKm . (A.5) The quantities S , V , and f for the shallower well are obtained simply by replacing the dc bias field parameter h by h in all the equations for S , V , and f . We remark that S and V in Eqs. (A.3) and (A.4) differ by a factor 2 from those given in Ref. [17]. This is so because S and V are now calculated between the saddle points and not over the precession period . When ( , )C , S in Eqs. (A.3) reduces to CES , Eq. (15). In the parameter ranges 01h and 1 , the integral in Eq. (A.2) can be accurately evaluated analytically using an interpolation function for /VS between t he two limiting values / AAVS and / CCVS at 1A h and 2 Ch , namely 11 C AA A C AA CAV VV V S S S S , (A.6) where 0.38 is an interpolation parameter yielding the best fit of /VS in the interval .AC These limiting values can be calculated from Eqs. (A.3) and (A.4) yielding after tedious algebra: 1 22A AV h S (A.7) and 2 2 1 12 2 1 1 2)(1 ) 1 (112 (1 (1 21 arct) )1an (1 (1 )(1 ) ) 1C CV h h Sh hh h h h h . (A.8) Hence with Eqs. (A.2) and (A.6), we have a simple a nalytic formula for the current -dependent parts of the exponentials in Eq. 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Meshkov, “Theory of activated rate processes: exact solution of the Kramers problem ”, J. Chem. Phys ., vol. 85, p. 1018 , 1986. [40] H. Grabert, “Escape from a metastable well: The Kramers turnover problem ”, Phys. Rev. Lett. , vol. 61, p. 1683 , 1988 . [41] E. Pollak, H. Grabert, and P. Hänggi, “Theory of activated rate processes for arbitrary frequency dependent friction: Solution of the turnover problem ”, J. Chem. Phys ., vol. 91, p. 4073 , 1989 . [42] W. T. Coffey, D. A. Garanin, and D. J. McCarthy, “Crossover formulas in the Kramers theory of thermally activated escape rates – application to spin systems ”, Adv. Chem. Phys . vol. 117, p. 483, 2001 ; P.M. Déjardin, D .S.F. Crothers, W.T. Coffey, and D.J. McCarthy, “Interpolation formula between very low and intermediate -to-high damping Kramers escape rates for single -domain ferromagnetic particles ”, Phys. Rev. E , vol. 63, p. 021102 , 2001 . [43] Yu. P. Kalmykov, W.T. Coffey, B. Ouari, and S. V. Titov, “Damping dependence of the magnetization relaxation time of single -domain ferromagnetic particles ”, J. Magn. Magn. Mater ., vol. 292, p. 372, 2005 ; Yu. P. Kalmykov and B. Ouari, “Longitudinal complex magnetic susceptibility and re laxation times of superparamagnetic particles with triaxial anisotropy ”, Phys. Rev. B , vol. 71, p. 094410 , 2005 ; B. Ouari and Yu. P. Kalmykov , “Dynamics of the magnetization of single domain particles having triaxial anisotropy subjected to a uniform dc ma gnetic field”, J. Appl. Phys ., vol. 100, p. 123912 , 2006 . [44] T. L. Gilbert, “A Lagrangian formulation of the gyromagnetic equation of the magnetic field ”, Phys. Rev ., vol. 100, p. 1243 , 1955 (Abstract only; full report in: Armour Research Foundation Project N o. A059, Supplementary Report, 1956). Reprinted in T.L. Gilbert , A phenomenological theory of damping in ferromagnetic materials, IEEE Trans. Magn ., vol. 40, p. 3443 , 2004 . [45] J. Z. Sun, “Spin-current interaction with a monodomain magnetic body: A model study ”, Phys. Rev. B , vol. 62, p. 570, 2000 . [46] D. A. Varshalovitch, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum . World Scientific, Singapore, 1988 . [47] Y.P. Kalmykov, “Evaluation of the smallest nonvanishing eigenvalue of the Fokker -Planck equation for the Brownian motion in a potential. II. The matrix continued fraction approach ”, Phys. Rev. E , vol. 62, p. 227, 2000. [48] M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical Functions . Dover, New York, 1964 . IEEE TRANSACTIONS ON MAGNETICS, VOL. 53, NO. 10, OCTOBER 2017 9

2017-03-06

Thermal fluctuations of nanomagnets driven by spin-polarized currents are treated via the Landau-Lifshitz-Gilbert equation as generalized to include both the random thermal noise field and Slonczewski spin-transfer torque terms. The magnetization reversal time of such a nanomagnet is then evaluated for wide ranges of damping by using a method which generalizes the solution of the so-called Kramers turnover problem for mechanical Brownian particles, thereby bridging the very low damping and intermediate damping Kramers escape rates, to the analogous magnetic turnover problem. The reversal time is then evaluated for a nanomagnet with the free energy density given in the standard form of superimposed easy-plane and in-plane easy-axis anisotropies with the dc bias field along the easy axis.

Damping dependence of spin-torque effects in thermally assisted magnetization reversal

1703.01879v5

Modeling coupled spin and lattice dynamics Mara Strungaru,1Matthew O A Ellis,2Sergiu Ruta,3Oksana Chubykalo-Fesenko,4Richard F L Evans,1and Roy W Chantrell1 1Department of Physics, University of York, York, United Kingdom 2Department of Computer Science, University of Sheffield, Sheffield, United Kingdom 3Department of Physics,University of York, York, United Kingdom 4Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid, Spain A unified model of molecular and atomistic spin dynamics is presented enabling simulations both in micro- canonical and canonical ensembles without the necessity of additional phenomenological spin damping. Trans- fer of energy and angular momentum between the lattice and the spin systems is achieved by a coupling term based upon the spin-orbit interaction. The characteristic spectra of the spin and phonon systems are analyzed for different coupling strength and temperatures. The spin spectral density shows magnon modes together with the uncorrelated noise induced by the coupling to the lattice. The effective damping parameter is investigated showing an increase with both coupling strength and temperature. The model paves the way to understanding magnetic relaxation processes beyond the phenomenological approach of the Gilbert damping and the dynamics of the energy transfer between lattice and spins. I. INTRODUCTION With the emergent field of ultrafast magnetisation dynamics1understanding the flow of energy and angular mo- mentum between electrons, spins and phonons is crucial for the interpretation of the wide range of observed phenom- ena2–5. For example, phonons strongly pumped in the THz regime by laser excitation can modulate the exchange field and manipulate the magnetisation as shown for the magnetic insulator YIG6or in Gd7. The excitation of THz phonons leads to a magnetic response with the same frequency in Gd7, proving the necessity of considering the dynamics of both lattice and spins. Phonon excitations can modulate both anisotropy and exchange which can successfully manipulate 8–10or potentially switch the magnetisation11,12, ultimately leading to the development of low-dissipative memories. Magnetisation relaxation is typically modeled using the phenomenological description of damping proposed by Lan- dau and Lifshitz13and later Gilbert14, where the precessional equation of motion is augmented by a friction-like term, re- sulting in the Landau-Lifshitz-Gilbert (LLG) equation. This represents the coupling of the magnetic modes (given pri- marily by the localised atomic spin) with the non-magnetic modes (lattice vibrations and electron orbits). The LLG equa- tion and its generalisations can be deduced from the quantum- mechanical approaches assuming an equilibrium phonon bath and the weak coupling of the spin to the bath degrees of freedom15–17. Thus the standard approach works on the sup- position that the time scales between the environmental de- grees of freedom and the magnetic degrees of freedom are well separated and reducing the coupling between the mag- netization and its environment to a single phenomenological damping parameter18,19. In reality, the lattice and magneti- sation dynamics have comparable time-scales, where the in- teraction between the two subsystems represents a source of damping, hence the necessity of treating spin and lattice dy- namics in a self-consistent way. To investigate these phenomena, and aiming at predictive power for the design of competitive ultrafast magnetic nano- devices, advanced frameworks beyond conventional micro-magnetics and atomistic spin dynamics20are needed21. A complete description of magnetic systems includes the inter- action between several degrees of freedom, such as lattice, spins and electrons, modeled in a self-consistent simulation framework. The characteristic relaxation timescales of elec- trons are much smaller ( fs) in comparison to spin and lattice (100fsps), hence magnetisation relaxation processes can be described via coupled spin and lattice dynamics, termed Spin- Lattice Dynamics (SLD)22–29. SLD models can be crucial in disentangling the interplay between these sub-systems. SLD models have so far considered either micro-canonical (NVE - constant particle number, volume and energy)27,28or canonical (NVT - constant particle number, volume and tem- perature) ensembles with two Langevin thermostats connected to both lattice and spin subsystems23,30. Damping due to spin- lattice interactions only within the canonical ensemble (NVT) has not yet been addressed, but is of interest in future mod- elling of magnetic insulators at finite temperature. Here we introduce a SLD model capable of describing both ensem- bles. Specifically, our model (i) takes into account the transfer of angular momentum from spin to lattice and vice-versa, (ii) works both in a micro-canonical ensemble (constant energy) and in a canonical ensemble (constant temperature), (iii) al- lows a fixed Curie temperature of the system independent of the spin-lattice coupling strength, (iv) disables uniform trans- lational motion of the system and additional constant energy drift, which can be produced by certain spin-lattice coupling forms. Furthermore, in this work, the characteristics of the in- duced spin-lattice noise, the magnon-phonon induced damp- ing and the equilibrium properties of the magnetic system has been systematically investigated. The paper is organised as follows. We start by describ- ing the computational model of Spin-Lattice Dynamics and the magnetic and mechanical energy terms used in this frame- work (Section II). We then explore the equilibrium properties of the system for both microcanonical and canonical simula- tions, proving that our model is able to efficiently transfer both energy and angular momentum between the spin and lattice degrees of freedom. In Section III we compute the equilibrium magnetisation as function of temperature for both a dynamic and static lattice and we show that the order parameter is notarXiv:2010.00642v1 [cond-mat.mtrl-sci] 1 Oct 20202 dependent on the details of the thermostat used. In Section IV we analyse the auto-correlation functions and spectral char- acteristics of magnon, phonons and the coupling term prov- ing that the pseudo-dipolar coupling efficiently mediates the transfer of energy from spins to the lattice and vice-versa. We then calculate the temperature and coupling dependence of the induced magnon-phonon damping and we conclude that the values agree well with damping measured in magnetic insula- tors, where the electronic contributions to the damping can be neglected ( Section V). II. COMPUTATIONAL MODEL In order to model the effects of both lattice and spin dy- namics in magnetic materials an atomistic system is adopted with localised atomic magnetic moments at the atomic coor- dinates. Within this framework there are now nine degrees of freedom; atomic magnetic moment (or spin) S, atomic posi- tionrand velocity v. The lattice and the magnetic system can directly interact with each other via the position and spin de- pendent Hamiltonians. The total Hamiltonian of the system consists of a lattice Hlatand magnetic Hmagparts: Htot=Hlat+Hmag: (1) The lattice Hamiltonian includes the classical kinetic and pairwise inter-atomic potential energies: Hlat=å imiv2 i 2+1 2å i;jU(ri j): (2) Our model considers a harmonic potential (HP) defined as: U(ri j) =( V0(ri jr0 i j)2=a2 0ri j<rc 0 ri j>rc:(3) where V0has been parametrised for BCC Fe in27anda0=1˚A is a dimension scale factor. To be more specific we consider the equilibrium distances r0 i jcorresponding to a symmetric BCC structure. The interaction cut-off is rc=7:8˚A. The pa- rameters of the potential are given in Table II. The harmonic potential has been used for simplicity, however it can lead to rather stiff lattice for a large interaction cutoff. Another choice of the potential used in our model is an an- harmonic Morse potential (MP) parameterised in31for BCC Fe and defined as: U(ri j) =( D[e2a(ri jr0)2ea(ri jr0)]ri j<rc 0 ri j>rc(4) The Morse potential approximates well the experimental phonon dispersion observed experimentally for BCC Fe32as shown in33. The phonon spectra for the choices of potential used in this work are given in Section IV. Other nonlinear choices of potential can be calculated via the embedded atom method34,35.The spin Hamiltonian ( Hmag) used in our simulations con- sists of contributions from the exchange interaction, Zeeman energy and a spin-lattice coupling Hamiltonian, given by the pseudo-dipolar coupling term ( Hc), which we will describe later: Hmag=1 2å i;jJ(ri j)(Si
Sj)å imiSi
Happ+Hc;(5) where miis the magnetic moment of atom i,Siis a unit vector describing its spin direction and Happis an external ap- plied magnetic field. The exchange interactions used in our simulations depend on atomic separation J(ri j). They were calculated from first principle methods for BCC Fe by Ma et al23and follow the dependence: J(ri j) =J0 1ri j rc3 Q(rcri j); (6) where rcis the cutoff and Q(rcri j)is the Heaviside step function, which implies no exchange coupling between spins situated at larger distance than rc. Several previous SLD models suffered from the fact that they did not allow angular momentum transfer between lattice and spin systems28. This happened for magnetisation dynam- ics in the absence of spin thermostat, governed by symmetric exchange only, due to total angular momentum conservation. To enable transfer of angular momentum, Perera et al26have proposed local anisotropy terms to mimic the spin-orbit cou- pling phenomenon due to symmetry breaking of the local en- vironment. Their approach was successful in thermalising the subsystems, however, single site anisotropy spin terms with a position dependent coefficients as employed in26can induce an artificial collective translational motion of the sample while the system is magnetically saturated, due to the force ¶Htot ¶riproportional to spin orientation. To avoid large collective mo- tion of the atoms in the magnetic saturated state, we consider a two-site coupling term, commonly known as the pseudo- dipolar coupling, described by Hc=å i;jf(ri j) (Si
ˆri j)(Sj
ˆri j)1 3Si
Sj : (7) The origin of this term still lies in the spin-orbit interaction, appearing from the dynamic crystal field that affects the elec- tronic orbitals and spin states. It has been employed previ- ously in SLD simulations22,27. It was initially proposed by Akhiezer36, having the same structure of a dipolar interac- tion, however with a distance dependence that falls off rapidly, hence the name pseudo-dipolar interaction. The exchange- like term1 3Si
Sjis necessary in order to preserve the Curie temperature of the system under different coupling strengths and to ensure no net anisotropy when the atoms form a sym- metric cubic lattice. For the mechanical forces, the exchange like term eliminates the anisotropic force that leads to a non- physical uniform translation of the system when the mag- netic system is saturated. The magnitude of the interactions3 is assumed to decay as f(ri j) =CJ0(a0=ri j)4as presented in27with Ctaken as a constant, for simplicity measured rel- ative to the exchange interactions and a0=1˚A is a dimen- sion scale factor. The constant Ccan be estimated from ab-initio calculations26, approximated from magneto-elastic coefficients27, or can be chosen to match the relaxation times and damping values, as in this work. Since the total Hamiltonian now depends on the coupled spin and lattice degrees of freedom ( vi,ri,Si), the following equations of motion (EOM) need to be solved concurrently to obtain the dynamics of our coupled system: ¶ri ¶t=vi; (8) ¶vi ¶t=hvi+Fi mi; (9) ¶Si ¶t=gSiHi; (10) Fi=¶Htot ¶ri+Gi; (11) Hi=1 mSm0¶Htot ¶Si; (12) where FiandHirepresent the effective force and field, Girep- resents the fluctuation term (thermal force) and hrepresents the friction term that controls the dissipation of energy from the lattice into the external thermal reservoir. The strength of the fluctuation term can be calculated by converting the dissi- pation equations into a Fokker-Planck equation and then cal- culating the stationary solution. The thermal force has the form of a Gaussian noise: hGia(t)i=0; (13) hGia(t)Gjb(t0)i=2hkBT midabdi jd(tt0): (14) To prove that the complete interacting many-body spin- lattice framework presented in here is in agreement with the fluctuation-dissipation theorem, we have followed the ap- proach presented by Chubykalo et al37based on the Onsager relations. Linearising the equation of motion for spins, we find that the kinetic coefficients for the spin system are zero, due to the fact that the spin and internal field are thermodynamic conjugate variables. Hence, if the noise applied to the lattice obeys the fluctuation dissipation theory, the coupled system will obey it as well, due to the precessional form of the equa- tion of motion for the spin. We compare the SLD model presented here with other ex- isting model that do not take into account the lattice degrees of freedom (Atomistic Spin Dynamics - ASD). Particularly, in our case we assume a fixed lattice positions. The summary of the comparison is presented in Table I. Atomistic spin dy- namics simulations (ASD)18,20,38,39have been widely used to study finite size effects, ultrafast magnetisation dynamics and numerous other magnetic phenomena. Here the intrinsic spin damping (the Gilbert damping - aG) is phenomenologically included. In our case since the lattice is fixed it is assumedModel Lattice Lattice Spin Intrinsic Spin thermostat thermostat damping SLD Dynamic On Off Phonon induced ASD Fixed Off On Electronic mainly TABLE I. Summary comparison of the SLD model developed here against other spin dynamics models. Quantity Symbol Value Units Exchange23J0 0:904 eV rc 3:75 ˚A Harmonic potential27V0 0:15 eV rc 7:8 ˚A Morse potential31D 0:4174 eV a 1:3885 ˚A r0 2:845 ˚A rc 7:8 ˚A Magnetic moment ms 2:22 mB Coupling constant C 0:5 Mass m 55:845 u Lattice constant a 2:87 ˚A Lattice damping h 0:6 s1 TABLE II. Parameters used in the spin-lattice model to simulate BCC Fe. to come from electronic contributions. Consequently, only 3 equations of motion per atom describing the spin dynamics are used: ¶Si ¶t=g (1+a2 G)Si(Hi+aGSiHi) (15) with an additional field coming from the coupling to the fixed lattice positions. The temperature effects are introduced in spin variables by means of a Langevin thermostat. The spin thermostat is modeled by augmenting the effective fields by a thermal stochastic field ( Hi=xi¶H=¶Si) and its proper- ties also follow the fluctuation-dissipation theorem: hxia(t)i=0; (16) hxia(t)xjb(t0)i=2aGkBT gmSda;bdi jd(tt0): (17) The characteristics of the above presented models are sum- marised in Table I. To integrate the coupled spin and lattice equations of mo- tion we used a Suzuki-Trotter decomposition (STD) method40 known for its numerical accuracy and stability. The scheme can integrate non-commuting operators, such as is the case of spin-lattice models and conserves the energy and space-phase volume. The conservation of energy is necessary when deal- ing with microcanonical simulations. Considering the gener- alized coordinate X=fr;v;Sgthe equations of motion can be4 re-written using the Liouville operators: ¶X ¶t=ˆLX(t)= (ˆLr+ˆLv+ˆLS)X(t): (18) The solution for the Liouville equation is X(t+Dt) = eLDtX(t). Hence, following the form of this solution and ap- plying a Suzuki-Trotter decomposition as in Tsai’s work41,42, we can write the solution as: X(t+Dt) =eˆLsDt 2eˆLvDt 2eˆLrDteˆLvDt 2eˆLsDt 2X(t)+O(Dt3);(19) where Ls;Lv;Lrare the Liouville operators for the spin, veloc- ity and position. This update can be abbreviated as (s,v,r,v,s) update. The velocity and position are updated using a first order update, however the spin needs to be updated using a Cayley transform43,44, due to the fact that the norm of each individual spin needs to be conserved. Thus we have eˆLvDtvi=vi+Dt miFi; (20) eˆLrDtri=ri+Dtvi; (21) eˆLSDtSi=Si+DtHiSi+Dt2 2 (Hi
Si)Hi1 2H2 iSi 1+1 4Dt2H2 i:(22) The spin equations of motions depend directly on the neigh- bouring spin orientations (through the effective field) hence individual spins do not commute with each other. We need to further decompose the spin system ˆLs=åiˆLsi. The following decomposition will be applied for the spin system: eˆLs(Dt=2)=eˆLs1(Dt=4):::eˆLsN(Dt=2):::eˆLs1(Dt=4)+O(Dt3)(23) Tests of the accuracy of the integration have been per- formed by checking the conservation of energy within the mi- crocanonical ensemble. To ensure that the spin and lattice sub-systems have reached equilibrium, we calculate both the lattice temperature (from the Equipartition Theorem) and spin temperature45. These are defined as: TL=2 3NkBå ip2 i 2m;TS=åi(SiHi)2 2kBåiSi
Hi: (24) III. SPIN-LATTICE THERMALISATION As an initial test of our model we look at the thermalisation process within micro-canonical (NVE) and canonical (NVT) simulations for a periodic BCC Fe system of 101010 unit cells. No thermostat is applied directly to the spin system and its thermalisation occurs via transfer of energy and angular momentum from the lattice, i.e. via the magnon-phonon inter- action. In the case of the NVE simulations, the energy is de- posited into the lattice by randomly displacing the atoms from an equilibrium BCC structure positions within a 0 :01˚A radius sphere and by initialising their velocities with a Boltzmann FIG. 1. NVE (top) and NVT (bottom) simulations for a 10 1010 unit cell BCC Fe system. The spin system is randomly initialised with a temperature of 1900 K, while the lattice velocities are initialised by a Boltzmann distribution at T=300 K. In both cases we obtain equilibration of the two subsystems on the ps timescale. distribution at T=300 K. The spin system is initialised ran- domly in the xyplane with a constant component of mag- netisation of 0.5 in the out of plane ( z) direction. In the case of NVT simulations, the lattice is connected to a thermostat at a temperature of T=300 K. The parameters used in the simulations are presented in Table II. Fig. 1 shows the thermalisation process for the two types of simulation. In both cases the spin system has an initial temper- ature of T=1900 K due to the random initialisation. For the NVE simulations, the two subsystems are seen to equilibrate at a temperature of T=600 K, this temperature being depen- dent on the energy initially deposited into the system. In the NVT simulations, the lattice is thermalised at T=300 K fol- lowed by the relaxation of the spin towards the same temper- ature. In both cases we observe that the relaxation of the spin system happens on a 100 ps timescale, corresponding to typi- cal values for spin-orbit relaxation. The corresponding change in the magnetisation is emphasized by the green lines in Fig. 1 showing a transfer of angular momentum between the spin and lattice degrees of freedom. To gain a better understanding of properties at thermal equi- librium within the Spin-Lattice Dynamics model, we have in- vestigated the temperature dependence of the magnetic order parameter in different frameworks that either enable or dis- able lattice dynamics, specifically: SLD or ASD. Tab. I il- lustrates the differences between the models. Since reaching joint thermal equilibrium depends strongly on the randomness already present in the magnetic system this process is acceler- ated by starting with a reduced magnetisation of M=MS=0:95 FIG. 2. Magnetisation versus temperature curves for the SLD model (with different choices of lattice potential: MP-Morse Potential, HP- Harmonic Potential) and fixed lattice ASD model. The inset zooms around the ferromagnetic to paramagnetic phase transition tempera- ture. forT>300 K. Fig. 2 shows the comparison of the equilibrium magnetisa- tion using either the harmonic potential (HP), Morse potential (MP) or fixed lattice (ASD) simulations. The magnetisation is calculated by averaging for 200 ps after an initial equilibra- tion for 800 ps (for SLD type simulations) or 100 ps (for ASD) simulations. We observe that even without a spin thermostat (in SLD model) the magnetisation reaches equilibrium via the thermal fluctuations of the lattice, proving that both energy and angular momentum can be successfully transferred be- tween the two sub-systems. Additionally, both the SLD and ASD methods give the same equilibrium magnetisation over the temperature range considered. This confirms that the equi- librium quantities are independent of the details of the thermo- stat used, in agreement with the fact that both SLD and ASD models obey the fluctuation-dissipation theorem. In principle, since the strength of the exchange interaction depends on the relative separation between the atoms, any thermal expansion of the lattice could potentially modify the Curie temperature. However, as highlighted in the inset of Fig. 2, the same Curie temperature is observed in each model. We attribute this to fact that the thermal lattice expansion is small in the temperature range considered due to two reasons: i) the Curie temperature of the system is well below the melt- ing point of Fe (1800K) and ii) we model a bulk, constant- volume system with periodic boundary conditions that does not present strong lattice displacements due to surfaces. We note that Evans et al46found a reduction of TCin nanoparticles due to an expansion of atomic separations at the surface that consequently reduces the exchange interactions. For systems with periodic boundary conditions we anticipate fluctuations in the exchange parameter due to changes in interatomic spac- ings to be relatively small. Although the equilibrium proper- ties are not dependent on the details of the thermostat or thepotential, the magnetisation dynamics could be strongly influ- enced by these choices. The strength of the pseudo-dipolar coupling parameters C determines the timescale of the thermalisation process. Its value can be parametrised from magneto-elastic simulations via calculations of the anisotropy energy as a function of strain. The magneto-elastic Hamiltonian can be written for a continuous magnetisation Mand elastic strain tensor eas47,48: Hme=B1 M2 Så iM2 ieii+B2 M2så iMiMjei j (25) where constants B1;B2can be measured experimentally49. The pseudo-dipolar term acts as a local anisotropy, however, for a lattice distorted randomly, this effective anisotropy is av- eraged out to zero. At the same time under external strain effects, an effective anisotropy will arise due to the pseudo- dipolar coupling which is the origin of the magneto-elastic effects. To calculate the induced magnetic anisotropy energy (MAE), the BCC lattice is strained along the zdirection whilst fixed in the xyplane. The sample is then uniformly rotated and the energy barrier is evaluated from the angular dependence of the energy. Fig. 3 shows MAE for different strain values and coupling strengths, with the magneto-elastic energy densities constants B1obtained from the linear fit. The values of the ob- tained constants B1are larger than the typical values reported forBCC FeB1=3:43 MJ m3=6:2415106eV A349 measured at T=300 K. Although the obtained magneto- elastic coupling constants for BCC Fe are larger than experi- mental values, it is important to stress that, as we will see later, a large coupling is necessary in order to obtain damping pa- rameters comparable to the ones known for magnetic insula- tors where the main contribution comes from magnon-phonon scattering. In reality, in BCC Fe there is an important contribu- tion to the effective damping from electronic sources, which if considered, can lead to the smaller coupling strengths, consis- tent in magnitude with experimental magneto-elastic parame- ters. Indeed, as we will show later, our finding suggests that phonon damping is a very small contribution in metallic sys- tems such as BCC Fe . IV . DYNAMIC PROPERTIES AT THERMAL EQUILIBRIUM Section III showed that the equilibrium magnetisation does not depend on the details of the thermostat used and a success- ful transfer of both energy and angular momentum is achieved between the spin and lattice sub-systems by the introduction of a pseudo-dipolar coupling term. In this section, we inves- tigate the properties of the magnons, phonons and the cou- pling term that equilibrates the spin and phonon systems in the absence of a phenomenological spin damping. Two types of simulations are presented here: i)magnon and phonon spectra calculated along the high symmetry path of a BCC lattice and ii)averaged temporal Fourier transform (FT) of individual atoms datasets (spin, velocity, pseudo-dipolar cou- pling field). The phonon - Fig. 4 and magnon - Fig. 5 spec-6 FIG. 3. Magnetic anisotropy energy as function of strain for different coupling strengths for T=0K. tra are calculated by initially equilibrating the system for 10 ps with a spin thermostat with aG=0:01 and a coupling of C=0:5, followed by 10 ps of equilibration in the absence of a spin thermostat. For the method i)the correlations are computed for a runtime of 20 ps after the above thermalisa- tion stage. For each point in k-space, the first three maxima of the auto-correlation function are plotted for better visual- isation. The auto-correlation function is projected onto the frequency space and the average intensity is plotted for dif- ferent frequencies. The phonon spectra are calculated from the velocity auto-correlation function defined in Fourier space as33,50: Ap(k;w) =Ztf 0hvp k(t)vp k(tt0)ieiwtdt (26) where p=x;y;z,tfis the total time and vp k(t)is the spatial Fourier Transform calculated numerically as a discrete Fourier Transform: vp k(t) =å ivp ieik
ri (27) The same approach is applied for the magnon spectra, us- ing the dynamical spin structure factor, which is given by the space-time Fourier transform of the spin-spin correlation function defined as Cmn(rr0;tt0) =<Sm(r;t)Sn(r0;t0)>, with m;ngiven by the x,y,z components51: Smn(k;w) =å r;r0eik
(rr0)Ztf 0Cmn(rr0;tt0)eiwtdt(28) The second method ( ii)) to investigate the properties of the system involves calculating temporal Fourier transform of in- dividual atoms datasets, and averaging the Fourier response over 1000 atoms of the system. This response represents an integrated response over the k-space. Hence, the projection of intensities on the frequency space presented by method i)has similar features as the spectra presented by method ii). For theresults presented in Fig. 6, a system of 10 1010BCC unit cells has been chosen. The system has been equilibrated for a total time of 20 ps with the method presented in i)and the Fast Fourier transform (FFT) is computed for the following 100 ps. Fig. 4 shows the phonon spectra for a SLD simulations at T=300K, C=0:5 for the Morse Potential - Fig. 4(a) and the Harmonic Potential - Fig. 4(b) calculated for the high sym- metry path of a BCC system with respect to both energy and frequency units. The interaction cutoff for both Morse and Harmonic potential is rc=7:8˚A. The Morse phonon spec- trum agrees well with the spectrum observed experimentally32 and with the results from33. The projection of the spectra onto the frequency domain shows a peak close to 10.5 THz, due to the overlap of multiple phonon branches at that frequency and consequently a large spectral density with many k-points excited at this frequency. Moving now to the harmonic poten- tial, parameterised as in Ref. 27, we first note that we observe that some of the phonon branches overlap - Fig.4b). Secondly, the projection of intensity onto the frequency domain shows a large peak at 8.6THz, due to a flat region in the phonon spec- tra producing even larger number of k-points in the spectrum which contribute to this frequency. Finally, the large cutoff makes the Harmonic potential stiffer as all interactions are defined by the same energy, V0, and their equilibrium posi- tions corresponding to a BCC structure. This is not the case for the Morse potential which depends exponentially on the difference between the inter-atomic distance and a constant equilibrium distance, r0. For a long interaction range, the har- monic approximation will result in a more stiff lattice than the Morse parameterisation. In principle, the harmonic potential with a decreased in- teraction cutoff and an increased strength could better repro- duce the full phonon spectra symmetry for BCC Fe. How- ever, in this work we preferred to use the parameterisation existing in literature27and a large interaction cutoff for sta- bility purposes. Although the full symmetry of the BCC Fe phonon spectra is not reproduced by this harmonic potential, the phonon energies/frequencies are comparable to the values obtained with the Morse potential. Nevertheless, we observed the same equilibrium magnetisation and damping (discussed later) for both potentials, hence the simple harmonic potential represents a suitable approximation, that has the advantage of being more computationally efficient. Fig. 5 shows the magnon spectrum obtained within the SLD framework using the Morse potential together with its pro- jection onto the frequency domain. The results agree very well with previous calculations of magnon spectra28,52. For the harmonic potential the magnon spectrum is found to be identical to that for the Morse potential with only very small changes regarding the projection of intensity onto the fre- quency domain. This is in line with our discussion in the pre- vious section where the choice of inter-atomic potential had little effect on the Curie temperature, which is closely linked to the magnonic properties. As the harmonic potential is more computationally efficient than the Morse, we next analyse the properties of the system for a 10 1010 unit cells system atT=300K with the harmonic potential.7 FIG. 4. Phonon spectra calculated for a 32 3232 unit cell system at T=300K, C=0:5 for a) Morse potential, b)Harmonic potential. The spectra are calculated via method i). Right figure includes the projection of the intensity of the spectra onto the frequency domain. Solid lines are the experimental data of Minkiewicz et al32. For the Minkiewicz et al data there is only 1 datapoint for the N- Gpath for the second transverse mode which does not show up on the line plots. FIG. 5. Magnon spectrum (x component) calculated for a 32 32 32 unit cell system at T=300K, C=0:5 for a Morse potential. The spectrum is calculated via method i). Right figure includes the projection of the intensity of the spectrum onto the frequency domain. The power spectral density (auto-correlation in Fourier space) of the magnon, phonons and coupling field at 300 K is shown in Fig. 6 computed using method iidetailed previ- ously. The amplitude of the FFT spectra of velocities and coupling field has been scaled by 0.12 and 0.05 respectively to allow for an easier comparison between these quantities. As shown in Fig. 6.a) the coupling term presents both magnon and phonon characteristics; demonstrating an efficient cou- pling of the two sub-systems. The large peak observed at a frequency of 8 :6 THz appears as a consequence of the flat phonon spectrum for a Harmonic potential, as observed in the spectrum and its projection onto the frequency domain in Fig. 4.b). Additionally, Fig. 6.a) can give us an insight into the in-duced spin noise within the SLD framework. The background of the FFT of the coupling field is flat for the frequencies plot- ted here, showing that the noise that acts on the spin is uncor- related. The inset shows a larger frequency domain where it is clear that there are no phonon modes for these frequencies, and only thermal noise decaying with frequency is visible. At the same time an excitation of spin modes are visible for fre- quencies up to ca .100 THz. The characteristics of the coupling field with respect to the coupling strength for a dynamic (SLD) and fixed lattice simu- lations (ASD) are presented in Fig. 6(b). The only difference between the ASD and SLD simulations is given by the pres- ence of phonons (lattice fluctuations) in the latter. Since the large peak at 8 :6 THz is due to the lattice vibrations, it is not present in the ASD simulations. The smaller peaks are present in both models since they are proper magnonic modes. With increasing coupling the width of the peaks increases suggest- ing that the magnon-phonon damping has increased. Moving towards the larger frequency regimes, Fig. 6.b) - (inset), we observe that large coupling gives rise to a plateau in the spec- tra at around 150 THz, which is present as well for the fixed- lattice simulations (ASD). The plateau arises from a weak an- tiferromagnetic exchange that appears at large distances due to the competition between the ferromagnetic exchange and the antiferromagnetic exchange-like term in the pseudo-dipolar coupling. We have also analysed the characteristics of the magnon and phonon spectra for different temperatures- Fig. 7. With8 FIG. 6. The power spectral density of the auto-correlation function in the frequency domain for magnons, phonons and coupling field for a SLD simulations with a Harmonic lattice, calculated by method ii). Panel a) shows the power density of the auto-correlation function of the x component of the velocity vx, spin Sxand coupling field Hcx. Panel b) presents the power density of the auto-correlation function for the x component of the coupling field for either static (ASD) or dynamic (SLD) lattice. The insets show the high-frequency spectra. For Panel a) the velocity and the coupling field have been multiplied by a factor of 0.12 and 0.05 respectively for easier graphical comparison. FIG. 7. The power spectral density of the auto-correlation function in the frequency domain for magnons - Panel a) and phonons - Panel b) for a SLD simulations with a Harmonic lattice, calculated by method ii), for three distinct temperatures and a coupling constant of C=0:5. increasing temperature, the peaks corresponding to magnons shift to smaller frequencies. This is a typical situation known as a softening of low-frequency magnon modes due to the in- fluence of thermal population, see e.g.53- Panel a). The same effect can be observed by calculating the magnon spectra via method ifor various temperatures. In Panel b), the peak cor- responding to phonons remains almost at the same frequency of about 8 :6 THz, as the phonon spectra is not largely affected by temperatures up to T=600K. The increase of the effec- tive damping (larger broadening) of each magnon mode with temperature is clearly observed. V . MACROSOPIC MAGNETISATION DAMPING In this section we evaluate the macroscopic damping pa- rameter experienced by magnetisation due to the magnon- phonon excitations for a periodic BCC system using our SLD model. This method for calculating the damping has been presented in54–56. The system is first thermalised at a non- zero temperature in an external field of Bext=50T applied in thezdirection, then the magnetisation is rotated coherently through an angle of 30. The system then relaxes back to equilibrium allowing the relaxation time to be extracted. The averaged zcomponent of magnetisation is then fitted to the function mz(t) =tanh(agBext(t+t0)=(1+a2))where arep-resents the macrosopic (LLG-like) damping, gthe gyromag- netic ratio and t0a constant related to the initial conditions. The model system consists of 10 1010 unit cells and the damping value obtained from fitting of mz(t)is averaged over 10 different simulations. Fig. 8 shows the dependence of the average damping pa- rameter together with the values obtained from individual simulations for different temperatures and coupling strengths for two choices of mechanical potential. In our model, the spin system is thermalised by the phonon thermostat, hence no electronic damping is present. With increasing coupling, the energy and angular momentum transfer is more efficient, hence the damping is enhanced. Since the observed value of induced damping is small, calculating the damping at higher temperature is challenging due to the strong thermal fluctua- tions that affect the accuracy of the results. Despite the low temperatures simulated here, the obtained damping values (at T=50K, a=4:9105) are of the same order as reported for magnetic insulators such as YIG (1 104to 110657,58 ) as well as in different SLD simulations (3 105,27). Gener- ally, the induced damping value depends on the phonon char- acteristics and the coupling term, that allows transfer of both energy and angular momentum between the two subsystems. Fig. 8(a) and (b) compare the calculated damping for the Morse and Harmonic potential for two values of the coupling strength. We observe that the values are not greatly affected9 FIG. 8. Damping parameter extracted from fitting the z component of the magnetisation for two different choices of potential: HP- Harmonic Potential (green open squares) and MP-Morse Potential (black open circles) as function of temperature Fig. a), b) and as function of the coupling strength Fig. c), d); Fig a) and b) are calculated for a constant coupling strength of C=0:3,C=0:5 respectively. Fig c) and d) are calculated for temperatures of T=100K,T=300Krespectively. The black and green lines represents the average damping parameter obtained from the simulations using the Morse and the Harmonic Potentials, respectively. by the choice of potential. This arises due to the fact that only the spin modes around Gpoint are excited and for this low k- vectors modes the inter-atomic distances between neighbour- ing atoms do not vary significantly. The extracted damping parameter as a function of coupling strength for 100 K and 300 K is presented in Fig. 8(c) and (d) respectively. The func- tional form of the variation is quadratic, in accordance with the form of the coupling term. Measurements of damping in magnetic insulators, such as YIG, show a linear increase in the damping with temperature,58which agrees with the relaxation rates calculated by Kasuya and LeCraw59and the relaxation rates calculated in the NVE SLD simulations in Ref. 27. How- ever, Kasuya and LeCraw suggest that the relaxation rate can vary as Tn, where n=12 with n=2 corresponding to larger temperature regimes. Nevertheless, the difference between the quadratic temperature variation of the damping observed in our simulations and the linear one observed in experiments for YIG can be attributed to the difference in complexity be- tween the BCC Fe model and YIG. The difference between the trends may as well suggest that the spin-orbit coupling in YIG could be described better by a linear phenomenological coupling term, such as the one used in Refs. 26 and 29, but we note that such forms can lead to a uniform force in the di- rection of the magnetisation and so might need further adap- tation before being suitable. To test an alternate form of the coupling we have changed the pseudo-dipolar coupling to an on-site form, specifically Hc=åi;jf(ri j)((Si
ˆri j)21 3S2 i) i.e a N ´eel-like anisotropy term. This leads to much smallerdamping as shown in Fig. 9 ( T=300 K, a=3:3105, averaged over 5 realisations) making it difficult to accurately calculate the temperature dependence of the damping, espe- cially for large temperatures. The magnon-phonon damping can clearly have complex behavior depending on the proper- ties of the system, especially the coupling term, hence no uni- versal behaviour of damping as function of temperature can be deduced for spin-lattice models. Neglecting the lattice contribution, the temperature depen- dence of the macrosopic damping can be mapped onto the Landau-Lifshitz-Bloch formalism (LLB)54and theory17and ASD simulations60have shown it to vary inversely with the equilibrium magnetisation. The LLB theory shows that the macrosopic damping is enhanced for large temperatures due to thermal spin fluctuations. Using the equilibrium magneti- sation it is possible to approximate the variation of damping with temperature produced due to thermal fluctuations within the LLB model. From 100K to 400K the damping calculated via the LLB model increases within the order of 105, which is considerably smaller than the results obtained via the SLD model. This shows that within the SLD model the temperature increase of the damping parameter is predominantly due to magnon-phonon interaction, and not due to thermal magnon scattering, as this process is predominant at larger tempera- tures.10 FIG. 9. Temperature variation of the damping parameter for N ´eel- like on-site coupling, Hc=åi;jf(ri j)((Si
ˆri j)21 3S2 i). The val- ues are extracted from mz(t)fittings for 10 realisations; VI. CONCLUSIONS AND OUTLOOK To summarise, we have developed a SLD model that is able to transfer energy and angular momentum efficiently from the spin to lattice sub-systems and vice-versa via a pseudo- dipolar coupling term. Our approached takes the best fea- tures from several previously suggested models and general- ize them which allows modelling in both canonical and mi- crocanonical ensembles. With only the lattice coupled to a thermal reservoir and not the spin system, we reproduce the temperature dependence of the equilibrium magnetisation, which agrees with the fact that the spin-lattice model obeys the fluctuation-dissipation theorem. We are able to study the dynamic properties such as phonon and spin spectrum and macrosopic damping, showing that the magnetic damping isnot greatly influenced by the choice of potential, however it is influenced by the form of the coupling term. This enables the possibility of tailoring the form of the coupling term so it can reproduce experimental dependencies of damping for dif- ferent materials. In future, the addition of quantum statistics for Spin Lattice Dynamics models61,62may also yield better agreement with experimental data. The SLD model developed here opens the possibility of the investigation of ultrafast dynamics experiments and theoret- ically studies of the mechanism through which angular mo- mentum can be transferred from spin to the lattice at ultrafast timescales. As we have demonstrated that the model works well in the absence of an phenomenological Gilbert damping, which consists mainly of electronic contributions, the SLD model can be employed to study magnetic insulators, such as YIG, where the principal contribution to damping is via magnon-phonon interactions. Future application of this model includes controlling the magnetisation via THz phonons7 which can lead to non-dissipative switching of the magnetisa- tion11,12. With the increased volume of data stored, field-free, heat-free switching of magnetic bits could represent the future of energy efficient recording media applications. Another pos- sible application is more advanced modelling of the ultrafast Einstein-de-Haas effect2or phonon-spin transport63. VII. ACKNOWLEDGEMENTS We are grateful to Dr. Pui-Wai Ma and Prof. Matt Probert for helpful discussions. Financial support of the Ad- vanced Storage Research Consortium is gratefully acknowl- edged. MOAE gratefully acknowledges support in part from EPSRC through grant EP/S009647/1. The spin-lattice simula- tions were undertaken on the VIKING cluster, which is a high performance compute facility provided by the University of York. 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2020-10-01

A unified model of molecular and atomistic spin dynamics is presented enabling simulations both in microcanonical and canonical ensembles without the necessity of additional phenomenological spin damping. Transfer of energy and angular momentum between the lattice and the spin systems is achieved by a coupling term based upon the spin-orbit interaction. The characteristic spectra of the spin and phonon systems are analyzed for different coupling strength and temperatures. The spin spectral density shows magnon modes together with the uncorrelated noise induced by the coupling to the lattice. The effective damping parameter is investigated showing an increase with both coupling strength and temperature. The model paves the way to understanding magnetic relaxation processes beyond the phenomenological approach of the Gilbert damping and the dynamics of the energy transfer between lattice and spins.

Modeling coupled spin and lattice dynamics

2010.00642v1

1 Magnetic properties in ultra -thin 3d transition metal alloys II: Experimental verification of quantitative theories of damping and spin -pumping Martin A. W. Schoen,1,2* Juriaan Lucassen,3 Hans T. Nembach,1 Bert Koopmans,3 T. J. Silva,1 Christian H. Bac k,2 and Justin M. Shaw1 1Quantum E lectromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305 , USA 2Institute of Experimental and Applied Physics, University of Regensburg, 93053 Regensburg, Germany 3Department of Applied P hysics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands Dated: 01/05/2017 *Corresponding author: martin1.schoen@physik.uni -regensburg.de Abstract A systematic experimental study of Gilbert damping is performed via ferromagnet ic resonance for the disordered crystalline binary 3 d transition metal alloys Ni -Co, Ni -Fe and Co-Fe over the full range of alloy compositions. After accounting for inhomogeneous linewidth broadening , the damping shows clear evidence of both interfacial da mping enhancement (by spin pumping) and radiative damping. We quantify these two extrinsic contributions and thereby determine the intrinsic damping. The comparison of the intrinsic damping to multiple theoretical calculation s yields good qualitative and q uantitative agreement in most cases . Furthermore, the values of the damping obtained in this study are in good agreement with a wide range of published experimental and theoretical values . Additionally, we find a compositional dependence of the spin mixing conductance. 2 1 Introduction The magnetization dynamics in f erromagnetic films are phen omenologically well described by the Landau -Lifshitz -Gilbert formalism (LLG) where the damping is described by a phenomenological damping parameter α.4,5 Over the past four decades, there have been considerable efforts to derive the phenomenological d amping parameter from first principles calculations and to do so in a quantitative manner. One of the early promising theories was that of Kamberský, who introduced the so -called breathing Fermi surface model6–8. The name “breathing Fermi surface ” stems from the picture that the precessing magnetization, due to spin -orbit coupling, distorts the Fermi surface. Re -populating the Fermi surface is delayed by the scattering time, resulting in a phase lag between the precession and the Fermi sur face distortion. This lag leads to a damping that is proportional to the scattering time. Although this approach describes the so-called conductivity -like behavior of the damping at low temperatures, it fails to describe the high temperature behavior of so me materials . The high temperature or resistivity -like behavior is described by the so-called “bubbling Fermi surface ” model. In the case of energetically shifted bands , thermal broaden ing can lead to a significant overlap of the spin-split bands in 3d ferromagnets. A precessing magnetization can induce elec tronic transitions between such overlapping bands , leading to spin-flip process es. This process scales with the amount of band overlap. Since s uch overlap is further increased with the band broadening th at results from the finite temperature of the sample, this contribution is expected to increase as the temperature is increased. This model for interband transition mediated damping describes the resistivity -like behavior of the damping at higher temperatu res (shorter scattering times). These two damping processes are combined in a torque correlation model by Gilmore , et al.9, as well as Thonig , et al.10, that describes both the low -temperature (intra band transitio ns) and high -temperature (inter band transitions) behavior of the damping . Another app roach via scattering theory was successfully implemented by Brataas , et al.11 to describe damping in transition metals. Starikov , et al. ,2 applied the scattering matrix approach to calculate the damping of NixFe1-x alloys and Liu, et al. ,12 expanded the formalism to include the influence of electron -phonon interaction s. A numerical realization of the torque correlation model was performed by Mankovsky , et al., for NixCo1-x, Ni xFe1-x, CoxFe1-x, and FexV1-x1. More recently , Turek , et al. ,3 calculated the damping for NixFe1-x and CoxFe1-x alloys with the torque -correlation model, u tilizing non -local torque correlators. It is important to stress that a ll of these approaches consider only the intrinsic damping. This complicates the quantitative comparison of calculated values for t he damping to experimental data since there are many extrinsic contributions to the damping that result from sample structure , measurement geometr y, and/or sample properties . While some extrinsic contributions to the damping and linewidth were discovered in the 1960 ’s and 1970 ’s, and are well described by the ory, e.g. eddy -current damping13,14, two -magnon scattering15–17, the slow rel axer mechanism18,19, or radiative damping20,21, interest in these mechanisms has been re -ignited recently22,23. Further contributions, such as spin-pumping, both extrinsic24,25 and intrinsic24,26, have 3 been discovered more recently and a re subject to extensive research27–31 for spintronics application. Therefore , in order to allow a quantitative comparison to theoretical calculations for intrins ic damping, both the measurement and sample geometry must be designed to allow both the determination and possibl e minimization of all additional contributions to the measured damping. In this study , we demonstrate methods to determine significant extrins ic contributions to the damping , which includ es a measurement of the effective spin mixing conductance for both the pure elements and select alloys. By precisely accounting for all of these extrinsic contributions, we determine the intrinsic damping parame ters of the binary alloys Ni xCo1-x, Ni xFe1-x and Co xFe1-x and compare them to the calculation s by Mankovsky , et al. ,1, Turek , et al. , and Starikov , et al.2. Furthermore, we present the concentration -dependence of the inhomogeneous linewidth broadening, which for most alloys shows exceptionally small values, indicative of the high homogeneity of our samples. 2 Samples and method We deposited NixCo1-x, Ni xFe1-x and Co xFe1-x alloys of varying composition (all compositio ns given in atomic percent) with a thickness of 10 nm on an oxidized (001) Si substrate with a Ta(3 nm)/Cu(3 nm) seed layer and a Cu(3 nm) /Ta(3 nm) cap layer. In order to investigate interface effects , we also deposited multiple thickness series at 10 nm, 7 nm, 4 nm, 3 nm , and 2 nm of both the pure elements and select alloy s. Structural characterization was performed using X-ray diffraction (XRD). Field swept vector -network -analyzer ferromagnetic resonance spectroscopy (VNA -FMR) was used in the out -of-plane geometry to determine the total damping parameter αtot. Further d etails of the deposition conditions, XRD, FMR measurement and fitting of the complex susceptibility to the measured S21 parameter are reported in Ref [66]. An example of susceptibility fits t o the complex S21 data is shown in Fig. 1 (a) and (b). All fits were constrained to a 3× linewidth ΔH field window around the resonance field in order to minimize the influence of measurement drifts on the error in the susceptibility fits. The total dampin g parameter αtot and the inhomogeneous linewidth broadening Δ H0 are then determined from a fit to the linewidth Δ H vs. frequency f plot22, as shown in Fig. 1 (c). ∆𝐻= 4𝜋𝛼𝑡𝑜𝑡𝑓 𝛾𝜇0+ ∆𝐻0, (1) where γ=gμB/ħ is the gyro -magnetic ratio, μ0 is the vacuum permeability, μB is the Bohr -magneton , ħ is the reduced Planck constant, and g is the spectroscopic g-factor reported in Ref [ 66]. 4 Figure 1: (a) and (b) show respectively the real and imaginary part of the S21 transmission parameter (black squares) measured at 20 GHz with the complex susceptibility fit (red lines) for the Ni 90Fe10 sample . (c) The linewidths from the suscept ibility fits (symbols) and linear fits (solid lines) are plotted against frequency for different Ni -Fe compositions. Concentrations are denoted on the right -hand axis. The damping α and the inhomogeneous linewidth broadening Δ H0 for each alloy can be extracted from the fits via Eq. (1). 3 Results The first contribution t o the linewidth we discuss is the inhomogeneous linewidth broadening ΔH0, which is presumably indicative of sample inhomogeneity32,33. We plot Δ H0 for all the alloy system s against the respective concentrations in Fig. 2. For all alloys , ΔH0 is in the range of a few mT to 10 mT . There are only a limited number of reports for ΔH0 in the literature with which to compare . For Permalloy (Ni 80Fe20) we measure Δ H0 = 0.35 mT, which is close to other reported values .34 For the other NixFe1-x alloys , ΔH0 exhibits a significant peak near the fcc-to-bcc (face - centered -cubic to body -centered -cubic) phase transition at 30 % Ni , (see Fig. 2 (b)) which is easily seen in the raw data in Fig. 1 (c). We speculate that this increase of inhomogeneous broadening in the NixFe1-x is caused by the coexistence of the bcc and fcc phases at the phase transition. However, the CoxFe1-x alloys do not exhibit an increase in ΔH0 at the equivalent phase transition at 70 % Co . This suggests that the bcc and fcc phases of NixFe1-x tend to segregate near the phase transition, whereas the same phases for CoxFe1-x remain intermixed throughout the transition. 5 One possible explanation for inhomogeneous broadening is magnetic anisotropy , as originally proposed in Ref. [35]. However, this explanation does not account for our measured dependence of ΔH0 on alloy concentration, since the perpendicular magnetic anisotropy , described in Ref [ 66] effectively exhibits opposite behavior with alloy concentration. For our alloys Δ H0 seems to roughly correlate to the inverse exchange constant36,37, which co uld be a starting point for future investigation of a quantitative theory of inhomogeneous broadening. Figure 2: The inhomogeneous linewidth -broadening Δ H0 is plotted vs. alloy composition for (a) Ni-Co, (b) Ni -Fe and (c) Co -Fe. The alloy phases are denoted by color code described in Ref [ 66] We plot the total measured damping αtot vs. composition for NixCo1-x, NixFe1-x and CoxFe1-x in Fig. 3 (red crosses). The total damping of the NixCo1-x system increases monotonically with increased Ni content . Such smooth behavior in the damping is not surprising owing to the absence of a phase transition for this alloy . In the NixFe1-x system , αtot changes very little from pure Fe to approximately 25 % Ni where the bcc to fcc phase transition occurs . At the phase transition, αtot exhibits a step, increasing sharply by approximately 30 %. For higher Ni concentrations , αtot increases monotonically with increasing Ni concentration . On the other hand, t he CoxFe1-x system shows a different behavior in the damping and d isplays a sharp minimum of (2.3 ± 0.1)×10-3 at 25 6 % Co as previously reported38. As the system changes to an fcc phase ( ≈ 70 % Co), αtot become almost constant. We compare our data to previously published values in Table I. However, direct c omparison of our data to previous report s is not trivial , owing to the variation in measurement conditions and sample characteristic s for all the reported measurements . For example , the damping can depend on the temperature .9,39 In addition, multiple intrinsic and intrinsic contributions to the total damping are not always accounted for in the literature . This can be seen in the fact that the reported damping in Ni80Fe20 (Permalloy) varies from α=0.0 055 to α=0.04 at room tem perature among studies . The large variation for these reported data is possibly the result of different uncontrolled contributions to the extrinsic damping that add to the total damping in the different experiments , e.g. spin- pumping40–42, or roughness41. Therefore , the value for the intrinsic damping of Ni 20Fe80 is expected to be at the low end of this scatter . Our measured value of α=0.007 2 lies within the range of reported values. Similarly , many of our measured damping values for different alloy compositions lie within the range of reported values22,43 –48. Our measured damping of the pure elements and the Ni 80Fe20 and Co 90Fe10 alloys is compared to room temperature values found in literature in Table 1, Col umns 2 and 3 . Column 5 contains theoretically calculated values . Table 1: The total measured damping α tot (Col. 2) and the intrinsic damping (C ol. 4) f or Ni 80Fe20, Co90Fe10, and the pure elements are compared to both experimental (Col. 3) and theoretical (Col. 5) values from the literature . All values of the damping are at room temperature if not noted otherwise . Material αtot (this study) Liter ature values αint (this study) Calculated literature values Ni 0.029 (fcc) 0.06444 0.04549 0.024 (fcc) 0.0179 (fcc) at 0K 0.02212 (fcc) at 0K 0.0131 (fcc) Fe 0.0036 (bcc) 0.001944 0.002746 0.0025 (bcc) 0.00139 (bcc) at 0K 0.001012 (bcc) at 0K 0.00121 (bcc) at 0K Co 0.0047 (fcc) 0.01144 0.0029 (fcc) 0.00119 (hcp) at 0K 0.0007312 (hcp) at 0K 0.0011 (hcp) Ni80Fe20 0.0073 (fcc) 0.00844 0.008 -0.0450 0.007848 0.00751 0.00652 0.00647 0.005553 0.0050 (fcc) 0.00462,54 (fcc) at 0K 0.0039 -0.00493 (fcc) at 0K Co90Fe10 0.0048 (fcc) 0.004344 0.004855 0.0030 (fcc) 7 Figure 3: (color online) The measured damping αtot of all the alloys is plotted against the alloy compositi on (red crosses) for (a) Ni -Co, (b) Ni -Fe and (c) Co -Fe (the data in (c) are taken from Ref.[38]). The black squares are the intrinsic damping αint after correction for spin pumping and radiative contributions to the measured damping. The blue line is the intr insic damping calculated from the Ebert -Mankovsky theory ,1 where the blue circles are the values for the pure elements at 300K . The green line is the calculated damping for the Ni -Fe alloys by Starikov , et al.2 The inset in (b) depicts the damping in a smaller concentration window in order to better depict the small features in the damping around the ph ase transition. The damping for the Co -Fe alloys, calculated by Turek et al.3 is plotted as the orange line. For the Ni -Co alloys the damp ing calculated by th e spin density of the respective alloy weighted bulk damping55 (purple dashed line). 8 This scatter in the experimental data reported in the literature and its divergence from calculated values of the damping shows th e necessity to determine the intrinsic damping αint by quantification of all extrinsic contributions to the measured total damping α tot. The first extrinsic contribution to the damping that we consider is the radiative damping α rad, which is caused by ind uctive coupling between sample and waveguide , which results in energy flow from the sample back into the microwave circuit.23 αrad depends directly on the measurement method and geometry. The effect is easily understood , since the strength of the inductive coupling depends on the inductance of the FMR mode itself , which is in turn determined by the saturation magnetization, sampl e thickness, sample length, and waveguide width. Assuming a homogeneous excitation field , a uniform magnetization profile throughout the sample , and negligible spacing between the waveguide and sample , αrad is well approximated by23 𝛼rad=𝛾𝑀𝑠𝜇02𝛿𝑙 16 𝑍0𝑤𝑐𝑐, (2) where l (= 10 mm in our case) is the sample length on the waveguide, wcc (= 100µm) is the width of the co -planar wave guide center conductor and Z0 (= 50 Ω), the impedance of the waveguide. Though inh erently small for most thin films , αrad can become significant for alloys with exceptionally small intrinsic damping and /or high saturation magnetization. For example, it plays a significant role (values of αrad ≈ 5x10-4) for the whole composition range of the Co -Fe alloy system and the Fe -rich side of the Ni -Fe system. On the other hand, for pure Ni and Permalloy (Ni 80Fe20) αrad comprises only 3 % and 5 % of αtot, respectively. The second non -negligible contribution to the damping that we consider is the interfacial contribution to the measured damping , such as spin-pumping into the adjacent Ta/Cu bilayers . Spin pumping is proportional to the reciprocal sample thickness as described in24 𝛼sp= 2𝑔eff↑↓𝜇𝐵𝑔 4𝜋𝑀s𝑡. (3) The spectroscopic g-factor and the saturation magnetization Ms of the alloys were reported in Ref [66] and the factor of 2 accounts for the presence of two nominally identical interfaces of the alloys in the cap and seed layers. In Fig. 4 (a)-(c) we plot the damping dependence on reciprocal thickness 1/t for select alloy concentrations, which allows us to determine the effective spin mixing conductance 𝑔eff↑↓ through fits to Eq. (3) . The effective spin m ixing conductance contains details of the spin transport in the adjacent non -magnetic layers, such as the interfacial spin mixing conductance, both the conductivity and spin diffusion for all the non -magnetic layers with a non - negligible spin accumulation, as well as the details of the spatial profile for the net spin accumulation .56,57 The values of 𝑔eff↑↓, are plotted versus the alloy concentration in Fig. 4 (d), and are in the range of previously reported values for samples prepare d under similar growth conditions55– 59. Intermediate values of 𝑔eff↑↓ are determined by a guide to the eye interpolation [ grey lines, Fig. 4 (d)] and αsp is calculated for all alloy concentrations utilizing those interpolated values. The data for 𝑔eff↑↓ in the NixFe1-x alloys shows approximately a factor two increase of 𝑔eff↑↓ between Ni concentrations of 30 % Ni and 50 % Ni, which we speculate to occur at the fcc to bcc phase transition around 30 % Ni. According to this line of speculation , the previously mentioned step increase in the measured total damping at the NixFe1-x phase transition can be fully attributed to the increase in spin pumping at the phase transition. In CoxFe1-x, the presence of a step in 𝑔eff↑↓ at the phase transition is not confirmed, given the measurement precision, although we do observe an increase in the effective spin mixing conductance when transitioning from the bcc to fcc phase. The 9 concentration dependence of 𝑔eff↑↓ requires further thorough investigation and we therefore restrict ourselves to reporting the expe rimental findings. Figure 4: The damping for the thickness series at select alloy compositions vs. 1/ t for (a) Ni -Co, (b) Ni -Fe and (c) Co -Fe (data points, concentrations denoted in the plots), with linear fits to Eq. (3) (solid lines). (d) The extracted effective spin mixing conductance 𝑔eff↑↓ for the measured alloy systems, where the gray lines show the linear interpolations for intermediate alloy concentrations. The data for the Co -Fe system are taken from Ref.[38]. Eddy -current damping13,14 is estimated by use of the equations given in Ref. [23] for films 10 nm thick or less . Eddy currents are neglected because they are found to be less than 5 % of the total damping. Two -magnon scattering is disregarded because the mechanism is largely e xcluded in the out -of-plane measurement geometry15–17. The total measured damping is therefore well approximated as the sum 𝛼tot≅𝛼int+𝛼rad+𝛼sp, (4) We determine the intrinsic damping of the material by subtracting α sp and α rad from the measured total damping , as shown in Fig. 3 . 10 The intrinsic damping increases monotonically with Ni concentration for the NixCo1-x alloys . Indicative of the importance of extrinsic sources of damping, αint is approximately 40 % smaller than αtot for the Fe -rich alloy, though the difference decreases to only 15 % for pure Ni. This behavior is expected, given that both αrad and αsp are proportional to Ms. A comparison of αint to the calculations by Mankovsky , et al. ,1 shows excellent quantitative agreement to within 30 %. Furthermore, w e compare αint of the NixCo1-x alloys to the spin density weighted average of the intrinsic damping of Ni and C o [purple dashed line in Fig. 3 (a)] , which gives good agreement with our data, as previously reported .55 αint for NixFe1-x (Fig. 3 (b)) also increases with Ni concentration after a small initial decrease from pure Fe to the first NixFe1-x alloys. The step increase found in αtot at the bcc to fcc phase transition is fully attributed to αsp, as detailed in the previous section, and therefore does not occur in αint. Similar to the NixCo1-x system αint is significantly lower than αtot for Fe -rich alloys. With in error bars, a comparison to the calculations by Mankovsky , et al.1 (blue line) and Starikov , et al.2 (green line) exhibit excellent agreement in the fcc phase, with marginally larger deviations in the Ni rich regime. Starikov , et al.2 calculated the damping over the ful l range of compositions, under the assumption of continuous fcc phase. This calcu lation deviates further from our measured αint in the bcc phase exhibiting qualitatively different behavior. As previously reported, t he dependence of αint on alloy compositio n in the CoxFe1-x alloys exhibits strongly non -monotonic behavior, differing from the two previously discussed alloys.38 αint displays a minimum at 25 % Co concentration with a, for conducting ferromagnets unprecedented, low value of int (5±1.8) × 10-4. With increasing Co concentration , αint grows up to the phase transition, at which point it increases by 10 % to 20 % unt il it reaches the value for pure Co. It was shown that αint scales with the density of states (DOS) at the Fermi energy n( EF) in the bcc phase38, and the DOS also exhibits a sharp minimum for Co 25Fe75. This scaling is expected60,61 if the damping is dominated by the breathing Fermi surface process. With the breathing surface model, the intraband scattering that leads to damping directly scales with n( EF). This scaling is particularly pronounc ed in the Co -Fe alloy system due to the small concentration dependence of the spin -orbit coupling on alloy composition. The special properties of the CoxFe1-x alloy system are discussed in greater detail in Ref.[38]. Comparing αint to the calculations by Mankovsky et al.1, we find good quantitative agreement with the value of the minimum. However, t he concentration of the minimum is calculated to occur at approximately 10 % to 20% Co, a slightly lower value than 25 % Co t hat we find in this study. Furthermore , the strong concentration dependence around the minimum is not reflected in the calculations. More recent calculations by Turek et al.3, for the bcc CoxFe1-x alloys [orange line in Fig. 3 (c)] find the a minimum of the damping of 4x10-4 at 25 % Co concentration in good agreement with our experiment, but there is some deviation in concentration dependenc e of the damping around the minimum. Turek et al.3 also reported on the damping in the NixFe1-x alloy system, with similar qualitative and quantitative results as the other two presented quantitative theories1,2 and the results are therefore not plotted in Fig. 3 (b) for the sake of comprehensibility of the figure. For both NixFe1-x and the CoxFe1-x alloys , the calculated spin density weighte d intrinsic damping of the pure elements (not plotted) deviates significantly from the determined intrinsic damping of the alloys, in contrary to the good agreement archived for the CoxNi1-x alloys. We speculate that this difference between the alloy syste ms is caused by the non -monotonous dependence of the density of states at the Fermi Energy in the CoxFe1-x and NixFe1-x systems. Other calculated damping values for the pure elements and the Ni80Fe20 and Co 90Fe10 alloys are compared to the determined intr insic damping in Table 1. Generally , the calculations underestimate the damping significantly, but our data are in good agreement with more recent calculations for Permalloy ( Ni80Fe20). 11 It is important to point out that n one of the theories considered he re include thermal fluctuations . Regardless, we find exceptional agreement with the calculations to αint at intermediate alloy concentrations . We speculate that the modeling of atomic disorder in the alloys in the calculations, by the coherent potential approximation (CPA) could be responsible for this exceptional agreement. The effect of disorder on the electronic band structure possibly dominates any effect s due to nonzero temperature. Indeed, both effects cause a broadening of the bands due to enhanced momentum scattering rates. This directly correlates to a change of the damping parameter according to the theory of Gilmore and Stiles9. Therefore , the inclusion of the inherent disorder of solid -solution alloys in the calculations by Mankovsky et al1 mimic s the effects of temperature on damping to some extent . This argument is corroborated by the fact that the calculations by Mankovsky et al1 diverge for diluted alloys and pure elements (as shown in Fig. 2 (c) for pure Fe) , where no or to little disorder is introduced to account for temperature effects. Mankovsky et al.1 performed temperature dependent calculations of the damping for pure bcc Fe, fcc Ni and hcp Co and the values for 300 K are shown in Table 1 and Fig. 3. These calculations for αint at a temperature of 300 K are approximately a factor of two less than our measured values , but the agreement is significantly improved relative to those obtained by calculations that neglect thermal fluctuations . Figure 5: The intrinsic damping α int is plotted against ( g-2)2 for all alloys. We do not observe a proportionality between α int and (g-2)2. 12 Finally, i t has been reported45,64 that there is a general proportionality between αint and (g- 2)2 , as contained in the original microscopic BFS model proposed by Kambersky .62 To examine this relationship, w e plot αint versus (g-2)2 (determined in Ref [66]) for all samples measured here in Figure 5 . While some samples with large values for ( g-2)2 also exhib it large αint, this is not a general trend for all the measured samples . Given that the damping is not purely a function of the spin-orbit strength, but also depends on the details of the band structure , the result in Fig. 5 is expected . For example , the amount of band overlap will determine the amount of interband transition leading to that damping channel. Furthermore, the density of states at the Fermi energy will affect the intraband contribution to the damping9,10. Finally , the ratio of inter - to intra -band scattering that mediate s damping contributions at a fixed temperature (RT for our measurements) changes for different elements9,10 and therefore with alloy concentration. None of these f actors are necessarily proportional to the spin -orbit coupling . Therefore , we conclude that this simple relation, which originally traces to an order of magnitude estimate for the case of spin relaxation in semiconductors65, does not hold for all magnetic systems in general. 4 Summary We determined the damping for the full compositi on range of the binary 3d transition metal all oys Ni-Co, Ni -Fe, and Co -Fe and showed that the measured damping can be explained by three contributions to the damping: Intrinsic damping, radiative damping and damping due to spin pumping. By quantifying all extrinsic contributions to the measured damping, we determine the intrinsic damping over the whole range of alloy compositions . These values are compared to multiple theoretical calculations and yield excellent qualitative and good quantitative agreement for intermediate alloy concentrations. For pure elements or diluted alloys, the effect of temperature seems to play a larger role for the damping and calculations including temperature effects give significantly better agreement to our data. Furthermore, w e demonstrated a compositional dependence of the spin mixing conductance , which can vary by a factor of two . Finally , we showed that the often postulated dependence of the damping on the g-factor does not apply to the investigated binary alloy systems, as their damping cannot be described solely by the strength of the spin -orbit interaction . 13 5 References 1. Mankovsky, S., Ködderitzsch, D., Woltersdorf, G. & Ebert, H. First -principles calculation of the Gilbert d amping parameter via the linear response formalism with application to magnetic transition metals and alloys. Phys. Rev. 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Celinski, Z. & Heinrich, B. Ferromagnetic resonance linewidth of Fe ultrathin films grown on a bcc Cu substrate. Journal of Applied Physics 70, 5935 –5937 (1991). 33. Platow, W., Anisimov, A. N., Dunifer, G. L., Farle, M. & Baberschke, K. Correlations between ferromagnetic - resonance linewidths and sample quality in the study of metallic ultrathin films. Phys. Rev. B 58, 5611 –5621 (1998). 34. Twisselmann, D. J. & McMichael, R. D. Intrinsic damp ing and intentional ferromagnetic resonance broadening in thin Permalloy films. Journal of Applied Physics 93, 6903 –6905 (2003). 35. McMichael, R. D., Twisselmann, D. J. & Kunz, A. Localized Ferromagnetic Resonance in Inhomogeneous Thin Films. Phys. Rev. L ett. 90, 227601 (2003). 36. Hennemann, O. D. & Siegel, E. Spin - Wave Measurements of Exchange Constant A in Ni -Fe Alloy Thin Films. phys. stat. sol. 77, 229 (1976). 37. Wilts, C. & Lai, S. Spin wave measurements of exchange constant in Ni -Fe alloy films. 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Temperature variation of ferromagnetic relaxation in the 3d transition metals. Phys. Rev. B 10, 179 (1974). 44. Oogane, M. et al. Magnetic Damping in Ferromagnetic Thin Films. Japanese Journal of Applied Physics 45, 3889 –3891 (2006). 45. Pelzl, J. et al. Spin-orbit -coupling effects on g -value and damping factor of the ferromagnetic res onance in Co and Fe films. Journal of Physics: Condensed Matter 15, S451 (2003). 46. Scheck, C., Cheng, L. & Bailey, W. E. Low damping in epitaxial sputtered iron films. Applied Physics Letters 88, (2006). 47. Shaw, J. M., Silva, T. J., Schneider, M. L. & McMichael, R. D. Spin dynamics and mode structure in nanomagnet arrays: Effects of size and thickness on linewidth and damping. Phys. Rev. B 79, 184404 (2009). 48. Yin, Y. et al. Tunable permalloy -based films for magnonic devices. Phys. Rev. B 92, 024427 ( 2015). 49. Walowski, J. et al. Intrinsic and non -local Gilbert damping in polycrystalline nickel studied by Ti:sapphire laser fs spectroscopy. 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T., Shaw, J. M. & Silva, T. J. Spin transport parameters in metallic multilayers determined by ferromagnetic resonance measurements of spin -pumping. Journal of Applied Physics 113, 15, 101063 (2013). 57. Boone, C. T., Shaw, J. M., Nembach, H. T. & Silva, T. J. Spin -scattering rates in metallic thin films measured by ferromagnetic resonance damping enhanced by spin -pumping. Journal of Applied Physics 117, 22, 1 01063 (2015). 58. Czeschka, F. D. et al. Scaling Behavior of the Spin Pumping Effect in Ferromagnet -Platinum Bilayers. Phys. Rev. Lett. 107, 046601 (2011). 59. Weiler, M., Shaw, J. M., Nembach, H. T. & Silva, T. J. Detection of the DC Inverse Spin Hall Eff ect Due to Spin Pumping in a Novel Meander -Stripline Geometry. Magnetics Letters, IEEE 5, 1–4 (2014). 60. Ebert, H., Mankovsky, S., Ködderitzsch, D. & Kelly, P. J. Ab Initio Calculation of the Gilbert Damping Parameter via the Linear Response Formalism. Phys. Rev. Lett. 107, 66603 –66607 (2011). 61. 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Magnetic properties of ultra -thin 3d transition -metal binary alloys I: spin and orbital mo ments, anisotropy, and confirmation of Slater -Pauling behavior . arXiv:1701.02177 (2017 ).

2017-01-10

A systematic experimental study of Gilbert damping is performed via ferromagnetic resonance for the disordered crystalline binary 3d transition metal alloys Ni-Co, Ni-Fe and Co-Fe over the full range of alloy compositions. After accounting for inhomogeneous linewidth broadening, the damping shows clear evidence of both interfacial damping enhancement (by spin pumping) and radiative damping. We quantify these two extrinsic contributions and thereby determine the intrinsic damping. The comparison of the intrinsic damping to multiple theoretical calculations yields good qualitative and quantitative agreement in most cases. Furthermore, the values of the damping obtained in this study are in good agreement with a wide range of published experimental and theoretical values. Additionally, we find a compositional dependence of the spin mixing conductance.

Magnetic properties in ultra-thin 3d transition metal alloys II: Experimental verification of quantitative theories of damping and spin-pumping

1701.02475v1

arXiv:1510.06793v1 [cond-mat.mtrl-sci] 23 Oct 2015Laser-induced THz magnetization precession for a tetragon al Heusler-like nearly compensated ferrimagnet S. Mizukami,1,a)A. Sugihara,1S. Iihama,2Y. Sasaki,2K. Z. Suzuki,1and T. Miyazaki1 1)WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Sendai 980-8577, Japan 2)Department of Applied Physics, Tohoku University, Sendai 9 80-8579, Japan (Dated: 23 July 2018) Laser-inducedmagnetizationprecessional dynamicswasinvestiga tedinepitaxialfilms of Mn 3Ge, which is a tetragonal Heusler-like nearly compensated ferrimag net. The ferromagnetic resonance (FMR) mode was observed, the preces sion frequency for which exceeded 0.5 THz and originated from the large magnetic anisot ropy field of approximately 200 kOe for this ferrimagnet. The effective damping c onstant was approximately 0.03. The corresponding effective Landau-Lifshitz c onstant of approx- imately 60 Mrad/s and is comparable to those of the similar Mn-Ga mate rials. The physical mechanisms for the Gilbert damping and for the laser-induc ed excitation of the FMR mode were also discussed in terms of the spin-orbit-induced damping and the laser-induced ultrafast modulation of the magnetic anisotropy , respectively. a)Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp 1Among the various types of magnetization dynamics, coherent mag netization precession, i.e.,ferromagneticresonance(FMR),isthemostfundamentaltype, andplaysamajorrolein rf spintronics applications based on spin pumping1–5and the spin-transfer-torque (STT).6,7 Spin pumping is a phenomenon through which magnetization precessio n generates dc and rf spin currents in conductors that are in contact with magnetic films. The spin current can be converted into anelectric voltage throughthe inverse spin-Hall eff ect.8The magnitude of the spin current generatedvia spinpumping is proportionaltothe FMRf requency fFMR;4,5thus, the output electric voltage is enhanced with increased fFMR. In the case of STT oscillators and diodes, the fFMRvalue for the free layer of a given magnetoresistive devices primarily determines the frequency range for those devices.9,10An STT oscillator and diode detector at a frequency of approximately 40 GHz have already been demonst rated;11–13therefore, one of the issues for consideration as regards practical applications is the possibility of increasing fFMRto hundreds of GHz or to the THz wave range (0.1-3 THz).11,14 Onesimple methodthroughwhich fFMRcanbeincreased utilizes magneticmaterials with large perpendicular magnetic anisotropy fields Heff kand small Gilbert damping constants α.13,15,16This is because fFMRis proportional to Heff kand, also, because the FMR quality factor and critical current of an STT-oscillator are inversely and d irectly proportional to α, respectively. The Heff kvalue is determined by the relation Heff k= 2Ku/Ms−4πMsfor thin films, where KuandMsare the perpendicular magnetic anisotropy constant and saturat ion magnetization, respectively. Thus, materials with a small Ms, largeKu, and low αare very favorable; these characteristics are similar to those of mate rials used in the free layers of magnetic tunnel junctions integrated in gigabit STT memory applic ation.17We have previously reported that the Mn-Ga metallic compound satisfies the above requirements, and that magnetization precession at fFMRof up to 0.28 THz was observed in this case.18 A couple of research groups have studied magnetization precessio n dynamics in the THz wave range for the FePt films with a large Heff k, and reported an αvalue that is a factor of about 10 larger than that of Mn-Ga.19–21Thus, it is important to examine whether there are materialsexhibiting properties similartothoseofMn-Gaexist, inorde r tobetter understand the physics behind this behavior. In this letter, we report on observed magnetization precession at fFMRof more than 0.5 THz for an epitaxial film of a Mn 3Ge metallic compound. Also, we discuss the relatively small observed Gilbert damping. Such THz-wave-range dynamics ca n be investigated by 2means of a THz wave22or pulse laser. Here, we use the all-optical technique proposed previously;23therefore, the mechanism of laser-induced magnetization preces sion is also dis- cussed, because this is not very clearly understood. Mn3Ge has a tetragonal D0 22structure, and the lattice constants are a= 3.816 and c= 7.261˚A in bulk materials [Fig. 1(a)].24,25The Mn atoms occupy at two non-equivalent sites in the unit-cell. The magnetic moment of Mn I(∼3.0µB) is anti-parallel to that of MnII(∼1.9µB), because of anti-ferromagnetic exchange coupling, and the net magnetic moment is ∼0.8µB/f.u. In other words, this material is a nearly compensated ferrima gnet with a Curie temperature Tcover 800 K.26The tetragonal structure induces a uniaxial magnetic anisotropy, where the c-axis is the easy axis.24The D0 22structure is identical to that of tetragonally-distorted D0 3, which is a class similar to the L2 1Heusler structure; thus, D0 22Mn3Ge is also known as a tetragonal Heusler-like compound, as is Mn 3Ga.27 The growth of epitaxial films of D0 22Mn3Ge has been reported quite recently, with these films exhibiting a large Kuand small Ms, similar to Mn-Ga.28–30Note that Mn 3Ge films with a single D0 22phase can be grown for near stoichiometric compositions.29,30Further, an extremely large tunnel magnetoresistance is expected in the magn etic tunnel junction with Mn3Ge electrodes, owing to the fully spin-polarized energy band with ∆ 1symmetry and the Bloch wave vector parallel to the c-axis at the Fermi level.29,31These properties constitute the qualitative differences between the Mn 3Ge and Mn 3Ga compounds from the material perspective. All-optical measurement for the time-resolved magneto-optical K err effect was employed using a standard optical pump-probe setup with a Ti: sapphire laser and a regenerative amplifier. The wavelength and duration of the laser pulse were appro ximately 800 nm and 150 fs, respectively, while the pulse repetition rate was 1 kHz. The p ulse laser beam was divided into an intense pump beam and a weaker probe beam; both bea ms weres-polarized. The pump beam was almost perpendicularly incident to the film surface , whereas the angle of incidence of the probe beam was ∼6◦with respect to the film normal [Fig. 1(b)]. Both laser beams were focused on the film surface and the beam spo ts were overlapped spatially. The probe and pump beams had spot sizes with 0.6 and 1.3 mm, respectively. The Kerr rotation angle of the probe beam reflected at the film surf ace was analyzed using a Wollaston prism and balanced photodiodes. The pump beam intensity was modulated by a mechanical chopper at a frequency of 360 Hz. Then, the volta ge output from the 3FIG. 1. (a) Illustration of D0 22crystal structure unit cell for Mn 3Ge. (b) Diagram showing coordinate system used for optical measurement and ferroma gnetic resonance mode of magnetiza- tion precession. The net magnetization (= MII−MI) precesses about the equilibrium angle of magnetization θ, whereMI(MII) is the magnetization vector for the Mn I(MnII) sub-lattice. (b) Out-of-plane normalized hysteresis loop of the Kerr rotati on angle φkmeasured for the sample. photodiodes was detected using a lock-in amplifier, as a function of d elay time of the pump- probe laser pulses. The pump pulse fluence was ∼0.6 mJ/cm2. Note that the weakest possible fluence was used in order to reduce the temperature incre ase while maintaining the signal-to-noise ratio. A magnetic field Hof 1.95 T with variable direction θHwas applied using an electromagnet [Fig. 1 (b)]. Thec-axis-oriented Mn 3Ge epitaxial films were grown on a single-crystalline (001) MgO substrate with a Cr seed layer, and were capped with thin MgO/Al lay ers at room tempera- ture using a sputtering method with a base pressure below 1 ×10−7Pa. The characteristics of a 130-nm-thick film with slightly off-stoichiometric composition (74 a t% Mn) deposited at 500◦C are reported here, because this sample showed the smallest coer civity (less than 1 T) and the largest saturation magnetization (117 emu/cm3) of a number of films grown with various thicknesses, compositions, and temperatures. Thes e properties are important to obtaining the data of time-resolved Kerr rotation angle φkwith a higher signal-to-noise ratio, because, as noted above, Mn 3Ge films have a large perpendicular magnetic anisotropy 4field and a small Kerr rotation angle.30Figure 1(c) displays an out-of-plane hysteresis loop ofφkobtained for a sample without pump-beam irradiation. The loop is norm alized by the saturation value φk,sat 1.95 T. The light skin depth is considered to be about 30 nm for the employed laser wavelength, so that the φkvalue measured using the setup described above was almost proportional to the out-of-plane component of the ma gnetization Mzwithin the light skin depth depth. The loop shape is consistent with that measur ed using a vibrating sample magnetometer, indicating that the film is magnetically homogen eous along the film thickness and that value of φk/φk,sapproximates to the Mz/Msvalue. Figure 2(a) shows the pump-pulse-induced change in the normalized Kerr rotation angle ∆φk/φk,s(∆φk=φk−φk,s) as a function of the pump-probe delay time ∆ twith an applied magnetic field Hperpendicular to the film plane. ∆ φk/φk,sdecreases quickly immediately after the pump-laser pulse irradiation, but it rapidly recovers within ∼2.0 ps. This change is attributed to the ultrafast reduction and ps restoration of Mswithin the light skin depth region, and is involved in the process of thermal equilibration among t he internal degrees of freedom, i.e., the electron, spin, and lattice systems.32. After the electron system absorbs light energy, the spin temperature increases in the sub-ps timesca le because of the heat flow from the electron system, which corresponds to a reduction in Ms. Subsequently, the electron and spin systems are cooled by the dissipation of heat into t he lattices, which have a high heat capacity. Then, all of the systems reach thermal equilib rium. This process is reflected in the ps restoration of Ms. Even after thermal equilibrium among these systems is reached, the heat energy remains within the light skin depth region a nd the temperature is slightly higher than the initial value. However, this region gradually co ols via the diffusion of this heat deeper into the film and substrate over a longer timesca le. Thus, the remaining heat causing the increased temperature corresponds to the sma ll reduction of ∆ φk/φk,safter ∼2.0 ps. With increasing θHfrom out-of-plane to in-plane, a damped oscillation becomes visible in the ∆φk/φk,sdata in the 2-12 ps range [Fig. 2(b)]. Additionally, a fast Fourier tran sform of this data clearly indicates a single spectrum at a frequency of 0.5- 0.6 THz [Fig. 2(c)]. These damped oscillations are attributed to the temporal oscillation ofMz, which reflects the damped magnetization precession,23because the zcomponent of the magnetization precession vector increases with increasing θH. Further, the single spectrum apparent in Fig. 2(c) indicates that there are no excited standing spin-waves ( such as those observed in 5thick Ni films), even though the film is thicker than the optical skin de pth.23 Ferrimagnets generally have two magnetization precession modes, i.e., the FMR and exchange modes, because of the presence of sub-lattices.33In the FMR mode, sub-lattice magnetization vectors precess while maintaining an anti-parallel dire ction, as illustrated in Fig. 1(b), such that their frequency is independent of the exchan ge coupling energy between the sub-lattice magnetizations. On the other hand, the sub-lattic e magnetization vectors are canted in the exchange mode; therefore, the precession fre quency is proportional to the exchange coupling energy between them and is much higher than tha t of the FMR mode. As observed in the case of amorphous ferrimagnets, the FMR mode is preferentially excited when the pump laser intensity is so weak that the increase in tempera ture is lower than the ferrimagnet compensation temperature.34No compensation temperature is observed in the bulk Mn 3Ge.25,26Also, the temperature increase in this experiment is significantly sma ller thanTcbecause the reduction of Msis up to 4 %, as can be seen in Fig. 2(a). Therefore, the observed magnetization precession is attributed to the FMR mo de. Further, as the mode excitation is limited to the light skin depth, the amplitude, freque ncy, and etc., for the excited mode are dependent on the film thickness with respect t o the light skin depth. This is because the locally excited magnetization precession propaga tes more deeply into the film as a spin wave in cases where fFMRis in the GHz range.23Note that it is reasonably assumed that such a non-local effect is negligible in this study, becau se the timescale of the damped precession discussed here ( ∼1-10 ps) is significantly shorter than that relevant to a spin wave with wavelength comparable to the light skin depth ( ∼100 ps). The FMR mode in the THz-wave range is quantitatively examined below. When the ex- changecouplingbetween thesub-latticemagnetizationsissufficient ly strongandthetemper- ature is well below both Tcand the compensation temperature, the magnetization dynamics for a ferrimagnet can be described using the effective Landau-Lifs hitz-Gilbert equation35 dm dt=−γeffm×/bracketleftbig H+Heff k(m·z)z/bracketrightbig +αeffm×dm dt, (1) wheremis the unit vector of the net magnetization parallel (anti-parallel) to the magnetiza- tion vector MII(MI) for the Mn II(MnI) sub-lattice [Fig. 1(b)]. Here, the spatial change of mis negligible, as mentioned above. Heff kis the effective value of the perpendicular magnetic anisotropyfieldincluding thedemagnetizationfield, even thoughthe demagnetizationfieldis negligibly small for thisferrimagnet (4 πMs= 1.5 kOe). Further, γeffandαeffaretheeffective 6FIG. 2. Change in Kerr rotation angle ∆ φknormalized by the saturation value φk,sas a function of pump-probe delay time ∆ t: (a) for a short time-frame at θH= 0◦and (b) for a relatively long time-frame and different values of θH. The solid curves in (a) and (b) are a visual guide and values fitted to the data, respectively. The data in (b) are plotted w ith offsets for clarity. (c) Power spectral density as a function of frequency fand magnetic field angle θH. 7values of the gyromagnetic ratio and the damping constant, respe ctively, which are defined asγeff= (MII−MI)/(MII/γII−MI/γI) andαeff= (αIIMII/γII−αIMI/γI)/(MII/γII−MI/γI), respectively, using the gyromagnetic ratio γI(II)and damping constant αI(II)for the sub- lattice magnetization of Mn I(II). In the case of Heff k≫H,fFMRand the relaxation time of the FMR mode τFMRare derived from Eq. (1) as fFMR=γeff/2π/parenleftbig Heff k+Hz/parenrightbig , (2) 1/τFMR= 2παefffFMR. (3) Here,Hzis the normal component of H. Figure 3(a) shows the Hzdependence of the precession frequency fp. This is obtained using the experimental data on the oscillatory part of the change in ∆ φk/φk,svia least-square fitting to the damped sinusoidal func- tion, ∆φk,p/φk,sexp(−t/τp)sin(2πfp+φp), with an offset approximating the slow change of ∆φk/φk,s[solid curves, Fig. 2(b)]. Here, ∆ φk,p/φk,s,τp, andφpare the normalized am- plitude, relaxation time, and phase for the oscillatory part of ∆ φk/φk,s, respectively. The least-square fitting of Eq. (2) to the fpvs.Hzdata yields γeff/2π= 2.83 GHz/kOe and Heff k= 183 kOe [solid line, Fig. 3(a)]. The γeffvalue is close to 2.80 GHz/kOe for the free electron. The value of Heff kis equal to the value determined via static measurement (198 kOe)30within the accepted range of experimental error. Thus, the analy sis confirms that the THz-wave range FMR mode primarily results from the large magne tic anisotropy field in the Mn 3Ge material. The αeffvalues, which are estimated using the relation αeff= 1/2πfpτp following Eq. (3), are also plotted in Fig. 3(a). The experimental αeffvalues are indepen- dent ofHzwithin the accepted range of experimental error, being in accorda nce with Eq. (3); the mean value is 0.03. This value of αefffor D0 22Mn3Ge is slightly larger than the previously reported values for for D0 22Mn2.12Ga (∼0.015) and L1 0Mn1.54Ga (∼0.008).18 In the case of metallic magnets, the Gilbert damping at ambient tempe rature is primarily caused by phonon and atomic-disorder scattering for electrons a t the Fermi level in the Bloch states that are perturbed by the spin-orbit interaction. Th is mechanism, the so- called Kambersky mechanism,36,37predicts α∝M−1 s, so that it is more preferable to use the Landau-Lifshitz constants λ(≡αγMs) for discussion of the experimental values of α for different materials. Interestingly, λeff(≡αeffγeffMs) for Mn 3Ge was estimated to be 61 Mrad/s, which is almost identical to the values for D0 22Mn2.12Ga (∼81 Mrad/s) and L1 0 Mn1.54Ga(∼66Mrad/s). The λfortheKamberkymechanism isapproximatelyproportional 8FIG. 3. (a) Normal component of magnetic field Hdependence on precession frequency fpand effective dampingconstant αeffforMn 3Gefilm. (b)Oscillation amplitudeoftheKerrrotation angle ∆φk,p/φk,scorresponding to the magnetization precession as a functio n of the in-plane component ofH. The solid line and curve are fit to the data. The dashed line de notes the mean value of αeff. toλ2 SOD(EF), whereλSOisthespin-orbitinteractionconstant and D(EF)isthetotaldensity of states at the Fermi level.37The theoretical values of D(EF) for the above materials are roughly identical, because of the similar crystal structures and co nstituent elements, even though the band structures around at the Fermi level differ slight ly, as mentioned at the beginning.18,29Furthermore, the spin-orbit interactions for Ga or Ge, depending on the atomic number, may not differ significantly. Thus, the difference in αefffor these materials can be understood qualitatively in terms of the Kambersky mechanis m. Further discussion based on additional experiments is required in order to obtain more p recise values for αeff and to examine whether other relaxation mechanisms, such as extr insic mechanisms (related to the magnetic inhomogeneities), must also be considered. Finally, the excitation mechanism of magnetization precession in this s tudy is discussed below, in the context of a previously proposed scenario for laser-in duced magnetization 9precession in Ni films.23The initial equilibrium direction of magnetization θis determined by thebalance between HandHeff k[Fig. 1(b)]. Duringtheperiodinwhich thethree internal systems are not in thermal equilibrium for ∆ t <∼2.0 ps after the pump-laser irradiation [Fig. 2(a)], not only the value of Ms, but also the value of the uniaxial magnetic anisotropy, i.e.,Heff k, is altered. Thus, the equilibrium direction deviates slightly from θand is restored, which causes magnetization precession. This mechanism may be exam ined by considering the angular dependence of the magnetization precession amplitude . Because the precession amplitudemaybeproportionaltoanimpulsive torquegeneratedfro mthemodulationof Heff k in Eq. (1), the torque has the angular dependence |m0×(m0·z)z|, wherem0is the initial direction of the magnetization. Consequently, the z-component of the precession amplitude, i.e., ∆φk,p/φk,s, is expressed as ∆ φk,p/φk,s=ζcosθsin2θ∼ζ/parenleftbig Hx/Heff k/parenrightbig2, whereζis the proportionalityconstant and Hxisthe in-plane component of H. The experimental values of ∆φk,p/φk,sare plotted as a function of Hxin Fig. 3(b). The measured data match the above relation, which supports the above-described scenario. Although ζcould be determined via the magnitude and the period of modulation of Heff k, it is necessary to consider the ultrafast dynamics of the electron, spin, and lattice in the non-equilibrium stat e in order to obtain a more quantitative evaluation;38,39this is beyond the scope of this report. In summary, magnetization precessional dynamics was studied in a D 022Mn3Ge epitaxial film using an all-optical pump-probe technique. The FMR mode at fFMRup to 0.56 THz was observed, which was caused by the extremely large Heff k. A relatively small damping constant of approximately 0.03 was also obtained, and the corresp onding Landau-Lifshitz constant for Mn 3Ge were shown to be almost identical to that for Mn-Ga, being in quali- tatively accordance with the prediction of the Kambersky spin-orb it mechanism. The field dependence of the amplitude of the laser-induced FMR mode was qua litatively consistent with the model based on the ultrafast modulation of magnetic anisot ropy. This work was supported in part by a Grant-in-Aid for Scientific Rese arch for Young Researchers (No. 25600070) and for Innovative Area (Nano Spin Conversion Science, No. 26103004), NEDO, and the Asahi Glass Foundation. REFERENCES 1R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,4382 (1979). 102S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys. 40,580 (2001). 3R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. 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Nazrul Islam, M. Oogane, S. Mizukami, and Y. Ando, Nano Lett. 15,623 (2015). 14M. A. Hoefer, M. J. Ablowitz, B. Ilan, M. R. Pufall, and T. J. Silva, Phy s. Rev. Lett. 95, 267206 (2005). 15W. H. Rippard, A. M. Deac, M. R. Pufall, J. M. Shaw, M. W. Keller, S. E. Russek, G. E. W. Bauer, and C. Serpico, Phys. Rev. B 81,014426 (2010). 16T. Taniguchi, H. Arai, S. Tsunegi, S. Tamaru, H. Kubota, and H. Ima mura, Appl. Phys. Express6,123003 (2013). 17K. Yamada, K. Oomaru, S. Nakamura, T. Sato, and Y. Nakatani, Ap pl. Phys. Lett. 106, 042402 2015. 18S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota , X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106,117201 (2011). 19P. He, X. Ma, J. W. Zhang, H. B. Zhao, G. Lpke, Z. Shi, and S. M. Zho u, Phys. Rev. Lett. 110,077203 (2013). 20S. Iihama, S. Mizukami, N. Inami, T. Hiratsuka, G. Kim, H. Naganuma, M. Oogane, T. Miyazaki, and Y. Ando, Jpn. J. Appl. Phys. 52,073002 (2013). 21J. Becker, O. Mosendz, D. 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2015-10-23

Laser-induced magnetization precessional dynamics was investigated in epitaxial films of Mn$_3$Ge, which is a tetragonal Heusler-like nearly compensated ferrimagnet. The ferromagnetic resonance (FMR) mode was observed, the precession frequency for which exceeded 0.5 THz and originated from the large magnetic anisotropy field of approximately 200 kOe for this ferrimagnet. The effective damping constant was approximately 0.03. The corresponding effective Landau-Lifshitz constant of approximately 60 Mrad/s and is comparable to those of the similar Mn-Ga materials. The physical mechanisms for the Gilbert damping and for the laser-induced excitation of the FMR mode were also discussed in terms of the spin-orbit-induced damping and the laser-induced ultrafast modulation of the magnetic anisotropy, respectively.

Laser-induced THz magnetization precession for a tetragonal Heusler-like nearly compensated ferrimagnet

1510.06793v1

arXiv:1909.09838v1 [math.AP] 21 Sep 2019STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN-VOIGT DAMPING FATHI HASSINE AND NADIA SOUAYEH Abstract. We consider a coupled wave system with partial Kelvin-Voigt damping in the interval (−1,1), where one wave is dissipative and the other does not. When the damping is effective in the whole domain ( −1,1) it was proven in [17] that the energy is decreasing over the time with a rate equal to t−1 2. In this paper, using the frequency domain method we show the effect of the coupling and the non smoothness of the damping coefficient on the energy decay. Actually, as expected we show the lack of exponential stability, that the semigroup l oses speed and it decays polynomially with a slower rate then given in [17], down to zero at least as t−1 12. Contents 1. Introduction 1 2. Well-posedness 2 3. Strong stability 4 4. Lack of exponential stability 6 5. Polynomial stabilization 13 References 16 1.Introduction When a vibrating source disturbs the first particular of a med ium, a wave is created. This phenomena begins to travel from particle to particle along t he medium, which is typically modelled by a wave equation. In order to suppress those vibrations, th e most common approach is adding damping. It’s more likely to use one of two types: 1) The linear viscous damping or ”external damping”, it does mostly model an external frictional force, such that the auto-mobile shock absorber. 2) The Kelvin-Voigt damping, it’s also called the ”internal damping” or the ”material damping”, which is originated from the extension or compression of the vibrating particles. In therecent years, many researchers showed interest in pro blems involving this kind of damping. In control theory for instance it was shown that when the Kelvin -Voigt dampingcoefficient is satisfying somegeometrical controlconditionsthesemigroupcorresp ondingtothissystemisexponentialstable (see [14, 19]). Nonetheless, when the damping is arbitrary l ocalized with singular coefficient, it’s not the case any more (see [2, 13]). Actually, in one-dimensi onal case we can consider the following problem (1.1) utt−[ux(x,t)+b(x)uxt]x= 0−1< x <1, t≥0, u(t,−1) =u(t,1) = 0 t≥0, u(0,x) =u0(x),ut(0,x) =u1(x)−1≤x≤1, withb∈L∞(−1,1) And b(x) =/braceleftbigg 0 for x∈[0,1) a(x) forx∈(−1,0). Under the assumption that the damping coefficient has a singul arity at the interface of the damped and undamped regions, and behaves like xαnear the interface, it was proven by Liu abd Zhang [15] that the semigroup corresponding to the system is polyn omially or exponentially stable and 2010Mathematics Subject Classification. 35B35, 35B40, 93D20. Key words and phrases. Coupled system, Kelvin-Voigt damping, frequency domain ap proach. 12 FATHI HASSINE AND NADIA SOUAYEH the decay rate depends on the parameter α∈(0,1]. When α= 0, Liu and Rao [13] showed that system (1.1) is polynomially stable with an order equal to 2 w here few years ago Liu and Liu [12] proved the lack of the exponential stability. When dealing with systems involving quantities described b y several components, pretending to control or observe all the state variables it turns out tha t certain systems possess an internal structure that compensates the lack of control variables. S uch a phenomenon is referred to as indirect stabilization or indirect control. For instance A labo et al. did study in [1] the coupled waves with partial frictional damping /braceleftbigg utt−∆u+αv= 0 x∈Ω, t≥0, vtt−∆v+αu+βvt= 0x∈Ω, t≥0, subjected to Dirichlet boundary conditions. It was proven t hen the semigroup corresponding to this system is not exponentially stable, but it’s polynomially w ith the rate t−1 2. In 2016, Oquendo and Pacheco studied the wave equation with internal coupled ter ms where the Kelvin-Voigt damping is global in one equation and the second equation is conservati ve. Although the damping is stronger than the frictional one, they had shown that the semigroup lo ses speed with a slower rate of t−1 4. For this kind of coupled visco-elastic models we distingui sh what is called the transmission problems which have been intensively studied by the first aut hor, Ammari and their collaborators in [2, 6, 7, 8, 9, 3] (see also [4]) where the systems studied in these papers are the wave or the plate equation or a coupled wave-plate equation. Assuming a non smooth and singular damping coefficient it was shown in these works a uniform and a non-unif orm decay rates of the energy. In thiswork, weexaminethebehaviourofacoupledwaves system withapartialKelvin-Voigt damping, namely we consider the following system where the first wave i s dissipative and the second one is conservative (1.2) utt(x,t)−[ux(x,t)+a(x)uxt(x,t)]x+vt(x,t) = 0 (x,t)∈(−1,1)×(0,+∞), vtt(x,t)−cvxx(x,t)−ut(x,t) = 0 ( x,t)∈(−1,1)×(0,+∞), u(0,t) =v(0,t) = 0,u(1,t) =v(1,t) = 0 ∀t >0, u(x,0) =u0(x),ut(x,0) =u1(x) ∀x∈(−1,1), v(x,0) =v0(x),vt(x,0) =v1(x) ∀x∈(−1,1), wherec >0 anda∈L∞(−1,1) is non-negative function. In this paper we assume that the damping coefficient is piecewise function in particular we suppose th atahave the following form a=d1[0,1], wheredis a strictly positive constant. Since the damping is singul ar, this system can be seen as a coupling of a conservative wave equation and a transmission wave equation. The natural energy of (u,v) solution of (1.2) at an instant tis given by E(t) =1 2/integraldisplay1 −1/parenleftbig |ut(x,t)|2+|vt(x,t)|2+|ux(x,t)|2+c|vx(x,t)|2/parenrightbig dx,∀t >0. Multiplying the first equation of (1.2) by ¯ u, the second by ¯ v, integrating over (-1,1) and then taking the real part leads to E′(t) =−/integraldisplay1 −1a(x)|uxt(x,t)|2dx,∀t >0. Therefore, the energy is a non-increasing function of the ti me variable t. We show the lack of the exponential stability and prove that the semigroup corresp onding to this system is polynomially stable for regular initial data and with a slower rate, down t ot−1 12. This paper is organized as follows. In section 2, we prove tha t system (1.2) is well-posed. In section 3, we show that the energy of the system is the strong s tability. In section 4, we prove the lack of exponential stability. In section 5, we prove a polyn omial stability decay of the energy. 2.Well-posedness In this section, we discuss the well-posesness of the proble m (1.2) using the semigroup theory. LetH= (H1 0(−1.1))2×(L2(−1,1))2be the Hilbert space endowed with the inner product define,STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 3 forU1= (u1,v1,w1,z1)∈ HandU2= (u2,v2,w2,z2)∈ H, by /an}bracketle{tU1,U2/an}bracketri}htH=/angbracketleftbig u1 x,u2 x/angbracketrightbig L2(−1,1)+/angbracketleftbig√cv1 x,√cv2 x/angbracketrightbig L2(−1,1)+/angbracketleftbig w1,w2/angbracketrightbig L2(−1,1)+/angbracketleftbig z1,z2/angbracketrightbig L2(−1,1). By setting y(t) = (u(t),v(t),ut(t),vt(t)) andy0= (u0,v0,u1,v1) we can rewrite system (1.2) as a first order differential equation as follow (2.1) ˙ y(t) =Ay(t), y(0) =y0, where A(u1,v1,u2,v2) =/parenleftbig u2,v2,/parenleftbig u1 x+au2 x/parenrightbig x−v2,cv1 xx+u2/parenrightbig , with (u1,v1,u2,v2)∈ D(A) =/braceleftbig (u1,v1,u2,v2)∈ H,(u2,v2)∈(H1 0(−1,1))2, v1∈H2(−1,1)∩H1 0(−1,1),/parenleftbig u1 x+au2 x/parenrightbig x∈L2(−1,1)/bracerightbig . For the well-posedness of system (2.1) we have the following proposition: Proposition 2.1. For an initial datum y0= (u0,v0,u1,v1)∈ H, there exists a unique solution y= (u,v,ut,vt)∈C([0,+∞),H)to problem (2.1). Moreover, if y0∈ D(A), then y= (u,v,ut,vt)∈C([0,+∞),D(A))∩C1([0,+∞),H). Proof.By Lumer-Phillips’ theorem (see [16]), it suffices to show tha tAis dissipative and maximal. (1) We first prove that Ais dissipative. Take Z= (u,v,w,z)∈ D(A). Then /an}bracketle{tAZ,Z/an}bracketri}htH=/an}bracketle{twx,ux/an}bracketri}htL2(−1,1)+c/an}bracketle{tzx,vx/an}bracketri}htL2(−1,1)+/an}bracketle{t(ux+awx)x,w/an}bracketri}htL2(−1,1) +/an}bracketle{tcvxx+w,z/an}bracketri}htL2(−1,1). By integration by parts and using the boundary conditions, i t holds: (2.2) ( AZ,Z)H=−/an}bracketle{tawx,wx/an}bracketri}htL2(−1,1)=−/integraldisplay1 −1a|wx|2dx≤0. This shows that Ais the dissipative. (2) Let us now prove that Ais maximal, i.e., that λI−Ais surjective for some λ >0. So, for any given ( f,g,f1,g1)∈ H, we solve the equation A(u,v,w,z) = (f,g,f1,g1), which is recast on the following way (2.3) w=f z=g uxx+(afx)x=f1+g cvxx=g1−f. It is well known that by Lax-Milgram’s theorem the system (2. 3) admits a unique solution ( u,v)∈ H1 0(−1,1)×H1 0(−1,1). Moreover by multiplying the second and the third lines of (2.3) by u,vre- spectively andintegrating over ( −1,1) andusingPoincar´ e inequality andCauchy-Schwarz inequ ality we find that there exists a constant C >0 such that/integraldisplay1 −1/parenleftbig |ux(x)|2+|vx(x)|2/parenrightbig dx≤C/integraldisplay1 −1/parenleftbig |fx(x)|2+|gx(x)|2+|f1(x)|2+|g1(x)|2/parenrightbig dx. It follows that ( u,v,w,z)∈ D(A) and we have /bardbl(u,v,w,z)/bardblH≤C/bardbl(f,g,f1,g1)/bardblH. This imply that 0 ∈ρ(A) and by contraction principle, we easily get R(λI−A) =Hfor sufficient smallλ >0. The density of the domain of Afollows from [16, Theorem 1.4.6]. Then thanks to Lumer-Phillips Theorem (see [16, Theorem 1.4.3]), the oper atorAgenerates a C0-semigroup of contractions on the Hilbert Hdenoted by ( etA)t≥0. /square4 FATHI HASSINE AND NADIA SOUAYEH 3.Strong stability Theorem 3.1. The semigroup (etA)t≥0is strongly stable in the energy space Hi.e., lim t→+∞/bardbletAy0/bardbl= 0,∀y0∈ D(A). Proof.To show that the semigroup ( etA)t≥0is strongly stable we only have to prove that the intersection of σ(A) withiRis an empty set. Since the resolvent of the operator Ais not compact (see [14]) but 0 ∈ρ(A) we only need to prove that ( iµI−A) is a one-to-one correspondence in the energy space Hfor allµ∈R∗. The proof will be done in two steps: In the first step we prove t he injective property of ( iµI−A) and in the second step we prove the surjective property of th e same operator. Step 1. Let ( u,v,w,z)∈ D(A) such that (3.1) A(u,v,w,z) =iµ(u,v,w,z). or equivalently, (3.2) w=iµu in (−1,1), z=iµv in (−1,1), (ux+awx)x−z=iµw in (−1,1), cvxx+w=iµz in (−1,1), u(−1) =u(1) = 0, v(−1) =v(1) = 0. Then taking the real part of the scalar product of (3.1) with ( u,v,w,z) we get Re(iµ/bardbl(u,v,w,z)/bardbl2 H) = Re/an}bracketle{tA(u,v,w,z),(u,v,w,z)/an}bracketri}htH=−d/integraldisplay1 0|wx|2dx= 0. Which implies that wx= 0 in (0 ,1). This implies that from the first equation (3.2) that ux= 0 in (0 ,1), which means that uis a constant in (0 ,1) and since u(1) = 0 we obtain that u=w= 0 in (0 ,1), Hence, from the third and the second equation of (3.2) one get s (3.3) u=w=v=z= 0 in (0 ,1), Using (3.3) then (3.2) is reduced to the following problem (3.4) w=iµu in (−1,0), z=iµv in (−1,0), µ2u+uxx−iµv= 0 in ( −1,0), µ2v+cvxx+iµu= 0 in ( −1,0), u(−1) =u(0) = 0, v(−1) =v(0) = 0. Lety= (u,v,ux,vx) andyx= (ux,vx,uxx,vxx) then (3.4) is recast as follow (3.5)/braceleftbigg yx=Aµyin(−1,0) Y(0) = 0 where Aµ= 0 0 1 0 0 0 0 1 −µ2iµ0 0 −iµ c−µ2 c0 0 . SinceAµis a bounded operator then the unique solution of (3.5) is y= 0 therefore u=v= 0 in (−1,0). Moreover, from the fist and the second equation of (3.4) we havew=z= 0 in (−1,1). Combining all this with(3.3), we deduce that u=v=w=z= 0 in (−1,1). This conclude the fist part of this proof.STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 5 Step 2. Now given ( f,g)∈ H, we solve the equation (iµI−A)(u,v,w,z) = (f,g,f1,g1). Or equivalently, (3.6) w=iµu−f z=iµv−g µ2u+uxx+iµ(aux)x−iµv= (afx)x−iµf−f1−g=F µ2v+cvxx+iµu=−µg+f−g1=G. Let’s define the operator A: (H1 0(−1,1))2−→(H−1(−1,1))2 (u,v)/ma√sto−→(−uxx−iµ(aux)x+iµv,−cvxx−iµu). First we are going to show that Ais an isomorphism. For this purpose we consider the two opera tor ˜AandCsuch that ˜A: (H1 0(−1,1))2−→(H−1(−1,1))2 (u,v)/ma√sto−→(−uxx−iµ(aux)x,−cvxx), andCsuch that A=C+˜A. It’s easy to show that ˜Ais an isomorphism, then we could rewrite A=˜A(Id−˜A−1(−C)). To begin with, thanks to the compact embedding H1 0(−1,1)2֒→L2(−1,1)2andL2(−1,1)2֒→H−1(−1,1)2, we see that ˜A−1is a compact operator. Secondly, it’s clear that Cis a bounded operator, therefore, thanks to Fredholms alternative, we only need to prove that ( Id−˜A−1(−C)) is injective. Let (u,v)∈(H1 0(−1,1))2such that ( Id−˜A−1(−C))(u,v) = 0, which implies that (˜A−(−C))(u,v) = 0. Or equivalently (3.7) uxx+iµ(aux)x−iµv= 0 in ( −1,1) cvxx+iµu= 0 in ( −1,1) u(−1) =u(1) = 0, v(−1) =v(1) = 0. Multiplying the first equation of (3.7) by ¯ uand the conjugate of the second by v, after integration over (−1,1), it follows −/integraldisplay1 −1|ux|2dx+c/integraldisplay1 −1|vx|2dx−iµ/integraldisplay1 −1a|ux|2dx= 0 Next, by taking the imaginary part, we can deduce that ux= 0 in (0 ,1) thenuis constant in (0 ,1) where with the boundary condition u(1) = 0 we have u= 0 in (0,1). Moreover, using the second equation of (3.7) we obtain v= 0 in (0 ,1), which implies that (3.7) that (3.8) uxx=iµv in(−1,1) vxx=−iµ cu in (−1,1) u(0) =u(−1) = 0, v(0) =v(−1) = 0 Lety= (u,v,ux,vx) andyx= (ux,vx,uxx,vxx), using the trace theorem we have: /braceleftbigg yx=Dµyin (−1,0) y(0) = 0, where Dµ= 0 0 1 0 0 0 0 1 0iµ0 0 −iµ2 c0 0 0 . With a same approach as in the first step, we can have the result that we are looking for (i.e. A is an isomorphism).6 FATHI HASSINE AND NADIA SOUAYEH Now, rewriting the third and the fourth lines of (3.6) one get s (u,v)−µ2A−1(u,v) =A−1(F,G). Let (u,v)∈ker(Id−µ2A−1), i.e.µ2(u,v)−A(u,v) = 0, so we can see that: (3.9)/braceleftbigg µ2u+uxx+iµ(aux)x−iµv= 0 in ( −1,1) µ2v+cvxx+iµu= 0 in ( −1,1). Furthermore, multiplying the first equation of (3.9) by ¯ uand the conjugate of the second by v, after integration over ( −1,1) and taking the imaginary part, we deduce that /integraldisplay1 −1a|ux|2dx=d/integraldisplay1 0|ux|2dx= 0. So, we get the same system as in the first step (see (3.2)). Thus , ker(I−µ2A−1) ={0(H−1(−1,1))2}. In another hand, thanks to the compact embeddings H1 0(−1,1)2֒→L2((−1,1))2andL2(−1,1)2֒→ H−1(−1,1)2, we see that A−1is a compact operator. Now, thanks to the Fredholm’s alterna tive, the operator ( Id−µ2A−1) is bijective in ( H1 0(−1,1))2.Finally, the equation (3.6) have a unique solution in H1 0(−1,1)2. This completes the proof. /square 4.Lack of exponential stability Now, we prove the lack of exponential stability given by the f ollowing theorem Theorem 4.1. The semigroup (etA)t≥0, is not exponentially stable in the energy space provided thatc >1and that (4.1) sin(2√cnπ)/ne}ationslash=O(n−1 2), Noting that the assumption c >1 is made here just to make the calculation readable. The seco nd assumption (4.1) can be fulfilled for instance by taking csuch that 2√cis an integer number. To prove (4.1) we mainly use the following theorem Theorem 4.2. (see[10, 18]) LetetBbe a bounded C0-semigroup on a Hilbert space Hwith generator Bsuch that iR⊂ρ(B). ThenetBis exponentially stable if and only if There exist a >0andM >0, such that /bardbletB/bardblL(H)≤Me−at,∀t≥0 if and only if limsup ω∈R,|ω|→∞/bardbl(iωI−B)−1/bardblL(H)<∞. Now, based on the Theorem 4.2 we prove the Theorem 4.1. Proof.Our main objective is to show that: (4.2) /bardbl(λI−A)−1/bardblL(H)is unbounded on the imaginary axis . Forn∈Nlarge enough let λ=λn=iωn, where (4.3)ωn=/radicaligg 8c(c+1)n2π2+2c+√ ∆ 4cwith ∆ = (8 c(c−1)π2n2)2+32(c+1)(cπn)2+4c2. It’s clear that ωn−→+∞and in particular we have (4.4) ωn=√c/parenleftbigg 2nπ+n−1 4π(c−1)−cn−3 32π3(c−1)3+o(n−4)/parenrightbigg and (4.5)1 ωn=1 2nπ√c−1 16√c(c−1)(πn)3+o(n−4). Define (F1,G1,F2,G2)∈(H1 0(0,1))2×(L2(0,1))2, such that F1=F1(x,n) = 0 ∀x∈(−1,1),STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 7 G1=G1(x,n) =/braceleftigg0 in (0 ,1) g1=sin(2nπx) 2nπin (−1,0), F2=F2(x,n) = 0 ∀x∈(−1,1), G2=G2(x,n) = 0 in (0 ,1) g2=csin(2nπx) i/radicalbigg 2c c+1+/radicalBig (c−1)2+4c ω2nin (−1,0). A straight forward calculation leads to (4.6) /bardbl(F1,G1,F2,G2)/bardbl2 H=1 2+1 2µ−−→1 2/parenleftbigg 1+1√c/parenrightbigg asnր+∞. Our goal is to prove that lim |λ|→∞/bardbl(λI−A)−1/bardblL(H)=∞. That’s why, we solve the resolvent equation (4.7) ( λI−A)(u1,v1,u2,v2) = (F1,G1,F2,G2). Step 1. For allx∈(0,1) , we have λu1−u2= 0 λv1−v2= 0 λu2−(1+λd)u1 xx+v2= 0 λv2−cv1 xx−u2= 0 v1(1) =u1(1) = 0(4.8) Let η+=−λ(1+λd−c)+ωn√reiφ 2 2(1+λd)and η−=−λ(1+λd−c)−ωn√reiφ 2 2(1+λd) where r=/radicalbig a2+b2,cos(φ) =a rand sin( φ) =b r, with a=−(1−c)2+d2ω2−4c ω2n b=−2d/parenleftbigg (1−c)ωn+2c ωn/parenrightbigg . It is important to note that √a=dω−(c−1)2 2dω−1−(c−1)4+16cd2 8d3ω−3+o(ω−3), b a=2(c−1) d+2(c−1)−4cd dω−3+o(ω−4) and i√r deiφ 2=λ−c−1 d−(c−1)3−d2(c−1)+2cd2 d3λ−2 +d2(c−1)2−(c−1)4−2cd3(c−1)−2cd2 d4λ−3+o(ω−3). Then we obtain η+=−λ+c d−c d2λ−1+(c−1)3+d2(c+1)+2c 2d3λ−2(4.9) +(c−1)4−(c−1)3−d2(c−1)(c−2)−2c 2d4+o(ω−3) and η−=−(c−1)3+d2(c+1) 2d3λ−2+(c−1)3(2−c)+d2(c−1)(c−2−2cd) 2d4λ−3+o(ω−3) (4.10)8 FATHI HASSINE AND NADIA SOUAYEH A straightforward calculation leads to (u1+η+v1)xx= (β+)2(u1+η+v1) (4.11) (u1+η−v1)xx= (β−)2(u1+η−v1), (4.12) where (β±)2=cλ2−λη±(1+λd) c(1+λd). So, fornlarge enough we get β±=ωn/radicalbig 2c(1+(dωn)2)√r±eiφ± 2, where r±=/radicalig a2 ±+b2 ±,cos(φ±) =a± r±and sin( φ±) =b± r± with a±=−(1+c)−(dωn)2±√r/parenleftbigg −dωncos/parenleftbiggφ 2/parenrightbigg +sin/parenleftbiggφ 2/parenrightbigg/parenrightbigg b±=cdωn±√r/parenleftbigg −cos/parenleftbiggφ 2/parenrightbigg −dωnsin/parenleftbiggφ 2/parenrightbigg/parenrightbigg . Noting that |a+|= 2(dω)2+c2−3c+6 2+o(ω−1), /radicalbig |a+|=√ 2dω+c2−3c+6 4√ 2dω−1+o(ω−1), b+=/parenleftbigg d(2cd+1−c)+(c−1)3 d/parenrightbigg ω−1+o(ω−1), and b+ a+=o(ω−2). Then we obtain (4.13) β+=λ√c−(c−1)(c−2) 8√ 2d2+o(ω−1), and (4.14) β2 +=λ2 c+o(1). Similarly we have b−= 2cdω+/parenleftbigg d(c−1−2cd)−(c−1)3 d/parenrightbigg ω−1+o(ω−1), /radicalbig b−=√ 2cdω/parenleftbigg 1+/parenleftbiggc−1−2cd 4c−(c−1)3 4cd2/parenrightbigg ω−2/parenrightbigg +o(ω−2) a−=−2c+o(ω−1), and a− b−=−ω−1 d+o(ω−2), then consequently we obtain (4.15) β−=/radicalbiggω deiπ 4−ω−1 2 2d3 2e−iπ 4+o(ω−1),STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 9 and β2 −=λ d−1 d2+(c−1)3+d(c−1)+2c 2cd3λ−1(4.16) −(c−1)3(2−c)+d(c−1)(c−2−2cd)+2c 2cd4λ−2+o(ω−2) Next, from (4.11), we get (u1+η+v1) =c1exβ++c2e−xβ+ and (u1+η−v1) =c3exβ−+c4e−xβ−. Recalling that u1(1) =v1(1) = 0 we can rewrite the last two equations as follow (4.17) ( u1+η+v1) =c1(exβ+−e(2−x)β+), (4.18) ( u1+η−v1) =c3(exβ−−e(2−x)β−). Hence by combining (4.17) and (4.18) we obtain (4.19) u1(x) =−c1η− η+−η−/parenleftig eβ+x−eβ+(2−x)/parenrightig +c3η+ η+−η−/parenleftig eβ−x−eβ−(2−x)/parenrightig , and (4.20) v1(x) =c1 η+−η−/parenleftig eβ+x−eβ+(2−x)/parenrightig −c3 η+−η−/parenleftig eβ−x−eβ−(2−x)/parenrightig . Step 2. For allx∈(−1,0) we have λu1−u2= 0 λv1−v2=g1 λu2−u1 xx+v2= 0 λv2−cv1 xx−u2=g2 v1(−1) =u1(−1) = 0.(4.21) Following to the third and the fourth equation of (4.8) and of (4.21) we can deduce, thanks to the regularity of the stats, that (1+λd)u1 x(0+) =u1 x(0−), (4.22) v1 x(0+) =v1 x(0−). (4.23) and (1+λd)u1 xx(0+) =u1 xx(0−), (4.24) v1 xx(0+) =v1 xx(0−). (4.25) We denote by (4.26) α+=λ 2/parenleftigg c−1+/radicaligg (1−c)2+4c ω2n/parenrightigg = (c−1)λ−c c−1λ−1−c2 (c−1)3+o(ω−3), and (4.27) α−=λ 2/parenleftigg c−1−/radicaligg (1−c)2+4c ω2n/parenrightigg =c c−1λ−1+o(ω−1) and we define for nlarge enough µ±as follow µ±=√ 2c/radicalbigg c+1−/parenleftig ±/radicalig (c−1)2+4c ω2n/parenrightig, in particular with the chose of ωnin (4.3) one get µ2 ±=λ λ−α± c.10 FATHI HASSINE AND NADIA SOUAYEH Besides, we have (4.28) µ+=√c/parenleftbigg 1−c 2(c−1)λ−2+o(ω−2)/parenrightbigg , (4.29) µ−= 1+λ−2 2(c−1)+o(ω−2), and (4.30)µ+ µ−=√c/parenleftbigg 1−c+1 2(c−1)λ−2+o(ω−2)/parenrightbigg . We set ω+ 1(x) = (u2+α+v2+µ+(u1 x+α+v1 x), (4.31) ω− 1(x) = (u2+α+v2−µ+(u1 x+α+v1 x)), (4.32) ω+ 2(x) = (u2+α−v2+µ−(u1 x+α−v1 x)), (4.33) ω− 2(x) = (u2+α−v2−µ−(u1 x+α−v1 x)). (4.34) Now, define Y= (ω+ 1,ω− 1,ω+ 2,ω− 2)tandZ= (g1x,g2)t. Then we have (4.35) Yx=AY+BZ where A= µ+(λ−α+ c) 0 0 0 0µ+(−λ+α+ c) 0 0 0 0 µ−(λ−α− c) 0 0 0 0 µ−(−λ+α− c) and B= −α+−µ+α+ c −α+µ+α+ c −α−−µ−α− c −α−µ−α− c . Then, a straightforward calculation leads to: µ+(λ−α+ c) = 2inπ. Using the boundary condition at −1 we get (4.36) ω+ 1(−1) =−ω− 1(−1) andω+ 2(−1) =−ω− 2(−1), Taking into account of (4.36) then the solution of (4.35) is w ritten as follow ω+ 1(x) =ω+ 1(−1)e2inπx−α+ 2/bracketleftbigg/parenleftbigg 1−µ+ µ−/parenrightbigg (x+1)e2inπx+1 2nπ/parenleftbigg 1+µ+ µ−/parenrightbigg sin(2nπx)/bracketrightbigg , (4.37) ω− 1(x) =−ω+ 1(−1)e−2inπx−α+ 2/bracketleftbigg/parenleftbigg 1−µ+ µ−/parenrightbigg (x+1)e−2inπx+1 2nπ/parenleftbigg 1+µ+ µ−/parenrightbigg sin(2nπx)/bracketrightbigg , (4.38) ω+ 2(x) =ω+ 2(−1)eµ−(λ−α− c)(x+1)+α− 2inπ+µ−/parenleftbig λ−α− c/parenrightbig/bracketleftig e−2inπx+eµ−(λ−α− c)(x+1)/bracketrightig , (4.39) ω− 2(x) =−ω+ 2(−1)e−µ−(λ−α− c)(x+1)−α− 2inπ+µ−/parenleftbig λ−α− c/parenrightbig/bracketleftig e2inπx+e−µ−(λ−α− c)(x+1)/bracketrightig . (4.40) Taking the trace of ω+ 1andω− 1in (4.37)-(4.38) and in (4.31)-(4.32) on the boundary 0 and u sing the continuity of the states u2andv2we obtain (ω+ 1+ω− 1)(0−) =α+/parenleftbiggµ+ µ−−1/parenrightbigg = 2u2(0−)+2α+v2(0−) = 2λ(u1(0−)+α+v1(0−)) = 2λ(u1(0+)+α+v1(0+)) =2λ η+−η−/parenleftig c1(1−e2β+)(α+−η−)+c3(1−e2β−)(η+−α+)/parenrightig ,STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 11 where we have used the the expressions of u1andv1in (4.19) and (4.20). This implies that (4.41) c3=1−e2β+ 1−e2β−Anc1+Bn 1−e2β− where An=η−−α+ η+−α+=c−1 c/parenleftbigg 1+λ−1 d−λ−2 c−1+o(ω−2)/parenrightbigg =c−1 c/parenleftbigg 1+n−1 2iπd√c−n−2 4π2c(c−1)+o(ω−2)/parenrightbigg , (4.42) and Bn=α+(η+−η−)/parenleftig µ+ µ−−1/parenrightig 2λ(η+−α+) =(c−1)(√c−1) 2c/parenleftbigg 1−c−1 dλ−1−/parenleftbigg1 (c−1)2+√c(c+1) 2(√c−1)(c−1)/parenrightbigg λ−2+o(ω−2)/parenrightbigg =(c−1)(√c−1) 2c/parenleftbigg 1−c−1 2iπd√cn−1−/parenleftbigg1 (c−1)2+√c(c+1) 2(√c−1)(c−1)/parenrightbigg ×n−2 4π2c+o(n−2)/parenrightbigg .(4.43) where we used here (4.9), (4.10), (4.26), (4.27), (4.30) and (4.3). Using (4.37)-(4.38) and (4.24)-(4.25), one gets (ω+ 1−ω− 1)′(0−) = 2inπα+/parenleftbiggµ+ µ−−1/parenrightbigg = 2µ+(u1+α+v1)xx(0−) = 2µ+((1+λd)u1+α+v1)xx(0+) =2µ+/bracketleftbig c1β2 +(1−e2β+)(α+−(1+λd)η−)+c3β2 −(1−e2β−)((1+λd)η+−α+)/bracketrightbig η+−η−. Then we obtain (4.44) c1=1−e2β− 1−e2β+A′ nc3+B′ n 1−e2β+ where A′ n=β2 −(α+−(1+λd)η+) β2 +(α+−(1+λd)η−) =c c−1/parenleftbigg 1−λ−1 d+/parenleftbigg(c−1)3 2cd2+c−1 2cd+3−c 2d2+1 2/parenrightbigg λ−2+o(ω−2)/parenrightbigg =c c−1/parenleftbigg 1−n−1 2iπd√c+/parenleftbigg(c−1)3 2cd2+c−1 2cd+3−c 2d2+1 2/parenrightbiggn−2 4π2c+o(n−2)/parenrightbigg , (4.45) and B′ n=inπα+(η+−η−)/parenleftig µ+ µ−−1/parenrightig µ+β2 +(α+−(1+λd)η−) =nπ(c−√c) 2ω/parenleftbigg −1+c dλ−1+/parenleftbiggc+√c+3 2(c−1)2−c+1+d2 2d2/parenrightbigg λ−2/parenrightbigg +o(ω−2) =√c−1 2/parenleftbigg −1+√c 2iπdn−1+/parenleftbigg√c+4 c−1−c+1+d d2/parenrightbiggn−2 8cπ2+o(n−2)/parenrightbigg . (4.46) where we used here (4.9), (4.10), (4.14), (4.16), (4.26), (4 .27), (4.28), (4.30) and (4.3). Combining (4.41) and (4.44) then we find that (4.47) c1=1 1−e2β+×A′ nBn+B′ n 1−AnA′n=c′ 1 1−e2β+12 FATHI HASSINE AND NADIA SOUAYEH and (4.48) c3=1 1−e2β−×AnB′ n+Bn 1−AnA′n=c′ 3 1−e2β−, where following to (4.42), (4.43), (4.45) and (4.46) we have (4.49) c′ 1=O(1) and c′ 3=O(1). In another hand, by denoting θ=−iµ−/parenleftbig λ−α− c/parenrightbig and by using the same argument as previously, one gets (ω+ 2+ω− 2)(0−) = 2isin(θ)ω+ 2(−1)+2α− 2nπ−θsin(θ) = 2λ(u1+α−v1)(0−) = 2λ(u1+α−v1)(0+) =2λ η+−η−/parenleftbig c′ 1(α−−η−)+c′ 3(η+−α−)/parenrightbig . It’s clear that θ/ne}ationslash= 0[π] then we can write ω+ 2(−1) =λ isin(θ)(η+−η−)/bracketleftbig c′ 1(α−−η−)+c′ 3(η+−α−)/bracketrightbig −α− 2inπ+iθ. (4.50) Noting that from (4.3), (4.4), (4.27) and (4.29) we have (4.51) θ=ω/parenleftbigg 1−3 2(c−1)ω−2+o(ω−2)/parenrightbigg =√c/parenleftbigg 2nπ+cπ−12 4cπ2(c−1)n−1+o(n−1)/parenrightbigg Then from (4.4), (4.9), (4.10), (4.27), (4.29) and (4.51) we deduce that (4.52) ω+ 2(−1)∼2πn√cc′ 3 sin(θ) Using (4.22)-(4.23), (4.31)-(4.32) and (4.37)-(4.38) we g et ω+ 1(−1) =(ω+ 1−ω− 1)(0−) 2=µ+(u1+α+v1)x(0−) =µ+((1+λd)u1+α+v1)x(0+) =µ+ η+−η−/bracketleftig c1β+(1+e2β+)(α+−(1+λd)η−)+c3β−(1+e2β−)((1+λd)η+−α+)/bracketrightig . (4.53) Then from (4.4), (4.9), (4.10), (4.13), (4.15), (4.27) and ( 4.28) we deduce that (4.54) ω+ 1(−1)∼c′ 3/radicalbiggc de−iπ 4(2π√cn)3 2. Next, for all x∈(−1,0) we have v1 x(x) =1 2µ−µ+(α+−α−)/bracketleftbig α−(ω+ 1(x)−ω− 1(x))−α+(ω+ 2(x)−ω− 2(x))/bracketrightbig =1 2µ−µ+(α+−α−)/bracketleftigg µ−/parenleftbigg 2ω+ 1(−1)cos(2nπx)−iα+/parenleftbigg 1−µ+ µ−/parenrightbigg (x+1)sin(2 nπx)/parenrightbigg (4.55) −µ+/parenleftbigg 2ω+ 2(−1)cos(θ(x+1))+2α− 2inπ+iθ(cos(2nπx)+cos(θ(x+1)))/parenrightbigg/bracketrightigg , where we have used (4.31)-(4.34) and (4.37)-(4.40). Thus fu rther leads to /bardblv1 x/bardbl2 L2(−1,0)≥max/braceleftbigg|ω+ 1(−1)|2 2µ2 +|α+−α−|2,|ω+ 2(−1)|2 µ2 −|α+−α−|2/bracerightbigg −|α+|2(µ+−µ−)2 4µ2 −µ2 +|α+−α−|2(4.56) −min/braceleftbigg|ω+ 1(−1)|2 2µ2 +|α+−α−|2,|ω+ 2(−1)|2 µ2 −|α+−α−|2/bracerightbigg −2|α−|2 µ2 −µ2 +(2nπ+θ)2|α+−α−|2. Since, sin(θ)/ne}ationslash=O(n−1 2), asngoes to the infinity (by (4.51) assumption (4.1)) then by usin g (4.3), (4.28), (4.29), (4.26), (4.27), (4.52) and (4.54) we can show that the second and the f ourth terms of the right hand sideSTABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 13 of (4.56) are bounded while the sum of the fist and the third ter ms tends to the infinity as ngoes to +∞, therefore we obtain (4.57) /bardblv1 x/bardbl2 L2(−1,0)asnր+∞. Last but not least, we have (4.58) /bardbl(iωnI−A)−1(F1,G1,F2,G2)/bardblH=/bardbl(u1,v1,u2,v2)/bardbl2 H≥/integraldisplay0 −1|v1 x(x)|2dx−→+∞,asnր+∞. Finally we conclude, using (4.58) and (4.6) that limsup ω∈R,|ω|→∞/bardbl(iωI−A)−1/bardblL(H)= +∞. So,etAis not exponentially stable in the energy space. This comple tes the proof. /square 5.Polynomial stabilization This subsection aims to prove the polynomial stability give n by the following theorem: Theorem 5.1. The semigroup of contraction (eTA)t≥0is polynomially stable of order1 12. Our method is based on the Borichev and Tomilov result given b y the following: Theorem 5.2. [5, Theorem 2.4] LetBbe a generator of a C0-semigroup of contraction in a Hilbert spaceXwith domain D(B)such that iR⊂σ(B)thenetBis polynomially stable with order1 γ,γ >0 i.e. there exists C >0such that /bardbletBU0/bardblX≤C (1+t)1 γ/bardblU0/bardblD(B),∀t≥0,∀U0∈ D(B), if and only if limsup β∈R,|β|→∞/bardblβ−γ(iβ−B)−1/bardblL(X)<+∞. Based on Theorem 5.2 we are able now to prove our main result gi ven in Theorem 5.1 of this section. For this purpose, let’s consider the following: Proposition 5.1. The operator Adefined in (2.1)satisfies: (5.1) limsup β∈R,|β|→∞/bardblβ−12(iβ−A)−1/bardblL(H)<+∞. Proof.To prove (5.1) we use an argument of contradiction. In fact, i f (5.1) is false, then, there exist βn∈R+andYn= (u1 n,v1 n,u2 n,v2 n)∈ D(A) such that (5.2) /bardblYn/bardblH= 1, βnր+∞andβγ(iβnI−A)Yn:= (f1 n,g1 n,f2 n,g2 n)−→0 inHasnր+∞. Equivalently, we have (5.3) βγ n/parenleftbig iβnu1 n−u2 n/parenrightbig =f1 n−→0 inH1 0(−1,1), (5.4) βγ n/parenleftbig iβnv1 n−v2 n/parenrightbig =g1 n−→0 inH1 0(−1,1), (5.5) βγ n/parenleftbig iβnu2 n−/parenleftbig u1 nx+au2 nx/parenrightbig x+v2 n/parenrightbig =f2 n−→0 inL2(−1,1), (5.6) βγ n/parenleftbig iβnv2 n−cv1 nxx−u2 n/parenrightbig =g2 n−→0 inL2(−1,1). We denote by Tn=u1 nx+au2 nx. Taking the real part of /an}bracketle{tβγ(iβnI−A)Yn,Yn/an}bracketri}htHthen by the dissipation property of the semigroup of the operator Awe get βγ n/integraldisplay1 0d.|u2 nx|2dx−→0,14 FATHI HASSINE AND NADIA SOUAYEH which leads to (5.7) βγ 2n/bardblu2 nx/bardblL2(0,1)−→0. Now thanks to (5.3) and (5.7), we obtain (5.8) βγ 2+1 n/bardblu1 nx/bardblL2(0,1)−→0. From (5.7) and (5.8), it follows (5.9) βγ 2n/bardblTn/bardblL2(0,1)−→0. Taking the inner product of (5.5) with u2 ninL2(0,1) we get (5.10) β3γ 4n/parenleftig iβn/bardblu2 n/bardbl2 L2(0,1)+/an}bracketle{tTn,u2 nx/an}bracketri}htL2(0,1)+Tn(0+)u2n(0+)+/an}bracketle{tv2 n,u2 n/an}bracketri}htL2(0,1)/parenrightig =o(1). Thanks to (5.2), (5.7) and (5.9), it’s clear that the second a nd the last terms converge to zero. Furthermore, we have β3γ 4nTn(0+)u2n(0+)≤Cβγ 2n/parenleftbigg /bardblTn/bardbl1 2 L2(0,1)./bardblu2 nx/bardbl1 2 L2(0,1)./bardblT′ n/bardbl1 2 L2(0,1)./bardblu2 n/bardbl1 2 L2(0,1)/parenrightbigg . From (5.5) we can see that /bardblβnu2 n+v2 n/bardblL2(0,1)∼ /bardblT′ n/bardblL2(0,1)which implies that β3γ 4n|Tn(0+)|.|u2n(0+)| ≤Cβ3γ 4n/bardblTn/bardbl1 2 L2(0,1)./bardblu2 nx/bardbl1 2 L2(0,1)× /parenleftbigg /bardblβnu2 n/bardbl1 2 L2(0,1)+/bardblv2 n/bardbl1 2 L2(0,1)+o(1)/parenrightbigg ./bardblu2 n/bardbl1 2 L2(0,1) ≤C/bardblβγ 2nTn/bardbl1 2 L2(0,1)./bardblβγ 2nu2 nx/bardbl1 2 L2(0,1)× /parenleftbigg /bardblβnu2 n/bardbl1 2 L2(0,1)+/bardblv2 n/bardbl1 2 L2(0,1)/parenrightbigg /bardblβγ 2nu2 n/bardbl1 2 L2(0,1)+o(1) ≤/parenleftbigg 1+β1 2+γ 4n./bardblu2 n/bardblL2(0,1)/parenrightbigg o(1). (5.11) Combining (5.10) and (5.11), one follows (5.12) β1 2+3γ 8n/bardblu2 n/bardblL2(0,1)−→0. Moreover, multiplying (5.5) by β−γ 2n(1−x)Tnand integrating over the interval (0 ,1) then by taking account of (5.9), an integration by parts leads to Re/an}bracketle{tiβ1 2+3γ 8nu2 n,(1−x)β1 2−3γ 8+γ 2nTn/an}bracketri}htL2(0,1)+βγ 2n 2/parenleftig |Tn(0+)|2−/bardblTn/bardbl2 L2(0,1)/parenrightig (5.13) +βγ 2nRe/an}bracketle{tv2 n,(1−x)Tn/an}bracketri}htL2(0,1)=o(1). We suppose that γ≥4 3. It’s clear from (5.2), (5.9) and (5.12) that the first, the th ird and the last terms of (5.13) converge to zero then one gets (5.14) βγ 4n.|Tn(0+)| −→0. Taking into account to (5.8) then the trace formula gives (5.15) βγ 2+1 n.|u1 n(0+)| −→0. Substituting(5.4) into(5.5) andtakingtheinnerproductw ithβ3−γ nv1 ninL2(0,1) thenbyintegrating by parts we have (5.16) iβ4 n/angbracketleftbig u2 n,v1 n/angbracketrightbig L2(0,1)+β3 n/angbracketleftbig Tn,v1 n/angbracketrightbig L2(0,1)+iβ4 n/bardblv1 n/bardbl2 L2(0,1)+β3 nTn(0+)v1n(0+) =o(1). Takingγ≥12 and using (5.2), (5.9), (5.12) and (5.14) we can see that th e first, the second and the fourth terms of (5.16) converge to zero, therefore (5.17) β2 n./bardblv1 n/bardblL2(0,1)−→0.STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 15 From (5.4) and (5.17) it follows (5.18) βn/bardblv2 n/bardblL2(0,1)−→0. Multiplying (5.6) with β−γ n(1−x)v1nxand integrating over (0 ,1) then by taking the real part we find c 2/parenleftig |v1 nx(0+)|2−/bardblv1 nx/bardbl2 L2(0,1)/parenrightig = Re/angbracketleftbig u2 n,(1−x)v1 nx/angbracketrightbig L2(0,1) −Re/an}bracketle{tiβnv2 n,(1−x)v1 nx/an}bracketri}htL2(0,1)+o(1). Using (5.2), (5.12) and (5.18) leads to (5.19) |v1 nx(0+)|2−/bardblv1 nx/bardbl2 L2(0,1)−→0. We take the inner product of (5.6) with β−γ nxv1 ninL2(0,1) then we have c/parenleftbigg/integraldisplay1 0x|v1 nx(x)|2dx+/angbracketleftbig v1 nx,v1 n/angbracketrightbig L2(0,1)/parenrightbigg =/angbracketleftbig u2 n,xv1 n/angbracketrightbig L2(0,1)−iβn/an}bracketle{tv2 n,xv1 n/an}bracketri}htL2(0,1)+o(1). Using (5.2), (5.12) and (5.18) we deduce that /integraldisplay1 0x|v1 nx(x)|2dx−→0. This implies in particular that for every εin (0,1) we have (5.20) /bardblv1 nx/bardblL2(ε,1)−→0 asnր+∞. Multiplying (5.6) with β−γ n(1−x)v1nxand integrating over (0 ,ε) then by taking the real part we find c 2/parenleftig |v1 nx(ε)|2−/bardblv1 nx/bardbl2 L2(ε,1)/parenrightig = Re/angbracketleftbig u2 n,(1−x)v1 nx/angbracketrightbig L2(ε,1)−Re/an}bracketle{tiβnv2 n,(1−x)v1 nx/an}bracketri}htL2(ε,1)+o(1). Besides, from (5.2), (5.12), (5.18) and (5.20) we follow |v1 nx(ε)| −→0 asnր+∞. Then we deduce that (5.21) v1 nx(x)−→0 a.e. in [0 ,1] asnր+∞. Now, (5.2) and (5.21) allows the use of the dominated converg ence theorem and lead to (5.22) /bardblv1 nx/bardblL2(0,1)−→0. Therefore, we obtain (5.23) |v1 n(0+)| −→0. By combining (5.19) and (5.22) we find (5.24) |v1 nx(0+)| −→0. Furthermore, taking the inner product of (5.4) with β1−γ n(1−x)v1 nxand then considering the imaginary part one gets β2 nRe(v1 nx,(1−x)v1 n)−Imβn(v2 n,(1−x)v1 nx) =o(1) =1 2(β2 n|v1 n(0+)|2−β2 n/bardblv1 n/bardbl2)−βnIm/an}bracketle{tv2 n,(1−x)v1 nx/an}bracketri}ht Adding to this (5.23), (5.17) and (5.18) we can deduce that : (5.25) βn|v1 n(0+)| −→016 FATHI HASSINE AND NADIA SOUAYEH Thanks to (5.14), (5.15), (5.23) and (5.24) one gets βγ 2+1 n.u1 n(0−)−→0, (5.26) βγ 4.u1 nx(0−)−→0, (5.27) βnv1 n(0−)−→0, (5.28) v1 nx(0−)−→0. (5.29) Next, inserting (5.3) into (5.5) and inserting (5.4) into (5 .6) and consider both equations in the interval (0 ,1), leads to (5.30) −β2 nu1 n−u1 nxx+v2 n=β−γ nf2 n+iβ1−γ nf1 n, and (5.31) −β2 nv1 n−cv1 nxx−u2 n=β−γ ng2 n+iβ1−γ ng1 n. A straightforward calculation shows that the real part of th e inner productof (5.30) with ( x+1).u1 nx and that the real part of the inner of (5.31) with ( x+1).v1 nxleads to 1 2/integraldisplay0 −1/parenleftbig |βnu1 n|2+|u1 nx|2/parenrightbig dx=1 2/parenleftbig |u1 nx(0−)|2+β2 n|u1 n(0−)|2/parenrightbig (5.32) −Re/an}bracketle{tv2 n,(x+1)u1 nx/an}bracketri}htL2(−1,0)+o(1), and 1 2/integraldisplay0 −1/parenleftbig |βnv1 n|2+c|v1 nx|2/parenrightbig dx=1 2/parenleftbig c|v1 nx(0−)|2+β2 n|v1 n(0−)|2/parenrightbig (5.33) +Re/an}bracketle{tu2 n,(x+1)v1 nx/an}bracketri}htL2(−1,0)+o(1). Where we have used (5.2)-(5.6). In another hand, from (5.2), (5.12), (5.18) and (5.26)-(5.29) we get (5.34)/integraldisplay0 −1/parenleftbig |βnu1 n|2+|u1 nx|2/parenrightbig dx−→0, and (5.35)/integraldisplay0 −1/parenleftbig |βnv1 n|2+c|v1 nx|2/parenrightbig dx−→0. Now by summing (5.8) (5.12), (5.17), (5.18), (5.34) and (5.3 5) we can see that (5.36) /bardblYn/bardblH−→0. This contradicts (5.2) and so (5.1) holds true with γ≥12. This completes the proof. /square References [1] F. Alabau, P. Cannarsa, V. Komornik, Indirect internal s tabilization of weakly coupled evolution equations, J. Evol. Equ.2 (2002) 127150. [2]K. Ammari, F. Hassine and L. Robbiano , Stabilization for the wave equation with singular Kelvin- Voigt damping, arXiv:1805.10430. [3] K. Ammari, Z. Liu and F. Shel, Stabilization for the wave e quation with singular Kelvin-Voigt damping, arXiv:1805.10430. [4]K. Ammari and S. Nicaise, Stabilization of elastic systems by collocated feedback, 2124, Springer, Cham, 2015. [5] A. Borichev and Y. Tomilov, Optimal polynomial decay of f unctions and operator semigroups, Math. Ann., 347 (2010), 455–478. [6]F. Hassine, Stability of elastic transmission systems with a local Kelv inVoigt damping, European Journal of Control, 23(2015), 84–93. [7]F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli p late and wave equation with a localized Kelvin-Voigt damping, Discrete and Continuous Dynamical Systems - Series B, 21(2016), 1757–1774. [8]F. Hassine, Energy decay estimates of elastic transmission wave/beam s ystems with a local Kelvin-Voigt damp- ing,Internat. J. Control, 89(10) (2016), 1933-1950. [9]F. Hassine, Logarithmic stabilization of the Euler-Bernoulli plate eq uation with locally distributed Kelvin-Voigt damping, Evolution Equations and Control Theory, 455(2) (2017), 1765–1782. [10] F. Huang, Characteristic conditions for exponential s tability of linear dynamical systems in Hilbert space, Ann. Differential Equations ,1(1985), 43–56.STABILITY FOR COUPLED WAVES WITH LOCALLY DISTURBED KELVIN- VOIGT DAMPING 17 [11] F. Huang, On the mathematical model for linear elastic s ystems with analytic damping, SIAM J. Control Optim., 26(3) (1988), 714–724. [12]K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam wit h locally distributed Kelvin– Voigt damping, SIAM Journal on Control and Optimization, 36(1998), 1086–1098. [13]K. S. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Zeitschrift f¨ ur Angewandte Mathematik und Physik (ZAMP), 56(2005), 630–644. [14]K. S. Liu and B. Rao, Exponential stability for wave equations with local Kelvin -Voigt damping, Zeitschrift f¨ ur Angewandte Mathematik und Physik (ZAMP), 57(2006), 419–432. [15] Z. Liu and Q. Zhang, Stability of a string with local Kelv in-Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54(2016), 1859–1871. [16] A. Pazy, Semigroups of linear operators and applications to partial differential equations , Springer, New York, 1983. [17] H. Portillo Oquendo and P. Snez Pacheco, Optimal decay f or coupled waves with KelvinVoigt damping, Applied Mathematics Letters 67(2017), 16-20. [18] J. Pr¨ uss, On the spectrum of C0-semigroups, Trans. Amer. Math. Soc. ,248(1984), 847–857. [19]L. Tebou, A constructive method for the stabilization of the wave equa tion with localized KelvinVoigt damping, C. R. Acad. Sci. Paris , Ser. I,350(2012), 603–608. UR Analysis and Control of PDEs, UR 13ES64, Department of Mat hematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia E-mail address :fathi.hassine@fsm.rnu.tn UR Analysis and Control of PDEs, UR 13ES64, Department of Mat hematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia E-mail address :nadia.souayeh@fst.utm.tn

2019-09-21

We consider a coupled wave system with partial Kelvin-Voigt damping in the interval (-1,1), where one wave is dissipative and the other does not. When the damping is effective in the whole domain (-1,1) it was proven in H.Portillo Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt damping, Applied Mathematics Letters 67 (2017), 16-20. That the energy is decreasing over the time with a rate equal to $t^{-\frac{1}{2}}$. In this paper, using the frequency domain method we show the effect of the coupling and the non smoothness of the damping coefficient on the energy decay. Actually, as expected we show the lack of exponential stability, that the semigroup loses speed and it decays polynomially with a slower rate then given in, H.Portillo Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt damping, Applied Mathematics Letters 67 (2017), 16-20, down to zero at least as $t^{-\frac{1}{12}}$.

Stability for coupled waves with locally disturbed Kelvin-Voigt damping

1909.09838v1

Nonlocal feedback in ferromagnetic resonance Thomas Bose and Steffen Trimper Institute of Physics, Martin-Luther-University, D-06099 Halle, Germany (Dated: April 27, 2022) Abstract Ferromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback effects. The equation of motion for the magnetization dynamics is nonlocal in both space and time andincludesisotropic, anisotropicanddipolarenergycontributionsaswellastheconservedGilbert- and the non-conserved Bloch-damping. We derive an analytical expression for the peak-to-peak linewidth. It consists of four separate parts originated by Gilbert damping, Bloch-damping, a mixed Gilbert-Bloch component and a contribution arising from retardation. In an intermediate frequency regimetheresultsarecomparablewiththecommonlyusedLandau-Lifshitz-Gilberttheorycombined with two-magnon processes. Retardation effects together with Gilbert damping lead to a linewidth the frequency dependence of which becomes strongly nonlinear. The relevance and the applicability of our approach to ferromagnetic resonance experiments is discussed. PACS numbers: 76.50.+g; 76.60.Es; 75.70.Ak; 75.40.Gb thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de 1arXiv:1204.5342v1 [cond-mat.mes-hall] 24 Apr 2012I. INTRODUCTION Ferromagnetic resonance enables the investigation of spin wave damping in thin or ul- trathin ferromagnetic films. The relevant information is contained in the linewidth of the resonance signal [1–3]. Whereas the intrinsic damping included in the Gilbert or Landau- Lifshitz-Gilbert equation [4, 5], respectively, predicts a linear frequency dependence of the linewidth [6], the extrinsic contributions associated with two-magnon scattering processes show a nonlinear behavior. Theoretically two-magnon scattering was analyzed for the case that the static external field lies in the film plane [7, 8]. The theory was quantitatively validated by experimental investigations with regard to the film thickness [9]. Later the approach was extended to the case of arbitrary angles between the external field and the film surface [10]. The angular dependence of the linewidth is often modeled by a sum of contributions including angular spreads and internal field inhomogeneities [11]. Among oth- ers, two-magnon mechanisms were used to explain the experimental observations [12–17] whereas the influence of the size of the inhomogeneity was studied in [18]. As discussed in [3, 14] the two-magnon contribution to the linewidth disappears for tipping angles between magnetization and film plane exceeding a critical one crit M==4. Recently, deviations from this condition were observed comparing experimental data and numerical simulations [17]. Spin pumping can also contribute to the linewidth as studied theoretically in [19]. How- ever, a superposition of both the Gilbert damping and the two-magnon contribution turned out to be in agreement very well with experimental data illustrating the dependence of the linewidth on the frequency [16, 20–23]. Based on these findings it was put into question whether the Landau-Lifshitz-Gilbert equation is an appropriate description for ferromag- netic thin films. The pure Gilbert damping is not able to explain the nonlinear frequency dependence of the linewidth when two-magnon scattering processes are operative [3, 24]. Assuming that damping mechanisms can also lead to a non-conserved spin length a way out might be the inclusion of the Bloch equations [25, 26] or the the Landau-Lifshitz-Bloch equation [27, 28] into the concept of ferromagnetic resonance. Another aspect is the recent observation [29] that a periodic scattering potential can alter the frequency dependence of the linewidth. The experimental results are not in agreement with those based upon a combination of Gilbert damping and two-magnon scattering. It was found that the linewidth as function of the frequency exhibits a non monotonous be- 2havior. The authors [29] suggest to reconsider the approach with regard to spin relaxations. Moreover, it would be an advantage to derive an expression for the linewidth as a measure for spin damping solely from the equation of motion for the magnetization. Taking all those arguments into account it is the aim of this paper to propose a gener- alized equation of motion for the magnetization dynamics including both Gilbert damping and Bloch terms. The dynamical model allows immediately to get the magnetic susceptibil- ity as well as the ferromagnetic resonance linewidth which are appropriate for the analysis of experimental observations. A further generalization is the implementation of nonlocal effects in both space and time. This is achieved by introducing a retardation kernel which takes into account temporal retardation within a characteristic time and a spatial one with a characteristic scale . The last one simulates an additional mutual interaction of the magnetic moments in different areas of the film within the retardation length . Re- cently such nonlocal effects were discussed in a complete different context [30]. Notice that retardation effects were already investigated for simpler models by means of the Landau- Lifshitz-Gilbert equation. Here the existence of spin wave solutions were in the focus of the consideration [31]. The expressions obtained for the frequency/damping parameters were converted into linewidths according to the Gilbert contribution which is a linear function of the frequency [31, 32]. In the present approach we follow another line. The propagating part of the varying magnetization is supplemented by the two damping terms due to Gilbert and Bloch, compare Eq. (9). Based on this equation we derive analytical expressions for the magnetic susceptibility, the resonance condition and the ferromagnetic resonance linewidth. Due to the superposition of damping and retardation effects the linewidth exhibits a non- linear behavior as function of the frequency. The model is also extended by considering the general case of arbitrary angles between the static external field and the film surface. Moreover the model includes several energy contributions as Zeeman and exchange energy as well as anisotropy and dipolar interaction. The consequences for ferromagnetic resonance experiments are discussed. II. DERIVATION OF THE EQUATION OF MOTION In order to define the geometry considered in the following we adopt the idea presented in [10], i.e. we employ two coordinate systems, the xyz-system referring to the film surface 3ΘMey ex,eX ezMS eZeY ΘHH0 /Bullet /Bullet /BulletξM(z1) M(z2) M(z3)hrf d lxlzFIG. 1. (Color online) The geometry referring to the film and the magnetization. Further descrip- tion in the text. and the XYZ-system which is canted by an angle Mwith respect to the film plane. The situation for a film of thickness dis sketched in Fig. 1. The angle Mdescribing the direction of the saturation magnetization, aligned with the Z-axis, originates from the static external fieldH0which impinges upon the film surface under an angle H. Therefore, it is more convenient to use the XYZ-system for the magnetization dynamics. As excitation source we consider the radio-frequency (rf) magnetic field hrfpointing into the x= X-direction. It should fulfill the condition hrfH0. To get the evolution equation of the magnetization M(r;t),r= (x;y;z )we have to define the energy of the system. This issue is well described in Ref. [10], so we just quote the most important results given there and refer to the cited literature for details. Since we consider the thin film limit one can perform the average along the direction perpendicular to the film, i.e. M(rk;t) =1 dZd=2 d=2dyM(r;t); (1) where rk= (x;0;z)lies in the film plane. In other words the spatial variation of the magnetization across the film thickness dis neglected. The components of the magnetization point into the directions of the XYZ-system and can be written as [33] M(rk;t) =MX(rk)eX+MY(rk)eY+ MSM2 X(rk) +M2 Y(rk) 2MS eZ:(2) 4Typically the transverse components MX;Yare assumed to be much smaller than the satu- ration magnetization MS. Remark that terms quadratic in MX;Yin the energy will lead to linear terms in the equation of motion. The total energy of the system can now be expressed in terms of the averaged magnetization from Eq. (1) and reads H=Hz+Hex+Ha+Hd: (3) The different contributions are the Zeeman energy Hz=Z d3rH0sin ( HM)MY(rk) Z d3rH0cos ( HM) MSMX(rk)2+MY(rk)2 2MS ;(4) the exchange energy Hex=D 2MSZ d3r rMX(rk)2+ rMY(rk)2; (5) the surface anisotropy energy Ha=HSMSV 2sin2(M) +HS 2sin(2 M)Z d3rM Y(rk) +HS 2MScos(2 M)Z d3rM Y(rk)2sin2(M)Z d3rM X(rk)2;(6) and the dipolar energy Hd=2M2 SVsin2(M) +Z d3r 2MSsin(2 M)MY(rk) +dk2 z kksin2(M)(dkk2) cos2(M)2 sin2(M) MY(rk)2 +dk2 x kk2 sin2(M) MX(rk)22dkxkz kksin( M)MX(rk)MY(rk) :(7) In these expressions V=lxlzdis the volume of the film, Ddesignates the exchange stiffness andHS/d1represents the uniaxial out-of-plane anisotropy field. If HS<0the easy axis is perpendicular to the film surface. The in-plane anisotropy contribution to the energy is neglected but it should be appropriate for polycrystalline samples [16]. Moreover kk=jkkj is introduced where kk=kxex+kzezis the wave vector of the spin waves parallel to the film surface. Eqs. (3)-(7) are valid in the thin film limit kkd1. In order to derive Hdin Eq. (7) one defines a scalar magnetic potential and has to solve the corresponding boundary 5value problem inside and outside of the film [34]. As result [10] one gets the expressions in Eq. (7). In general if the static magnetic field is applied under an arbitrary angle Hthe mag- netization does not align in parallel, i.e. M6=  H. The angle Mcan be derived from the equilibrium energy Heq=H(MX= 0;MY= 0). Defining the equilibrium free energy density asfeq(M) =Heq=Vaccording to Eqs. (3)-(7) one finds the well-known condition sin( HM) =4M S+HS 2H0sin(2  M) (8) by minimizing feqwith respect to M. We further note that all terms linear in MYin Eqs. (3)-(7) cancel mutually by applying Eq. (8) as already pointed out in Ref. [10]. The energy contributions in Eqs. (3) and the geometric aspects determine the dynamical equation for the magnetization. The following generalized form is proposed @ @tM(rk;t) =ZZ dr0 kdt0(rkr0 k;tt0)(  Heff(r0 k;t0)M(r0 k;t0) + M(r0 k;t0)@ @t0M(r0 k;t0) 1 T2M?(r0 k;t0)) ;(9) where =gB=~is the absolute value of the gyromagnetic ratio, T2is the transverse relaxation time of the components M?=MXeX+MYeYanddenotes the dimensionless Gilbertdampingparameter. Thelatterisoftentransformedinto G= M Srepresentingthe corresponding damping constant in unit s1. The effective magnetic field Heffis related to the energy in Eqs. (3)-(7) by means of variational principles [35], i.e. Heff=H=M+hrf. Here the external rf-field hrf(t)is added which drives the system out of equilibrium. Regarding the equation of motion presented in Eq. (9) we note that a similar type was applied in [12] for the evaluation of ferromagnetic resonance experiments. In this paper the authors made use of a superposition of the Landau-Lifshitz equation and Bloch-like relaxation. Here we have chosen the part which conserves the spin length in the Gilbert form and added the non-conserving Bloch term in the same manner. That the combination of thesetwodistinctdampingmechanismsissuitablefortheinvestigationofultrathinmagnetic films was also suggested in [24]. Since the projection of the magnetization onto the Z-axis is not affected by T2this relaxation time characterizes the transfer of energy into the transverse components of the magnetization. This damping type is supposed to account for spin-spin relaxation processes such as magnon-magnon scattering [33, 36]. In our ansatz we introduce 6another possible source of damping by means of the feedback kernel (rkr0 k;tt0). The introduction of this quantity reflects the assumption that the magnetization M(rk;t2)is not independent of its previous value M(rk;t1)providedt2t1< . Hereis a time scale where the temporal memory is relevant. In the same manner the spatial feedback controls the magnetization dynamics significantly on a characteristic length scale , called retardation length. Physically, it seems to be reasonable that the retardation length differs noticeably from zero only in z-direction which is shown in Fig. 1. As illustrated in the figure M(x;z1;t)is affected by M(x;z2;t)while M(x;z3;t)is thought to have negligible influence onM(x;z1;t)sincejz3z1j>. Therefore we choose the following combination of a local and a nonlocal part as feedback kernel (rkr0 k;tt0) = 0(rkr0 k)(tt0) +0 4(xx0) expjzz0j  exp(tt0)  ; t>t0:(10) The intensity of the spatiotemporal feedback is controlled by the dimensionless retardation strength 0. The explicit form in Eq. (10) is chosen in such a manner that the Fourier- transform (kk;!)!0for!0and!0, and in case 0= 1the ordinary equation of motion for the magnetization is recovered. Further,R drkdt(rk;t) = 0<1, i.e. the integral remains finite. III. SUSCEPTIBILITY AND FMR-LINEWIDTH If the rf-driving field, likewise averaged over the film thickness, is applied in X-direction, i.e.hrf(rk;t) =hX(rk;t)eX, the Fourier transform of Eq. (9) is written as i! (kk;!)+1 T2+H21(kk) MX(kk;!) = H1(kk) +i!  MY(kk;!); i! (kk;!)+1 T2+H12(kk) MY(kk;!) = H2(kk) +i!  MX(kk;!)MShX(kk;!): (11) 7The effective magnetic fields are expressed by H1(kk) =H0cos( HM) + (4M S+HS) cos(2  M) + 2dkkMS k2 z k2 ksin2(M)cos2(M)! +Dk2 k H2(kk) =H0cos( HM)(4M S+HS) sin2(M) + 2dM Sk2 x kk+Dk2 k;(12) and H12(kk) = 2dM Skxkz kksin( M) =H21(kk): (13) The Fourier transform of the kernel yields (kk;!) =0(1 + i!) + 1 2 (1 + i!)(!221)'0+ 1 2i 21!; 1=0 1 +2;  =kz;(14) where the factor 1=2arises from the condition t > t0when performing the Fourier trans- formation from time into frequency domain. In Eq. (14) we discarded terms !221. This condition is fulfilled in experimental realizations. So, it will be turned out later the retardation time 10 fs. Because the ferromagnetic resonance frequencies are of the order 10:::100 GHz one finds!22108:::106. The retardation parameter =kz, introduced in Eq. (14), will be of importance in analyzing the linewidth of the resonance signal. With regard to the denominator in 1, compare Eq. (14), the parameter may evolve ponderable influence on the spin wave damping if this quantity cannot be neglected compared to 1. As known from two-magnon scattering the spin wave modes can be degenerated with the uniform resonance mode possessing wave vectors kk105cm1. The retardation length  may be estimated by the size of inhomogeneities or the distance of defects on the film sur- face, respectively. Both length scales can be of the order 10:::1000 nm, see Refs. [18, 29]. Consequently the retardation parameter could reach or maybe even exceed the order of 1. Let us stress that in case = 0,= 0,0= 1and neglecting the Gilbert damping, i.e.= 0, the spin wave dispersion relation is simply p H1(kk)H2(kk)H2 12(kk). This expression coincides with those ones given in Refs. [7] and [10]. Proceeding the analysis of Eq. (11) by defining the magnetic susceptibility as M(kk;!) =X (kk;!)h(kk;!);f;g=fX;Yg;(15) 8wherehplays the role of a small perturbation and the susceptibility exhibits the response of the system. Eq. (15) reflects that there appears no dependence on the direction ofkk. Since the rf-driving field is applied along the eX-direction it is sufficient to focus the following discussion to the element XXof the susceptibility tensor. From Eq. (11) we conclude XX(kk;!) =MSh H1(kk;!) +i! i h H1(kk;!) +i! ih H2(kk;!) +i! i +h i! (kk;!)+1 T2i2:(16) Because at ferromagnetic resonance a uniform mode is excited let us set kk= 0in Eqs. (12)- (13). Considering the resonance condition we can assume =kz= 0. For reasons men- tioned above we have to take =kz6= 0when the linewidth as a measure for spin damping is investigated. Physically we suppose that spin waves with non zero waves vectors are not excited at the moment of the ferromagnetic resonance. However such excitations will evolve during the relaxation process. In finding the resonance condition from Eq. (16) it seems to be a reasonable approximation to disregard terms including the retardation time . Such terms give rise to higher order corrections. In the same manner all the contributions orig- inated from the damping, characterized by andT2, are negligible. Let us justify those approximation by quantitative estimations. The fields H1,H2and!= are supposed to range in a comparable order of magnitude. On the other hand one finds 103:::102, !T2102and!104. Under these approximations the resonance condition reads !r 2 = 2 0H1(kk= 0)H2(kk= 0): (17) Thisresultiswellknownforthecasewithoutretardationwith 0= 1. Althoughtheretarda- tion timeand the retardation length are not incorporated in the resonance condition, the strength of the feedback may be important as visible in Eq. (17). Now the consequences for the experimental realization will be discussed. To address this issue the resonance condition Eq. (17) is rewritten in terms of the resonance field Hr=H0(!=!r)leading to Hr=1 2 cos( HM)8 < :s (4M S+HS)2cos4(M) +1 02!r 2 (4M S+HS)(13 sin2(M))9 = ;:(18) 9ΘM[deg] ΘH[deg]Γ0= 0 .7 Γ0= 1 .0 Γ0= 1 .3FIG. 2. (Color online) Dependence of the magnetization angle Mon the angle Hunder which the static external field is applied for !r=(2) = 10 GHz . The parameters are taken from [16]: 4MS= 16980 G,HS=3400 G; = 0:019 GHz=G. The result is arranged in the in the same manner as done in [16]. The difference is the occurrence of the parameter 0in the denominator. In [16] the gyromagnetic ratio and the sum (4M S+HS)were obtained from H-dependent measurements and a fit of the data according to Eq. (18) with 0= 1under the inclusion of Eq. (8). If the saturation magnetization can be obtained from other experiments [16] the uniaxial anisotropy field HS results. Thus, assuming 06= 1the angular dependence M(H)and the fitting parameters as well would change. In Fig. 2 we illustrate the angle M(H)for different values of 0and a fixed resonance frequency. If 0<1the curve is shifted to larger Mand for 0>1to smaller magnetization angles. To produce Fig. 2 we utilized quantitative results presented in [16]. They found for Co films grown on GaAs the parameters 4M S= 16980 G ,HS= 3400 Gand = 0:019 GHz=G. As next example we consider the influence of HSand denote H(0) S=3400 Gthe anisotropy field for 0= 1andH(R) Sthe anisotropy field for 06= 1. The absolute value of their ratio jH(R) S=H(0) Sj, derived from Hr(H(0) S;0= 1) =Hr(H(R) S;06= 1), isdepictedinFig.3forvariousfrequencies. Inthisgraphweassumedthatallotherquantities remain fixed. The effect of a varying retardation strength on the anisotropy field can clearly beseen. Thechangeinthesignoftheslopeindicatesthattheanisotropyfield H(R) Smayeven change its sign. From here we conclude that the directions of the easy axis and hard axis are interchanged. For the frequencies 4 GHzand10 GHzthis result is not observed in the range chosen for 0. Moreover, the effects become more pronounced for higher frequencies. 10/vextendsingle/vextendsingle/vextendsingleH(R) S/H(0) S/vextendsingle/vextendsingle/vextendsingle Γ04 GHz 10 GHz 35 GHz 50 GHz 70 GHzFIG. 3. (Color online) Effect of varying retardation strength on the uniaxial anisotropy field for various frequencies and M==3.4MS= 16980 G ,HS=3400 G; = 0:019 GHz=G, see [16]. In Fig. 3 we consider only a possible alteration of the anisotropy field. Other parameters like the experimentally obtained gyromagnetic ration were unaffected. In general this parameter may also experiences a quantitative change simultaneously with HS. Let us proceed by analyzing the susceptibility obtained in Eq. (16). Because the following discussion is referred to the energy absorption in the film, we investigate the imaginary part ofthesusceptibility 00 XX. SinceexperimentallyoftenaLorentziancurvedescribessufficiently the resonance signal we intend to arrange 00 XXin the form A0=(1 +u2), whereA0is the absolute value of the amplitude and uis a small parameter around zero. The mapping to a Lorentzian is possible under some assumptions. Because the discussion is concentrated on the vicinity of the resonance we introduce H=H0Hr, whereHris the static external field when resonance occurs. Consequently, the fields in Eq. (12) have to be replaced by H1;2!H(r) 1;2+Hcos( HM). Additionally, we take into account only terms of the order p in the final result for the linewidth where f;g/f!= [+!] + 1=( T2)g. After a lengthy but straightforward calculation we get for H=H(r) 1;21and using the resonance condition in Eq. (17) 00 XX(!) =A0 1 +h H0Hr
Ti2; A0=MS (1 +) cos( HM)
T; =H(r) 2 H(r) 1:(19) Here we have introduced the total half-width at half-maximum (HWHM)
Twhich can be 11brought in the form
T=1 cos( HM)q
2 G+
2 B+
2 GB+
2 R: (20) The HWHM is a superposition of the Gilbert contribution
G, the Bloch contribution
B, a joint contribution
GBarising from the combination of the Gilbert and Bloch damping parts in the equation of motion and the contribution
Rwhich has its origin purely in the feedback mechanisms introduced into the system. The explicit expressions are
G=! s  16p (1 +)01! (0+ 1)3 ; (21a)
B=4 0 (0+ 1)p (1 +)s 1 ( T2)24 1 (0+ 1)2! ! T2; (21b)
GB=s 80 (0+ 1)p (1 +)! 2T2; (21c)
R=8p (1 +)! 01! (0+ 1)3: (21d) The parameter 1is defined in Eq. (14). If the expressions under the roots in Eqs. (21a) and (21b) are negative we assume that the corresponding process is deactivated and does not contribute to the linewidth
HT. Typically, experiments are evaluated in terms of the peak-to-peak linewidth of the derivative d00 XX=dH0, denoted as
H. One gets
H=2p 3
; (22) where the index stands for G(Gilbert contribution), B(Bloch contribution), GB(joint Gilbert-Bloch contribution), R(pure retardation contribution) or Tdesignating the total linewidth according to Eq. (20) and Eqs. (21a)-(21d). Obviously these equations reveal a strong nonlinear frequency dependence, which will be discussed in the subsequent section. IV. DISCUSSION As indicated in Eqs. (20) - (22) the quantity
Hconsists of well separated distinct contributions. Thebehaviorof
HisshowninFigs.4-6asfunctionofthethreeretardation parameters, the strength 0, the spatial range and the time scale . In all figures the frequencyf=!=(2)is used. In Fig. 4 the dependence on the retardation strength 0is 12∆HT[G]4 GHz 10 GHz 35 GHz 50 GHz 70 GHz∆Hη[G] Γ0∆HG ∆HB ∆HGB ∆HR ∆HTf= 70 GHzFIG. 4. (Color online) Influence of the retardation strength 0on the peak-to-peak linewidth
HT for various frequencies (top graph) and on the single contributions
Hforf= 70 GHz (bottom graph).
B= 0is this frequency region. The parameters are: H=  M= 0,= 0:5,= 0:01, T2= 5108s;= 1:71014s. The other parameters are 4MS= 16980 G ,HS=3400 G; = 0:019 GHz=G, compare [16]. shown. As already observed in Figs. 2 and 3 a small change of 0may lead to remarkable effects. Hence we vary this parameter in a moderate range 0:502. The peak-to-peak linewidth
HTas function of 0remains nearly constant for f= 4 GHz andf= 10 GHz , whereas for f= 35 GHz a monotonous growth-up is observed. Increasing the frequency further tof= 50 GHz and70 GHzthe curves offers a pronounced kink. The subsequent enhancement is mainly due to the Gilbert damping. In the region of negative slope we set
HG(0) = 0, while in that one with a positive slope
HG(0)>0grows and tends to2!=(p 3 )for0!1. The other significant contribution
HR, arising from the retardation decay, offers likewise a monotonous increase for growing values of the retardation parameter 0. This behavior is depicted in Fig. 4 for f= 70 GHz . Now let us analyze the dependence on the dimensionless retardation length =kz. Becauseis only nonzero if 13∆HT[G]4 GHz 10 GHz 35 GHz 50 GHz 70 GHz∆Hη[G] β∆HG ∆HB ∆HGB ∆HR ∆HTf= 70 GHzFIG. 5. (Color online) Influence of the dimensionless retardation length =kzon the total peak-to-peak linewidth
HTfor various frequencies (top graph) and on the single contributions
Hforf= 70 GHz (bottom graph);
B= 0in this range. The parameters are: H=  M= 0, 0= 1:1,= 0:01,T2= 5108s;= 1:71014s. The other parameters: 4MS= 16980 G , HS=3400 Gand = 0:019 GHz=Gare taken from [16]. kz6= 0this parameter accounts the influence of excitations with nonzero wave vector. We argue that both nonzero wave vector excitations, those arising from two-magnon scattering and those originated from feedback mechanisms, may coincide. Based on the estimation in the previous section we consider the relevant interval 10210. The results are shown in Fig.5. Within the range of one recognizes that the total peak-to-peak linewidths
HTforf= 4 GHz andf= 10 GHz offer no alteration when is changed. The plotted linewidths are characterized by a minimum followed by an increase which occurs when  exceeds approximately 1. This behavior is the more accentuated the larger the frequencies are. The shape of the curve can be explained by considering the single contributions as is visible in the lower part in Fig. 5. While both quantities
HG()and
HR()remain constant for small ,
HG()tends to a minimum and increases after that. The quantity 14∆HT[G]4 GHz 10 GHz 35 GHz 50 GHz 70 GHz∆Hη[G] τ[fs]∆HG ∆HB ∆HGB ∆HR ∆HTf= 70 GHzFIG. 6. (Color online) Influence of the retardation time on the total peak-to-peak linewidth
HTfor various frequencies (top graph) and on the single contributions
Hforf= 70 GHz (bottom graph).
B= 0in this region. The parameters are H=  M= 0,= 0:5,= 0:01, T2= 5108s;0= 1:1; the other parameters are taken from [16]: 4MS= 16980 G ,HS= 3400 G; = 0:019 GHz=G.
HR()develops a maximum around 1. Thus, both contributions show nearly opposite behavior. The impact of the characteristic feedback time on the linewidth is illustrated in Fig. 6. In this figure a linear time scale is appropriate since there are no significant effects in the range 1 fs0. The total linewidth
HT()is again nearly constant forf= 4 GHz andf= 10 GHz . In contrast
HT()reveals for higher frequencies two regions with differing behavior. The total linewidth decreases until
HG()becomes zero. After that one observes a positive linear slope which is due to the retardation part
HR(). This linear dependency is recognizable in Eq. (21d), too. Below we will present arguments why the feedback time is supposed to be in the interval 0<  < 100 fs. Before let us study the frequency dependence of the linewidth in more detail. The general shape of the total linewidth
HT(!)is depicted in Fig. 7. Here both the single contribution to the 15∆Hη[G] f[GHz]∆HG ∆HB ∆HGB ∆HR ∆HTFIG. 7. (Color online) Frequency dependence of all contributions to the peak-to-peak linewidth for H=  M= 0,= 0:5,= 0:01,T2= 5108s,= 1:71014sand0= 1:2. Parameters taken from Ref. [16]: 4MS= 16980 G ,HS=3400 Gand = 0:019 GHz=G. The Bloch contribution
HBis shown in the inset. linewidth and the total linewidth are shown. Notice that the total linewidth is not simply the sum of the individual contributions but has to be calculated according to Eq. (20). One realizes that the Bloch contribution
HBis only nonzero for frequencies f6 GHzin the examples shown. Accordingly
HB= 0in Figs. 4-6 (lower parts) since these plots refer to f= 70 GHz . The behavior of the Gilbert contribution deviates strongly from the typically applied linear frequency dependence. Moreover, the Gilbert contribution will develop a maximum value and eventually it disappears at a certain frequency where the discriminant in Eq. (21a) becomes negative. Nevertheless, the total linewidth is a nearly monotonous increasing function of the frequency albeit, as mentioned before, for some combinations of the model parameters there might exist a very small frequency region where
HGreaches zero and the slope of
HTbecomes slightly negative. The loss due to the declining Gilbert part is nearly compensated or overcompensated by the additional line broadening originated bytheretardationpartandthecombinedGilbert-Blochterm. Thelatteroneis
HGB/pf and
HR/f2, see Eqs. (21c)-(21d). In the frequency region where
HG= 0only
HGB and
HRcontribute to the total linewidth, the shape of the linewidth is mainly dominated by
HR. Thispredictionisanewresult. Thebehavior
HR/f2, obtainedinourmodelfor high frequencies, is in contrast to conventional ferromagnetic resonance including only the sum of a Gilbert part linear in frequency and a two-magnon contribution which is saturated 16at high frequencies. So far, experimentally the frequency ranges from 1 GHzto225 GHz, see [21]. Let us point out that the results presented in Fig. 7 can be adjusted in such a manner that the Gilbert contribution will be inoperative at much higher frequencies by the appropriate choice of the model parameters. Due to this fact we suggest an experimental verification in more extended frequency ranges. Another aspect is the observation that excitations with a nonzero wave vector might represent one possible retardation mechanism. Regarding Eqs. (21a)-(21d) retardation can also influence the linewidth in case kz= 0 (i.e.= 0and1= 0). Only if= 0the retardation effects disappear. Therefore let us consider the time domain of retardation and its relation to the Gilbert damping. The Gilbert damping and the attenuation due to retardation can be considered as competing processes. So temporal feedback can cause that the Gilbert contribution disappears. In the same sense the Bloch contribution is a further competing damping effect. In this regard temporal feedback has the ability to reverse the dephasing process of spin waves based on Gilbert and Bloch damping. On the other hand the retardation part
Rin Eq. (21d) is always positive for > 0. Thus, the retardation itself leads to linewidth broadening in ferromagnetic resonance and consequently to spin damping. Whether the magnitude of retardation is able to exceed the Gilbert damping depends strongly on the frequency. With other words, the frequency of the magnetic excitation ’decides’ to which damping mechanisms the excitation energyistransferred. Ourcalculationsuggeststhatforsufficienthighfrequenciesretardation effects dominate the intrinsic damping behavior. Thus the orientation and the value of the magnetization within the retardation time plays a major role for the total damping. Generally, experimental data should be fit according to the frequency dependence of the linewidth in terms of Eqs. (20)-(22). To underline this statement we present Fig. 8. In this graph we reproduce some results presented in [7] for the case H=  M= 0. To be more specific, we have used Eq. (94) in [7] which accounts for the two-magnon scattering and the parameters given there. As result we find a copy of Fig. 4 in [7] except of the factor 2=p 3. Further, we have summed up the conventional Gilbert linewidth /fwith the Gilbert damping parameter 1= 0:003. This superposition yields to the dotted line in Fig. 8. The result is compared with the total linewidth resulting from our retardation model plotted as solid line. To obtain the depicted shape we set the Gilbert damping parameter according to the retardation model 2= 0:0075, i.e. to get a similar behavior in the same order of magnitude of
HTwithin both approaches we have to assume that 2is more than twice 17∆HT[G] f[GHz]retardation model Gilbert+2-magnonFIG. 8. (Color online) Comparison with the two-magnon model. Frequency dependence of the total peak-to-peak linewidth
HTforH=  M= 0,= 0:5,1= 0:003,2= 0:0075,T2= 5108s, = 1:221014sand0= 1:2. Parameters taken from [7]: 4MS= 21000 G ,HS=15000 Gand from [37]: = 0:018 GHz=G(derived from g= 2:09for bulk Fe). The dotted line is a superposition of Fig. 4 in [7] reflecting the two-magnon contribution and the Gilbert contribution (denoted as 1in the text) linear in the frequency. as large compared to 1. Finally we discuss briefly the H-dependence of the linewidth which is shown in Fig. 9. In the upper part of the figure one observes that
HT(H)exhibits a maximum which is shifted towards lower field angles as well as less pronounced for increasing frequencies. The lower part of Fig. 9, referring to f= 10 GHz , displays that the main contribution to the total linewidth arises from the Gilbert part
HG. This result for f= 10 GHz is in accordance with the results discussed previously, compare Fig. 7. For higher frequencies the retardation contribution
HRmay exceed the Gilbert part. V. CONCLUSIONS A detailed study of spatiotemporal feedback effects and intrinsic damping terms offers that both mechanisms become relevant in ferromagnetic resonance. Due to the superposi- tion of both effects it results a nonlinear dependence of the total linewidth on the frequency which is in accordance with experiments. In getting the results the conventional model in- cluding Landau-Lifshitz-Gilbert damping is extended by considering additional spatial and 18linewidth ∆ HT[G] 4 GHz 10 GHz 35 GHz 50 GHz 70 GHzlinewidth ∆ Hη[G] ΘH[deg]∆HB ∆HR ∆HGB ∆HG ∆HTf= 10 GHzFIG. 9. (Color online) Angular dependence of the total peak-to-peak linewidth
HTfor various frequencies (top graph) and all contributions
Hforf= 10 GHz (bottom graph) with = 0:5, = 0:01,T2= 5108s,= 1:71014sand0= 1:1. The parameters are taken from [16]: 4MS= 16980 G ,HS=3400 Gand = 0:019 GHz=G. temporal retardation and non-conserved Bloch damping terms. Our analytical approach enables us to derive explicit expressions for the resonance condition and the peak-to-peak linewidth. We were able to link our results to such ones well-known from the literature. The resonance condition is affected by the feedback strength 0. The spin wave damping is likewise influenced by 0but moreover by the characteristic memory time and the retar- dation length . As expected the retardation gives rise to an additional damping process. Furthermore, the complete linewidth offers a nonlinear dependence on the frequency which is also triggered by the Gilbert damping. From here we conclude that for sufficient high frequencies the linewidth is dominated by retardation effects. Generally, the contribution of thedifferentdampingmechanismstothelinewidthiscomprisedofwellseparatedrateswhich are presented in Eqs. (20)-(22). Since each contribution to the linewidth is characterized by adjustable parameters it would be very useful to verify our predictions experimentally. 19Notice that the contributions to the linewidth in Eqs. (20)-(22) depend on the shape of the retardation kernel which is therefore reasonable not only for the theoretical approach but for the experimental verification, too. One cannot exclude that other mechanisms as more-magnon scattering effects, nonlinear interactions, spin-lattice coupling etc. are likewise relevant. Otherwise, we hope that our work stimulates further experimental investigations in ferromagnetic resonance. We benefit from valuable discussions about the experimental background with Dr. Khali Zakeri from the Max-Planck-Institute of Microstructure Physics. 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2012-04-24

Ferromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback effects. The equation of motion for the magnetization dynamics is nonlocal in both space and time and includes isotropic, anisotropic and dipolar energy contributions as well as the conserved Gilbert- and the non-conserved Bloch-damping. We derive an analytical expression for the peak-to-peak linewidth. It consists of four separate parts originated by Gilbert damping, Bloch-damping, a mixed Gilbert-Bloch component and a contribution arising from retardation. In an intermediate frequency regime the results are comparable with the commonly used Landau-Lifshitz-Gilbert theory combined with two-magnon processes. Retardation effects together with Gilbert damping lead to a linewidth the frequency dependence of which becomes strongly nonlinear. The relevance and the applicability of our approach to ferromagnetic resonance experiments is discussed.

Nonlocal feedback in ferromagnetic resonance

1204.5342v1

The Cauchy problem for the Landau–Lifshitz–Gilbert equation in BMO and self-similar solutions Susana Gutiérrez1and André de Laire2 Abstract We prove a global well-posedness result for the Landau–Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any dimension. In the one-dimensional case, we characterize the self-similar solutions associated with an initial data given by some (S2-valued) step function and establish their stability. We also show the existence of multiple solutions if the damping is strong enough. Our arguments rely on the study of a dissipative quasilinear Schrödinger equation ob- tained via the stereographic projection and techniques introduced by Koch and Tataru. Keywords and phrases: Landau–Lifshitz–Gilbert equation, global well-posedness, discontin- uous initial data, stability, self-similar solutions, dissipative Schrödinger equation, complex Ginzburg–Landau equation, ferromagnetic spin chain, heat-flow for harmonic maps. 2010Mathematics Subject Classification: 35R05, 35Q60, 35A01, 35C06, 35B35, 35Q55, 35Q56, 35A02, 53C44. Contents 1 Introduction and main results 2 2 The Cauchy problem 6 2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation . . . . . . 6 2.2 The Cauchy problem for the LLG equation . . . . . . . . . . . . . . . . . . . . . 15 3 Applications 22 3.1 Existence of self-similar solutions in RN. . . . . . . . . . . . . . . . . . . . . . . 22 3.2 The Cauchy problem for the one-dimensional LLG equation with a jump initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2 . . . . . . . . . 24 3.2.2 Multiplicity of solutions. Proof of Theorem 1.3 . . . . . . . . . . . . . . . 28 3.3 A singular solution for a nonlocal Schrödinger equation . . . . . . . . . . . . . . . 30 4 Appendix 34 1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom. E-mail: s.gutierrez@bham.ac.uk 2Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France. E-mail: andre.de-laire@univ-lille.fr 1arXiv:1701.03083v2 [math.AP] 20 Mar 20191 Introduction and main results We consider the Landau–Lifshitz–Gilbert (LLG) equation @tm=m
mm(m
m);onRNR+; (LLG) where m= (m1;m2;m3) :RNR+!S2is the spin vector, 0,0;denotes the usual cross-product in R3, and S2is the unit sphere in R3. This model introduced by Landau and Lifshitz describes the dynamics for the spin in ferromagnetic materials [26, 16] and constitutes a fundamental equation in the magnetic recording industry [36]. The parameters 0and0 are respectively the so-called exchange constant and Gilbert damping, and take into account the exchange of energy in the system and the effect of damping on the spin chain. Note that, by performing a time-scaling, we assume w.l.o.g. that 2[0;1]and=p 12: The Landau–Lifshitz family of equations includes as special cases the well-known heat-flow for harmonic maps and the Schrödinger map equation onto the 2-sphere. In the limit case = 0 (and so= 1) the LLG equation reduces to the heat-flow equation for harmonic maps @tm
m=jrmj2m;onRNR+: (HFHM) The case when = 0(i.e. no dissipation/damping) corresponds to the Schrödinger map equation @tm=m
m;onRNR+: (SM) In the one-dimensional case N= 1, we established in [17] the existence and asymptotics of the familyfmc;gc>0of self-similar solutions of (LLG ) for any fixed 2[0;1], extending the results in Gutiérrez, Rivas and Vega [18] in the setting of the Schrödinger map equation and related binormal flow equation. The motivation for the results presented in this paper first originated from the desire to study further properties of the self-similar solutions found in [17], and in particular their stability. In the case = 0, the stability of the self-similar solutions of the Schrödinger map has been considered in the series of papers by Banica and Vega [5, 6, 7], but no stability result is known for these solutions in the presence of damping, i.e.  > 0. One of the key ingredients in the analysis given by Banica and Vega is the reversibility in time of the equation in the absence of damping. However, since (LLG ) is a dissipative equation for >0, this property is no longer available and a new approach is needed. In the one-dimensional case and for fixed 2[0;1], the self-similar solutions of (LLG ) constitute a uniparametric family fmc;gc>0where mc;is defined by mc;(x;t) =fxp t ; for some profile f:R!S2, and is associated with an initial condition given by a step function (at least when cis small) of the form m0 c;:=A+ c;R++A c;R; (1.1) where A c;are certain unitary vectors and Edenotes the characteristic function of a set E. In particular, when >0, the Dirichlet energy associated with the solutions mc;given by krmc;(
;t)k2 L2=c22 t1=2 ; t> 0; (1.2) 2diverges as t!0+3. A first natural question in the study of the stability properties of the family of solutions fmc;gc>0is whether or not it is possible to develop a well-posedness theory for the Cauchy problem for (LLG ) in a functional framework that allows us to handle initial conditions of the type (1.1). In view of (1.1) and (1.2), such a framework should allow some “rough” functions (i.e. function spaces beyond the “classical” energy ones) and step functions. A few remarks about previously known results in this setting are in order. In the case >0, global well-posedness results for (LLG ) have been established in N2by Melcher [31] and by Lin, Lai and Wang [30] for initial conditions with a smallness condition on the gradient in the LN(RN)and the Morrey M2;2(RN)norm4, respectively. Therefore these results do not apply to the initial condition m0 c;. When= 1, global well-posedness results for the heat flow for harmonic maps (HFHM) have been obtained by Koch and Lamm [22] for an initial condition L1-close to a point and improved to an initial data with small BMO semi-norm by Wang [35]. The ideas used in [22] and [35] rely on techniques introduced by Koch and Tataru [23] for the Navier–Stokes equation. Since m0 c;has a small BMO semi-norm if cis small, the results in [35] apply to the case = 1. Therearetwomainpurposesinthispaper. Thefirstoneistoadaptandextendthetechniques developed in [22, 23, 35] to prove a global well-posedness result for (LLG ) with2(0;1]for datam0inL1(RN;S2)with small BMO semi-norm. The second one is to apply this result to establish the stability of the family of self-similar solutions fmc;gc>0found in [17] and derive further properties for these solutions. In particular, a further understanding of the properties of the functions mc;will allow us to prove the existence of multiple smooth solutions of (LLG ) associated with the same initial condition, provided that is close to one. In order to state the first of our results, we introduce the function space Xas follows: X=fv:RnR+!R3:v;rv2L1 loc(RNR+)andkvkX:=supt>0kv(t)kL1+ [v]X<1g where [v]X:=supt>0p tkrvkL1+ sup x2RN r>0 1 rN Br(x)[0;r2]jrv(y;t)j2dtdy!1 2 ; andBr(x)denotes the ball with center xand radius r >0inRN. Let us remark that the first term in the definition of [v]Xallows to capture a blow-up rate of 1=p tforkrv(t)kL1, ast!0+. This is exactly the blow-up rate for the self-similar solutions (see (3.1) and (3.12)). The integral term in [v]Xis associated with the space BMO as explained in Subsection 2.1, and it is also well adapted to the self-similar solutions (see Proposition 3.4 and its proof). We can now state the following (global) well-posedness result for the Cauchy problem for the LLG equation: Theorem 1.1. Let2(0;1]. There exist constants M1;M2;M3>0, depending only on and Nsuch that the following holds. For any m02L1(RN;S2),Q2S2,2(0;2]and"0>0such that"0M16, inf RNjm0Qj22and [m0]BMO"0; (1.3) there exists a unique solution m2X(RNR+;S2)of(LLG)with initial condition m0such that inf x2RN t>0jm(x;t)Qj24 1 +M2 2(M34+1)2and [m]X4M2(M34+ 82"0):(1.4) 3We refer the reader to Theorem A.5 in the Appendix and to [17] for precise statements of these results. 4See footnote in Section 3.3 for the definition of the Morrey space M2;2(RN). 3In addition, mis a smooth function belonging to C1(RNR+;S2). Furthermore, assume that nis a solution to (LLG)fulfilling (1.4), with initial condition n0satisfying (1.3). Then kmnkX120M2 2km0n0kL1: (1.5) As we will see in Section 2, the proof of Theorem 1.1 relies on the use of the stereographic pro- jection to reduce Theorem 1.1 to establish a well-posedness result for the associated dissipative (quasilinear) Schrödinger equation (see Theorem 2.1). In order to be able to apply Theorem 1.1 to the study of both the initial value problem related to the LLG equation with a jump initial condition, and the stability of the self-similar solutions found in [17], we will need a more quanti- tative version of this result. A more refined version of Theorem 1.1 will be stated in Theorem 2.9 in Subsection 2.2. Theorem 1.1 (or more precisely Theorem 2.9) has two important consequences for the Cauchy problem related to (LLG ) in one dimension: 8 < :@tm=m@xxmm(m@xxm);onRR+; m0 A:=A+R++AR;(1.6) where Aare two given unitary vectors such that the angle between A+andAis sufficiently small: (a)From the uniqueness statement in Theorem 1.1, we can deduce that the solution to (1.6) provided by Theorem 1.1 is a rotation of a self-similar solution mc;for an appropriate value ofc(see Theorem 3.3 for a precise statement). (b)(Stability) From the dependence of the solution with respect to the initial data established in (1.5) and the analysis of the 1d-self-similar solutions mc;carried out in [17], we obtain the following stability result: For any given m02S2satisfying (1.3) and close enough to m0 A, the solution mof (LLG) associated with m0given by Theorem 1.1 must remain close to a rotation of a self-similar solution mc;, for somec>0. In particular, mremains close to a self-similar solution. The precise statement is provided in the following theorem. Theorem 1.2. Let2(0;1]. There exist constants L1;L2;L3>0,2(1;0),#>0such that the following holds. Let A+,A2S2with angle#between them. If 0<##; then there is c>0such that for every m0satisfying km0m0 AkL1cp 2p; there existsR2SO(3), depending only on A+,A,andc, such that there is a unique global smooth solution mof(LLG)with initial condition m0that satisfies inf x2R t>0(Rm)3(x;t)and [m]XL1+L2c: (1.7) Moreover, kmRmc;kXL3km0m0 AkL1: 4In particular, k@xm@xRmc;kL1L3p tkm0m0 AkL1; for allt>0. Notice that Theorem 1.2 provides the existence of a unique solution in the set defined by the conditions (1.7), and hence it does not exclude the possibility of the existence of other solutions not satisfying these conditions. In fact, as we will see in Theorem 1.3 below, one can prove the existence of multiple solutions of the initial value problem (1.6), at least in the case when is close to 1. We point out that our results are valid only for >0. If we let!0, then the constants M1 andM3in Theorem 1.1 go to 0 and M2blows up. Indeed, we use that the kernel associated with the Ginzburg–Landau semigroup e(+i)t
belongs toL1and its exponential decay. Therefore our techniques cannot be generalized (in a simple way) to cover the critical case = 0. In particular, we cannot recover the stability results for the self-similar solutions in the case of Schrödinger maps proved by Banica and Vega in [5, 6, 7]. As mentioned before, in [30] and [31] some global well-posedness results for (LLG ) with 2(0;1]were proved for initial conditions with small gradient in LN(RN)andM2;2(RN), respectively (see footnote in Subsection 3.3 for the definition of the space M2;2(RN)). In view of the embeddings LN(RN)M2;2(RN)BMO1(RN); forN2, Theorem 1.1 can be seen as generalization of these results since it covers the case of less regular initial conditions. The arguments in [30, 31] are based on the method of moving frames that produces a covariant complex Ginzburg–Landau equation. In Subsection 3.3 we give more details and discuss the corresponding equation in the one-dimensional case and provide some properties related to the self-similar solutions. Our existence and uniqueness result given by Theorem 1.1 requires the initial condition to be small in the BMO semi-norm. Without this condition, the solution could develop a singularity in finite time. In fact, in dimensions N= 3;4, Ding and Wang [13] have proved that for some smooth initial conditions with small (Dirichlet) energy, the associated solutions of (LLG ) blow up in finite time. In the context of the initial value problem (1.6), the smallness condition in the BMO semi- norm is equivalent to the smallness of the angle between A+andA. As discussed in [17], in the one dimensional case N= 1for fixed2(0;1]there is some numerical evidence that indicates the existence of multiple (self-similar) solutions associated with the same initial condition of the type in (1.6) (see Figures 2 and 3 in [17]). This suggests that the Cauchy problem for (LLG ) with initial condition (1.6) is ill-posed for general A+andAunitary vectors. The following result states that in the case when is close to 1, one can actually prove the existenceofmultiplesmoothsolutionsassociatedwiththesameinitialcondition m0 A. Moreover, given any angle #2(0;)between two vectors A+andA2S2, one can generate any number of distinct solutions by considering values of sufficiently close to 1. Theorem 1.3. Letk2N,A+,A2S2and let#be the angle between A+andA. If #2(0;), then there exists k2(0;1)such that for every 2[k;1]there are at least kdistinct smooth self-similar solutions fmjgk j=1inX(RR+;S2)of(LLG)with initial condition m0 A. These solutions are characterized by a strictly increasing sequence of values fcjgk j=1, withck!1 ask!1, such that mj=Rjmcj;; (1.8) 5whereRj2SO(3). In particular p tk@xmj(
;t)kL1=cj;for allt>0: (1.9) Furthermore, if = 1and#2[0;], then there is an infinite number of distinct smooth self- similar solutions fmjgj1inX(RR+;S2)of(LLG)with initial condition m0 A. These solutions are also characterized by a sequence fcjg1 j=1such that (1.8)and(1.9)are satisfied. This sequence is explicitly given by c2`+1=`p# 2p; c 2`=`p+# 2p;for`0: (1.10) It is important to remark that in particular Theorem 1.3 asserts that when = 1, given A+;A2S2such that A+=A, there exists an infinite number of distinct solutions fmjgj1 inX(RR+;S2)of (LLG) with initial condition m0 Asuch that [m0 A]BMO = 0. This particular case shows that a condition on the size of X-norm of the solution as that given in (1.4) in Theorem 1.1 is necessary for the uniqueness of solution. We recall that for finite energy solutions of (HFHM) there are several nonuniqueness results based on Coron’s technique [11] in dimensionN= 3. Alouges and Soyeur [2] successfully adapted this idea to prove the existence of multiple solutions of the (LLG ), with>0, for maps m: !S2, with a bounded regular domain of R3. In our case, since fcjgk j=1is strictly increasing, we have at least kgenuinely different smoothsolutions. Notice also that the identity (1.9) implies that the X-norm of the solution is large as j!1. Structure of the paper. This paper is organized as follows: in Section 2 we use the ste- reographic projection to reduce matters to the study the initial value problem for the resulting dissipative Schrödinger equation, prove its global well-posedness in well-adapted normed spaces, and use this result to establish Theorem 2.9 (a more quantitative version of Theorem 1.1). In Section 3 we focus on the self-similar solutions and we prove Theorems 1.2 and 1.3. In Section 3.3 we discuss some implications of the existence of explicit self-similar solutions for the Schrödinger equation obtained by means of the Hasimoto transformation. Finally, and for the convenience of the reader, we have included some regularity results for the complex Ginzburg–Landau equation and some properties of the self-similar solutions mc;in the Appendix. Notations. We write R+= (0;1). Throughout this paper we will assume that 2(0;1]and the constants can depend on . In the proofs A.Bstands forACBfor some constant C > 0depending only on andN. We denote in bold the vector-valued variables. Since we are interested in S2-valued functions, with a slightly abuse of notation, we denote byL1(RN;S2)(resp.X(RN;S2)) the space of function in L1(RN;R3)(resp.X(RN;R3)) such thatjmj=1 a.e. on RN. 2 The Cauchy problem 2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation Our approach to study the Cauchy problem for (LLG ) consists in analyzing the Cauchy prob- lem for the associated dissipative quasilinear Schrödinger equation through the stereographic projection, and then “transferring” the results back to the original equation. To this end, we introduce the stereographic projection from the South Pole P:S2nf(0;0;1)g!Cdefined for by P(m) =m1+im2 1 +m3: 6Letmbe a smooth solution of (LLG ) withm3>1, then its stereographic projection u= P(m)satisfies the quasilinear dissipative Schrödinger equation (see e.g. [25] for details) iut+ (i)
u= 2(i)u(ru)2 1 +juj2: (DNLS) At least formally, the Duhamel formula gives the integral equation: u(x;t) =S(t)u0+t 0S(ts)g(u)(s)ds; (IDNLS) whereu0=u(
;0)corresponds to the initial condition, g(u) =2i(i)u(ru)2 1 +juj2 andS(t)is the dissipative Schrödinger semigroup (also called the complex Ginzburg–Landau semigroup) given by S(t)=e(+i)t
, i.e. (S(t))(x) = RNG(xy;t)(y)dy;withG(x;t) =ejxj2 4(+i)t (4(+i)t)N=2:(2.1) One difficulty in studying (IDNLS) is to handle the term g(u). Taking into account that jij= 1anda 1 +a21 2;for alla0; (2.2) we see that jg(u)jjruj2; (2.3) so we need to control jruj2. Koch and Taratu dealt with a similar problem when studying the well-posedness for the Navier–Stokes equation in [23]. Their approach was to introduce some new spaces related to BMO and BMO1. Later, Koch and Lamm [22] and Wang [35] have adapted these spaces to study some geometric flows. Following these ideas, we define the Banach spaces X(RNR+;F) =fv:RNR+!F:v;rv2L1 loc(RNR+);kvkX<1gand Y(RNR+;F) =fv:RNR+!F:v2L1 loc(RNR+);kvkY<1g; where kvkX:= sup t>0kvkL1+ [v]X;with [v]X:= sup t>0p tkrvkL1+ sup x2RN r>0 1 rN Qr(x)jrv(y;t)j2dtdy!1 2 ;and kvkY= sup t>0tkvkL1+ sup x2RN r>01 rN Qr(x)jv(y;t)jdtdy: HereQr(x)denotes the parabolic ball Qr(x) =Br(x)[0;r2]andFis either CorR3. The absolute value stands for the complex absolute value if F=Cand for the euclidean norm if F=R3. We denote with the same symbol the absolute value in FandF3. Here and in the sequel we will omit the domain in the norms and semi-norms when they are taken in the whole space, for example k
kLpstands fork
kLp(RN), forp2[1;1]. 7The spaces XandYare related to the spaces BMO (RN)and BMO1(RN)and are well- adapted to study problems involving the heat semigroup S1(t) =et
. In order to establish the properties of the semigroup S(t)with2(0;1], we introduce the spaces BMO (RN)and BMO1 (RN)as the space of distributions f2S0(RN;F)such that the semi-norm and norm given respectively by [f]BMO:= sup x2RN r>0 1 rN Qr(x)jrS(t)fj2dtdy!1 2 ;and kfkBMO1 := sup x2RN r>0 1 rN Qr(x)jS(t)fj2dtdy!1 2 ; are finite. Ontheonehand, theCarlesonmeasurecharacterizationofBMOfunctions(see[34, Chapter4] and [27, Chapter 10]) yields that for fixed 2(0;1],BMO(RN)coincides with the classical BMO (RN)space5, that is for all 2(0;1]there exists a constant >0depending only on  andNsuch that [f]BMO[f]BMO1[f]BMO: (2.4) On the other hand, Koch and Tataru proved in [23] that BMO1(or equivalently BMO1 1, using our notation) can be characterized as the space of derivatives of functions in BMO. A straightforward generalization of their argument shows that the same result holds for BMO1  (see Theorem A.1). Hence, using the Carleson measure characterization theorem, we conclude that BMO1 coincides with the space BMO1and that there exists a constant ~>0, depending only onandN, such that ~kfkBMO1kfkBMO1 ~1kfkBMO1: (2.5) The above remarks allows us to use several of the estimates proved in [22, 23, 35] in the case = 1to study the integral equation (IDNLS) by using a fixed-point approach. Our first result concerns the global well-posedness of the Cauchy problem for (IDNLS) with small initial data in BMO (RN). Theorem 2.1. Let2(0;1]. There exist constants C;K1such that for every L0,">0, and>0satisfying 8C(+")2; (2.6) ifu02L1(RN;C), with ku0kL1Land [u0]BMO"; (2.7) then there exists a unique solution u2X(RNR+;C)to(IDNLS) such that [u]XK(+"): (2.8) Moreover, 5 BMO (RN) =ff:RN[0;1)!F:f2L1 loc(RN);[f]BMO <1g; with the semi-norm [f]BMO = sup x2RN r>0 Br(x)jf(y)fx;rjdy; where fx;ris the average fx;r= Br(x)f(y)dy=1 jBr(x)j Br(x)f(y)dy: 8(i)supt>0kukL1K(+L). (ii)u2C1(RNR+)and(DNLS) holds pointwise. (iii) lim t!0+u(
;t) =u0as tempered distributions. Moreover, for every '2S(RN), we have k(u(
;t)u0)'kL1!0;ast!0+: (2.9) (iv) (Dependence on the initial data) Assume that uandvare respectively solutions to (IDNLS) fulfilling (2.8)with initial conditions u0andv0satisfying (2.7). Then kuvkX6Kku0v0kL1: (2.10) Although condition (2.6) appears naturally from the fixed-point used in the proof, it may be no so clear at first glance. To better understand it, let us define for C > 0 S(C) =f(;")2R+R+:C(+")2g: (2.11) We see that if (;")2S(C), then;"> 0and "pp C: (2.12) Therefore the set S(C)is non-empty and bounded. The shape of this set is depicted in Figure 1. In particular, we infer from (2.12) that if (;")2S(C), then 1 4C 1 4C1 Cρε Figure 1: The shape of the set S(C). 1 Cand"1 4C: (2.13) In addition, if ~CC, then S(~C)S(C): (2.14) Moreover, taking for instance = 1=(32C), Theorem 2.1 asserts that for fixed 2(0;1], we can take for instance "= 1=(32C)(that depends on andN, but not on the L1-norm of the initial data) such that for any given initial condition u02L1(RN)with [u0]BMO", there exists a global (smooth) solution u2X(RNR+;C)of (DNLS). Notice that u0is allowed to have a largeL1-norm as long as [u0]BMOis sufficiently small; this is a weaker requirement that asking for theL1-norm ofu0to be sufficiently small, since [f]BMO2kfkL1;for allf2L1(RN): (2.15) 9Remark 2.2. The smallness condition in (2.8) is necessary for the uniqueness of the solution. As we will see in Subsection 3.2.2, at least in dimension one, it is possible to construct multiple solutions of (IDNLS) in X(RNR+;C), ifis close enough to 1. The aim of this section is to prove Theorem 2.1 using a fixed-point technique. To this pursuit we write (IDNLS) as u(t) =Tu0(u)(t); (2.16) where Tu0(u)(t) =S(t)u0+T(g(u))(t)andT(f)(t) =t 0S(ts)f(s)ds: (2.17) In the next lemmas we study the semigroup Sand the operator Tto establish that the appli- cationTu0is a contraction on the ball B(u0) =fu2X(RNR+;C) :kuS(t)u0kXg; for some>0depending on the size of the initial data. Lemma 2.3. There exists C0>0such that for all f2BMO1 (RN), sup t>0p tkS(t)fkL1(RN)C0kfkBMO1 : (2.18) Proof.The proof in the case = 1is done in [27, Lemma 16.1]. For 2(0;1), decomposing S(t) =S(ts)S(s)and using the decay properties of the kernel Gassociated with the operatorsS(t)(see (2.1)), we can check that the same proof still applies. Lemma 2.4. There exists C11such that for all f2Y(RNR+;C), kT(f)kXC1kfkY: (2.19) Proof.Estimate (2.19) can be proved using the arguments given in [23] or [35]. For the conve- nience of the reader, we sketch the proof following the lines in [35, Lemma 3.1]. By scaling and translation, it suffices to show that jT(f)(0;1)j+jrT(f)(0;1)j+  Q1(0)jrT(f)j2!1=2 .kfkY: (2.20) LetBr=Br(0). SettingW=T(f), we have W(0;1) =1 0 RNG(y;1s)f(y;s)dyds = 1 1=2 RN+1=2 0 B2+1=2 0 RNnB2! G(y;1s)f(y;s)dyds :=I1+I2+I3: SincejG(y;1s)j=ejyj2 4(1s) (4(1s))N=2;we obtain jI1j1 1=2 RNjG(y;1s)jjf(y;s)jdyds sup 1 2s1kf(s)kL1 1 1 2 RnjG(y;1s)jdyds! .kfkY; 10jI2j1=2 0 B2jG(y;1s)jjf(y;s)jdyds .sup 0s1 2kG(
;1s)kL1(RN) B2[0;1 2]jf(y; s)jdyds.kfkY and jI3j1=2 0 RNnB2jG(y;1s)jjf(y;s)jdyds C1 2 0 RNnB2ejyj2 4jf(y; s)jdyds C 1X k=2kn1ek2 4! sup y2RN Q1(y)jf(y; s)jdyds! .kfkY: The quantityjrT(f)(0;1)jcan be bounded in a similar way. The last term in the l.h.s. of (2.20) can be controlled using an energy estimate. Indeed, Wsatisfies the equation i@tW+ (i)
W=if (2.21) with initial condition W(
;0) = 0. Let2C1 0(B2)be a real-valued cut-off function such that 01onRNand= 1onB1. By multiplying (2.21) by i2W, integrating and taking real part, we get 1 2@t RN2jWj2+ RN2jrWj2+ 2 Re (+i) RNrWrW = RN2Re(fW): Using thatj+ij= 1and integrating in time between 0and1, it follows that 1 2 RN2jW(x;1)j2+ RN[0;1]2jrWj2 RN[0;1](2jrjjWjjrWj+2jfjjWj): From the inequality ab"a2+b2=(4");witha=jrWj,b= 2jrjjWjand"==2, we deduce that 2 RN[0;1]2jrWj2 RN[0;1]2 jrj2jWj2+2jfjjWj
: By the definition of , this implies that krWk2 L2(B1[0;1]).kWk2 L1(B2[0;1])+kWkL1(B2[0;1])kfkL1(B2[0;1]):(2.22) From the first part of the proof, we have kWkL1(B2[0;1])CkfkY: Using also that kfkL1(B2[0;1]).kfkY; we conclude from (2.22) that krWkL2(B1[0;1]).kfkY; which finishes the proof. 11Lemma 2.5. Let2(0;1]and;";L> 0. There exists C21, depending on andN, such that for all u02L1(RN) kS(t)u0kXC2(ku0kL1+ [u0]BMO ): (2.23) If in additionku0kL1Land[u0]BMO", then for all u2B(u0)we have sup t>0kukL1C2(+L)and [u]XC2(+"): (2.24) Proof.We first controlkS(t)u0kX. On the one hand, using the definition of Gand the relation 2+2= 1, we obtain kS(t)u0kL1=kGu0kL1kGkL1ku0kL1=N 2ku0kL1;8t>0: Thus sup t>0kS(t)u0kL1N 2ku0kL1: (2.25) On the other hand, using Lemma 2.3, Theorem A.1 and (2.4), [S(t)u0]X= sup t>0p tkrS(t)u0kL1+ sup x2RN r>0 1 rN Qr(x)jrS(t)u0j2dtdy!1 2 .kru0kBMO1 + [u0]BMO .[u0]BMO .[u0]BMO:(2.26) The estimate in (2.23) follows from (2.25) and (2.26), and we w.l.o.g. can choose C21. Finally, using (2.25), given u0such thatku0kL1Land[u0]BMO", for allu2B(u0)we have kukL1kuS(t)u0kL1+kS(t)u0kL1kuS(t)u0kX+kS(t)u0kL1C2(+L); and, using (2.26), [u]X[uS(t)u0]X+ [S(t)u0]XkuS(t)u0kX+ [S(t)u0]XC2(+"); which finishes the proof of (2.24). Now we proceed to bound the nonlinear term g(u) =2i(i)u(ru)2 1 +juj2: Lemma 2.6. For allu2X(RNR+;C), we have kg(u)kY[u]2 X: Proof.Letu2X(RNR+;C). Using (2.3) and the definitions of the norms in YandX, it follows that kg(u)kY sup t>0p tkrukL12 + sup x2RN r>01 rN Qr(x)jruj2dtdy[u]2 X: 12Now we have all the estimates to prove that Tu0is a contraction on B(u0). Proposition 2.7. Let2(0;1]and;"> 0. Given any u02L1(RN)with [u0]BMO", the operatorTu0given in (2.17)defines a contraction on B(u0), whenever and"satisfy 8C1C2 2(+")2: (2.27) Moreover, for all u;v2X(RNR+;C), kT(g(u))T(g(v))kXC1(2[u]2 X+ [u]X+ [v]X)kuvkX: (2.28) Here,C11andC21are the constants in Lemmas 2.4 and 2.5, respectively. Remark 2.8. Using the notation introduced in (2.11), the hypothesis (2.27) means that (;")2 S(8C1C2 2). Therefore, by (2.13), 1 8C1C2 2;and"1 32C1C2 2; (2.29) soand"are actually small. Since C1;C21, we have C2(+")5 32: (2.30) Proof.Letu02L1(RN)withku0kL1Land[u0]BMO", andu2B(u0). Using Lemma 2.4, Lemma 2.5 and Lemma 2.6, we have kTu0(u)S(t)u0kX=kT(g(u))kXC1kg(u)kYC1[u]2 XC1C2 2(+")2: ThereforeTu0mapsB(u0)into itself provided that C1C2 2(+")2: (2.31) Notice that by (2.14), the condition (2.27) implies that (2.31) is satisfied. To prove (2.28), we use the decomposition g(u)g(v) =2i(i)u 1 +juj2v 1 +jvj2 (ru)2+v 1 +jvj2((ru)2(rv)2)) : Since u 1 +juj2v 1 +jvj2juvj1 +jujjvj (1 +juj2)(1 +jvj2)juvj; and using (2.2), we obtain jg(u)g(v)j2juvjjruj2+jrurvj(jruj+jrvj): Therefore kg(u)g(v)kY2kjuvjjruj2kY+kjrurvj(jruj+jrvj)kY:=I1+I2:(2.32) ForI1, it is immediate that I12 sup t>0kuvkL12 64 sup t>0p tkrukL12 + sup x2RN r>01 rN Qr(x)jruj2dtdy3 752kuvkX[u]2 X: (2.33) 13Similarly, using the Cauchy–Schwarz inequality, I2 sup t>0p tkrurvkL1 sup t>0p t(krukL1+krvkL1) + sup x2RN r>01 rN krurvkL2(Qr(x))
krukL2(Qr(x))+krvkL2(Qr(x))
kuvkX([u]X+ [v]X):(2.34) Using Lemma 2.4, (2.32), (2.33) and (2.34), we conclude that kT(g(u))T(g(v))kXC1(2[u]2 X+ [u]X+ [v]X)kuvkX: (2.35) Letu;v2B(u0), by Lemma 2.5 and (2.30) [u]XC2(+")5 32; (2.36) so that 2[u]2 X+ [u]X+ [v]X37 16C2(+")<3C2(+"): (2.37) Then (2.35) implies that kTu0(u)Tu0(v)kX3C1C2(+")kuvkX: (2.38) From (2.29), we conclude that 3C1C2(+")15 321 2; (2.39) and then (2.38) yields that the operator Tu0defined in (2.17) is a contraction on B(u0). This concludes the proof of the proposition. Proof of Theorem 2.1. Let us setC=C1C2 2andK=C2, whereC1andC2are the constants in Lemma 2.4 and Lemma 2.5 respectively. Since satisfies (2.6), Proposition 2.7 implies that there exists a solution uof equation (2.16) in the ball B(u0), and in particular from Lemma 2.5 sup t>0kukL1K(+L)and [u]XK(+"): To prove the uniqueness part of the theorem, let us assume that uandvare solutions of (IDNLS) inX(RNR+;C)such that [u]X;[v]XK(+"); (2.40) with the same initial condition u0. By the definitions of CandK, (2.6) and (2.40), the estimates in (2.29) and (2.30) hold. It follows that (2.36), (2.37) and (2.39) are satisfied. Then, using (2.28), kuvkX=kT(g(u))T(g(v))kXC1(2[u]2 X+ [u]X+ [v]X)kuvkX 1 2kuvkX: From which it follows that u=v. To prove the dependence of the solution with respect to the initial data (part (iv)), consider uandvsolutions of (IDNLS) satisfying (2.40) with initial conditions u0andv0. Then, by definition,u=Tu0(u),v=Tv0(v)and kuvkX=kTu0(u)Tv0(v)kXkS(u0v0)kX+kT(g(u))T(g(v))kX: 14Using (2.15), (2.23) and (2.28) and arguing as above, we have kuvkXC2(ku0v0kL1+ [u0v0]BMO ) +C1(2[u]2 X+ [u]X+ [v]X)kuvkX 3C2ku0v0kL1+1 2kuvkX: This yields (2.10), since K=C2. The assertions in (ii)and(iii)follow from Theorem A.3. 2.2 The Cauchy problem for the LLG equation By using the inverse of the stereographic projection P1:C!S2nf0;0;1g, that is explicitly given by m= (m1;m2;m3) =P1(u), with m1=2 Reu 1 +juj2; m 2=2 Imu 1 +juj2; m 3=1juj2 1 +juj2; (2.41) we will be able to establish the following global well-posedness result for (LLG ). Theorem 2.9. Let2(0;1]. There exist constants C1andK4, such that for any 2(0;2],"0>0and>0such that 8K4C4(+ 82"0)2; (2.42) ifm0= (m0 1;m0 2;m0 3)2L1(RN;S2)satisfies inf RNm0 31 +and [m0]BMO"0; (2.43) then there exists a unique solution m= (m1;m2;m3)2X(RNR+;S2)of(LLG)such that inf x2RN t>0m3(x;t)1 +2 1 +K2(+1)2and [m]X4K(+ 82"0):(2.44) Moreover, we have the following properties. i)m2C1(RNR+;S2). ii)jm(
;t)m0j! 0inS0(RN)ast!0+. iii) Assume that mandnare respectively smooth solutions to (IDNLS) satisfying (2.44)with initial conditions m0andn0satisfying (2.43). Then kmnkX120K2km0n0kL1: (2.45) Remark 2.10. The restriction (2.42) on the parameters is similar to (2.27), but we need to include. To better understand the role of , we can proceed as before. Indeed, setting for a;> 0, S(a) =f(;"0)2R+R+:a4(+ 82"0)2g; we see that its shape is similar to the one in Figure 1. It is simple to verify that for any (;"0)2S(a), we have the bounds 4 aand"06 32a; (2.46) and the maximum value " 0=6 32ais attained at =4 4a. Also, the sets are well ordered, i.e. if ~aa>0, thenS(~a)S(a). 15We emphasize that the first condition in (2.43) is rather technical. Indeed, we need the essential range of m0to be far from the South Pole in order to use the stereographic projection. In the case= 1, Wang [35] proved the global well-posedness using only the second restriction in (2.43). It is an open problem to determinate if this condition is necessary in the case 2(0;1). The choice of the South Pole is of course arbitrary. By using the invariance of (LLG ) under rotations, we have the existence of solutions provided that the essential range of the initial condition m0is far from an arbitrary point Q2S2. Precisely, Corollary 2.11. Let2(0;1],Q2S2,2(0;2], and"0;> 0such that (2.42)holds. Given m0= (m0 1;m0 2;m0 3)2L1(RN;S2)satisfying inf RNjm0Qj22and [m0]BMO"0; there exists a unique smooth solution m2X(RNR+;S2)of(LLG)with initial condition m0 such that inf x2RN t>0jm(x;t)Qj24 1 +K2(+1)2and [m]X4K(+ 82"0):(2.47) For the sake of clarity, before proving Theorem 2.9, we provide a precise meaning of what we refer to as a weak and smooth global solution of the (LLG ) equation. The definition below is motivated by the following vector identities for a smooth function mwithjmj= 1: m
m= div( mrm); m(m
m) =
m+jrmj2m: Definition 2.12. LetT2(0;1]andm02L1(RN;S2). We say that m2L1 loc((0;T);H1 loc(RN;S2)) is a weak solution of (LLG)in(0;T)with initial condition m0if hm;@t'i=hmrm;r'ihrm;r'i+hjrmj2m;'i; and k(m(t)m0)'kL1!0;ast!0+;for all'2C1 0(RN(0;T)): (2.48) IfT=1, and in addition m2C1(RNR+), we say that mis a smooth global solution of (LLG)inRNR+with initial condition m0. Hereh
;
istands for hf1;f2i=1 0 RNf1
f2dxdt: With this definition, we see the following: Assume that mis a smooth global solution of (LLG) with initial condition m0and consider its stereographic projection P(m). IfP(m)and P(m0)are well-defined, then P(m)2C1(RNR+;C)satisfies (DNLS) pointwise, and lim t!0+P(m) =P(m0)inS0(RN): Therefore,ifinaddition P(m)2X(RNR;C),thenP(m)isasmoothglobalsolutionof (DNLS) with initial condition P(m0). Reciprocally, suppose that u2X(RNR+;C)\C1(RNR+) is a solution of (IDNLS) with initial condition u02L1(RN)such that (2.9) holds. If P1(u) andP1(u0)are in appropriate spaces, then P1(u)is a global smooth solution of (LLG ) with initial conditionP1(u0). The above (formal) argument allows us to obtain Theorem 2.9 from Theorem 2.1 once we have established good estimates for the mappings PandP1. In this context, we have the following 16Lemma 2.13. Letu;v2C1(RN;C),m= (m1;m2;m3);n= (n1;n2;n3)2C1(RN;S2). a) Assume that inf RNm31+andinf RNn31+for some constant 2(0;2]. Ifu=P(m) andv=P(n), then ju(x)v(x)j4 2jm(x)n(x)j; (2.49) [u]BMO8 2[m]BMO; (2.50) jru(x)j4 2jrm(x)j; (2.51) for allx2RN. b) Assume thatkukL1M,kvkL1M, for some constant M0. Ifm=P1(u)and n=P1(v), then inf RNm31 +2 1 +M2; (2.52) jm(x)n(x)j3ju(x)v(x)j; (2.53) jrm(x)j4jru(x)j; (2.54) jrm(x)rn(x)j4jru(x)rv(x)j+ 12ju(x)v(x)j(jru(x)j+jrv(x)j):(2.55) Proof.In the proof we will use the notation m:=m1+im2. To establish (2.49), we write u(x)v(x) =m(x)n(x) 1 +m3(x)+n(x)(n3(x)m3(x)) (1 +m3(x))(1 +n3(x)): Hence, sincejnj1,m3(x) + 1andn3(x) + 1,8x2RN, ju(x)v(x)jjm(x)n(x)j +jn3(x)m3(x)j 2: Using that jmnjjmnj (2.56) and that max1 a;1 a2 2 a2;for alla2(0;2]; we obtain (2.49). The same argument also shows that ju(y)u(z)j4 2jm(y)m(z)j;for ally;z2RN: (2.57) To verify (2.50), we recall the following inequalities in BMO (see [10]): [f]BMOsup x2RN Br(x) Br(x)jf(y)f(z)jdydz2[f]BMO: (2.58) Estimate (2.50) is an immediate consequence of this inequality and (2.57). To prove (2.51) it is enough to remark that jruj2 2(jrm1j+jrm2j+jrm3j)4 2jrmj: 17We turn into (b). Using the explicit formula for P1in (2.41), we can write m3=1 +2 1 +juj2: SincekukL1M, we obtain (2.52). To show (2.53), we compute mn=2u 1 +juj22v 1 +jvj2=2(uv) + 2uv(vu) (1 +juj2)(1 +jvj2); (2.59) m3n3=1juj2 1 +juj21jvj2 1 +jvj2=2(jvj2juj2) (1 +juj2)(1 +jvj2): (2.60) Using the inequalities a 1 +a21 2;1 +ab (1 +a2)(1 +b2)1;anda+b (1 +a2)(1 +b2)1;for alla;b0;(2.61) from (2.59) and (2.60) we deduce that jmnj2juvjandjm3n3j2juvj: (2.62) Hence jmnj=p jmnj2+jm3n3j2p 8juvj3juvj: To estimate the gradient, we compute rm=2ru 1 +juj24uRe(uru) (1 +juj2)2; (2.63) from which it follows that jrmjjruj2 1 +juj2+4juj2 (1 +juj2)2 3jruj; since4a (1+a)21;for alla0. Form3, we have rm3=2 Re(uru) 1 +juj22 Re(uru)(1juj2) (1 +juj2)2=4 Re(uru) (1 +juj2)2; and thereforejrm3j2jruj, since a (1 +a2)21 2;for alla0: (2.64) Hence jrmj=p jrm1j2+jrm2j2+jrm3j2p 13jruj4jruj; which gives (2.54). In order to prove (2.55), we start differentiating (2.59) rmrn=2r(uv) +r(uv)(vu) +uvr(vu) (1 +juj2)(1 +jvj2) 4((uv) +uv(vu))(Re(uru)(1 +jvj2) + Re(vrv)(1 +juj2)) (1 +juj2)2(1 +jvj2)2; 18Hence, setting R= maxfjru(x)j;jrv(x)jg, jrmrnj2jrurvj1 +jujjvj (1 +juj2)(1 +jvj2) + 2Rjuvjjuj+jvj (1 +juj2)(1 +jvj2) + 4Rjuvjjuj(1 +jujjvj) (1 +juj2)2(1 +jvj2)+jvj(1 +jujjvj) (1 +juj2)(1 +jvj2)2 : Using again (2.61), we get juj(1 +jujjvj) (1 +juj2)2(1 +jvj2)juj (1 +juj2)1 2: By symmetry, the same estimate holds interchanging ubyv. Therefore, invoking again (2.61), we obtain jrmrnj2jrurvj+ 6Rjuvj: (2.65) Similarly, writing juj2jvj2= (uv)u+ (uv)v, from (2.60) we have jrm3rn3j2jrurvj+ 6Rjuvj: (2.66) Therefore, sincep a2+b2a+b;8a;b0; inequalities (2.65) and (2.66) yield (2.55). Now we have all the elements to establish Theorem 2.9. Proof of Theorem 2.9. We continue to use the constants CandKdefined in Theorem 2.1. We recall that they are given by C=C1C2 2andK=C2, whereC11andC21are the constants in Lemmas 2.4 and 2.5, respectively. In addition, w.l.o.g. we assume that K=C24; (2.67) in order to simplify our computations. First we notice that by Remark 2.10, any and"0fulfilling the condition (2.42), also satisfy 8C(+ 82"0)2; (2.68) since4=K41(notice that K4and2(0;2]). Letm0as in the statement of the theorem and set u0=P(m0). Using (2.50) in Lemma 2.13, we have ku0kL1 1 1 +m0 3 L11 and [u0]BMO8"0 2: Therefore, bearing in mind (2.68), we can apply Theorem 2.1 with L:=1 and":= 82"0; to obtain a smooth solution u2X(RNR+;C)to (IDNLS) with initial condition u0. In particularusatisfies sup t>0kukL1K(+1)and [u]XK(+ 82"0): (2.69) 19Defining m=P1(u), we infer that mis a smooth solution to (LLG ) and, using the fact that k(u(
;t)u0)'kL1!0(see (2.9)) and (2.53), jm(
;t)m0j! 0inS0(RN);ast!0+: Notice also that applying Lemma 2.13 we obtain inf x2RN t>0m3(x;t)1 +2 1 +K2(+1)2and [m]X4[u]X4K(+ 82"0); which yields (2.44). Let us now prove the uniqueness. Let nbe a another smooth solution of (LLG ) with initial conditionu0satisfying inf x2RN t>0n3(x;t)1 +2 1 +K2(+1)2and [n]X4K(+ 82"0); (2.70) and letv=P(n)be its stereographic projection. Then by (2.51), [v]X 1 +K2(+1)22 [n]X: (2.71) We continue to control the upper bounds for [v]Xand[u]Xin terms of and the constants C11andC24. Notice that since and"0satisfy (2.42), from (2.46) with a= 8K4C, it follows that 4 8K4Cand"06 28K4C; or equivalently (recall that K=C2andC=C1C2 2) 4 8C1C6 2and8"0 24 32C1C6 2: (2.72) Hence K(+ 82"0)54 32C1C5 2: (2.73) Also, using (2.72), we have 1 +K2(+1)2=1 +C2 2 2(+ 1)2=C2 2 22 C2 2+ (+ 1)2 C2 2 2 2 C2 2+5 8C1C6 2+ 12! 2C2 2 2;(2.74) sinceC11,C24and2. From the bounds in (2.73) and (2.74), combined with (2.69), (2.70) and (2.71), we obtain [u]XK(+ 82"0)54 32C1C5 25 211C1 and [v]X(1+K2(+1)2)2[n]X(1+K2(+1)2)24K(+82"0) 2C2 2 22204 32C1C5 25 8C1; 20since2andC24. Finally, since uandvare solutions to (IDNLS) with initial condition u0, (2.28) and the above inequalities for [u]Xand[v]Xyield kuvkXC1(2[u]2 X+ [u]X+ [v]X)kuvkX C1 25 211C12 +5 211C1+5 8C1! kuvkX; which implies that u=v, bearing in mind that the constant on the r.h.s. of the above inequality is strictly less that one. This completes the proof of the uniqueness. It remains to establish (2.45). Let mandntwo smooth solutions of (LLG ) satisfying (2.44). As a consequence of the uniqueness, we see that mandnare the inverse stereographic projection of some functions uandvthat are solutions of (IDNLS) with initial condition u0=P(m0)and v0=P(n0), respectively. In particular, uandvsatisfy the estimates in (2.69). Using also (2.53) and (2.55), we deduce that kmnkX3 sup t>0kuvkL1+ 4[uv]X+ 12 sup t>0kuvkL1([u]X+ [v]X]) 4kuvkX+ 24C2(+ 82"0)kuvkX; 5kuvkX; where we have used (2.73) in obtaining the last inequality. Finally, using also (2.43) and (2.49), and applying (2.10) in Theorem 2.1, kmnkX30Kku0v0kL1 120K2km0n0kL1; which yields (2.45). Proof of Corollary 2.11. LetR2SO(3)such thatRQ= (0;0;1), i.e.Ris the rotation that mapsQto the South Pole. Let us set m0 R=Rm0. Then jm0Qj2=jR(m0Q)j2=jm0 R(0;0;1)j2= 2(1 +m0 3;R): Hence, inf x2RNm0 3;R1 + and [m0 R]BMO = [m0]BMO"0: Therefore, Theorem2.9providestheexistenceofauniquesmoothsolution mR2X(RNR+;S2) of (LLG) satisfying (2.44). Using the invariance of (LLG ) and setting m=R1mRwe obtain the existence of the desired solution. To establish the uniqueness, it suffices to observe that if n is another smooth solution of (LLG ) satisfying (2.47), then nR:=Rnis a solution of (LLG ) with initial condition m0 Rand it fulfills (2.44). Therefore, from the uniqueness of solution in Theorem 2.9, it follows that mR=nRand then m=n. Proof of Theorem 1.1. In Theorem 2.9 and Corollary 2.11, the constants are given by C=C1C2 2 andK=C2. As discussed in Remark 2.10, the value =4 32C1C2 2 21maximizes the range for "0in (2.27) and this inequality is satisfied for any "0>0such that "06 256C1C2 2: Taking M1=1 256C1C2 2; M 2=C2andM3=1 32C1C2 2; so that=M34, the conclusion follows from Theorem 2.9 and Corollary 2.11. Remark2.14. Wefinallyremarkthatispossibletostatelocal(intime)versionsofTheorems2.1 and 2.9 as it was done in [23, 22, 35]. In our context, the local well-posedness would concern solutions with initial condition m02VMO (RN), i.e. such that lim r!0+sup x2RN Br(x)jm0(y)m0 x;rjdy= 0: (2.75) Moreover, some uniqueness results have been established for solutions with this kind of initial data by Miura [32] for the Navier–Stokes equation, and adapted by Lin [29] to (HFHM). It is also possible to do this for (LLG ), for > 0. We do not pursuit here these types of results because they do not apply to the self-similar solutions mc;. This is due to the facts that the function m0 Adoes not belong to VMO (R)and that lim T!0+sup 0<t<Tp tk@xmc;kL16= 0: 3 Applications 3.1 Existence of self-similar solutions in RN The LLG equation is invariant under the scaling (x;t)!(x;2t), for > 0, that is if m satisfies (LLG ), then so does the function m(x;t) =m(x;2t); > 0: Therefore is natural to study the existence of self-similar solutions (of expander type), i.e. a solution msatisfying m(x;t) =m(x;2t);8>0; (3.1) or, equivalently, m(x;t) =fxp t ; for some f:RN!S2profile of m. In particular we have the relation f(y) =m(y;1), for all y2RN. From (3.1) we see that, at least formally, a necessary condition for the existence of a self-similar solution is that initial condition m0be homogeneous of degree 0, i.e. m0(x) =m0(x);8>0: Since the norm in X(RNR+;R3)is invariant under this scaling, i.e. kmkX=kmkX;8>0; where mis defined by (3.1), Theorem 2.9 yields the following result concerning the existence of self-similar solutions. 22Corollary 3.1. With the same notations and hypotheses as in Theorem 2.9, assume also that m0is homogeneous of degree zero. Then the solution mof(LLG)provided by Theorem 2.9 is self-similar. In particular there exists a smooth profile f:RN!S2such that m(x;t) =fxp t ; for allx2RNandt>0, andfsatisfies the equation 1 2y
rf(y) =f(y)
f(y)f(y)(f(y)
f(y)); for ally2RN. Herey
rf(y) = (y
rf1(y);:::;y
rfN(y)). Remark 3.2. Analogously, Theorem 2.1 leads to the existence of self-similar solutions for (DNLS), provided that u0is a homogeneous function of degree zero. For instance, in dimensions N2, Corollary 3.1 applies to the initial condition m0(x) =Hx jxj ; withHa Lipschitz map from SN1toS2\f(x1;x2;x3) :x31=2g, provided that the Lipschitz constant is small enough. Indeed, using (2.58), we have [m0]BMO4kHkLip; so that taking = 1=2;  =4 32K4C; " 0=6 256K4CandkHkLip"0; the condition (2.42) is satisfied and we can invoke Corollary 3.1. Other authors have considered self-similar solutions for the harmonic map flow (i.e. (LLG ) with= 1) in different settings. Actually, equation (HFHM) can be generalized for maps m:MR+!N, withMandNRiemannian manifolds. Biernat and Bizoń [8] established results whenM=N=Sdand3d6:Also, Germain and Rupflin [15] have investigated the caseM=RdandN=Sd, ind3. In both works the analysis is done only for equivariant solutions and does not cover the case M=RNandN=S2. 3.2 The Cauchy problem for the one-dimensional LLG equation with a jump initial data This section is devoted to prove Theorems 1.2 and 1.3 in the introduction. These two results con- cern the question of well-posedness/ill-posedness of the Cauchy problem for the one-dimensional LLG equation associated with a step function initial condition of the form m0 A:=A+R++AR; (3.2) where A+andAare two given unitary vectors in S2. 233.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2 As mentioned in the introduction, in [17] we proved the existence of the uniparametric smooth family of self-similar solutions fmc;gc>0of (LLG) for all2[0;1]with initial condition of the type (3.2) given by m0 c;:=A+ c;R++A c;R; (3.3) where A c;2S2are given by Theorem A.5. For the convenience of the reader, we collect some of the results proved in [17] in the Appendix. The results in this section rely on a further understanding of the properties of the self-similar solutions mc;. In Proposition 3.4 we show that mc;= (m1;c;;m2;c;;m3;c;)2X(RR+;S2); thatm3;c;is far from the South Pole and that [mc;]Xis small, if cis small enough. This will yield that mc;corresponds (up to a rotation) to the solution given by Corollary 3.1. More precisely, using the invariance under rotations of (LLG ), we can prove that, if the angle between A+andAis small enough, then the solution given by Corollary 3.1 with initial condition m0 A coincides (modulo a rotation) with mc;, for somec. We have the following: Theorem 3.3. Let2(0;1]. There exist L1;L2>0,2(1;0)and#>0such that the following holds. Let A+,A2S2and let#be the angle between them. If 0<##; (3.4) then there exists a solution mof(LLG)with initial condition m0 A. Moreover, there exists 0< c <p 2p, such that mcoincides up to a rotation with the self-similar solution mc;, i.e. there existsR2SO(3), depending only on A+,A,andc, such that m=Rmc;; (3.5) andmis the unique solution satisfying inf x2R t>0m3(x;t)and [m]XL1+L2c: (3.6) In order to prove Theorem 3.3, we need some preliminary estimates for mc;in terms of c and. To obtain them, we use some properties of the profile profile fc;= (f1;c;;f2;c;;f3;c;) constructed in [17] using the Serret–Frenet equations with initial conditions f1;c;(0) = 1; f 2;c;(0) =f3;c;(0) = 0: Also, jf0 j;c;(s)jces2=4;for alls2R; forj2f1;2;3gand mc;(x;t) =fc;xp t ;for all (x;t)2RR+: (3.7) Hence, for any x2R, jf3;c(x)j=jf3;c(x)f3;c(0)jjxj 0ce2=4dcpp: 24Since the same estimate holds for f2;c;, we conclude that jm2;c;(x;t)jcpp;andjm3;c;(x;t)jcppfor all (x;t)2RR+:(3.8) Moreover, since A c;= lim x!1fc;(x); we also get jA j;c;jcpp;forj2f2;3g: (3.9) We now provide some further properties of the self-similar solutions. Proposition 3.4. For2(0;1]andc>0, we have km0 2;c;kL1cpp;km0 3;c;kL1cpp;sup t>0km3;c;kL1cpp;(3.10) [m0 c;]BMO2cp 2p; (3.11) p tk@xmc;k1=c;for allt>0; (3.12) sup x2R r>01 r Qr(x)j@ymc;(y;t)j2dtdy2p 2c2 p: (3.13) In particular, mc;2X(RR+;S2)and [mc;]X4c 1 4: (3.14) Proof of Proposition 3.4. The estimates in (3.10) follow from (3.8) and (3.9). To prove (3.11), we use (2.58), (3.3), (3.10) and the fact that A c;= (A+ 1;c;;A+ 2;c;;A+ 3;c;); (3.15) (see Theorem A.5) to get [m0 c;]BMOsup x2RN Br(x) Br(x)jm0 c;(y)m0 c;(z)jdydz 2q (A+ 2;c;)2+ (A+ 3;c;)2sup x2RN Br(x) Br(x)dydz 2cp 2p: From (A.12) we obtain the equality in (3.12) and also Ir;x:=1 r Qr(x)j@ymc;(y;t)j2dtdy =c2 rx+r xrr2 0ey2 2t tdtdy: (3.16) Performing the change of variables z= (y2)=(2t), we see that r2 0ey2 2t tdt=E1y2 2r2 ; (3.17) 25whereE1is the exponential integral function E1(y) =1 yez zdz: This function satisfies that limy!0+E1(y) =1andlimy!1E1(y) = 0(see e.g. [1, Chapter 5]). Moreover, taking >0and integrating by parts, 1 E1(y2)dy=yE1(y2)1 + 21 ey2dy; (3.18) so L’Hôpital’s rule shows that the first term in the r.h.s. of (3.18) vanishes as !0+. Therefore, the Lebesgue’s monotone convergence theorem allows to conclude that E1(y2)2L1(R+)and 1 0E1(y2) =p: (3.19) By using (3.16), (3.17), (3.19), and making the change of variables z=py=(rp 2), we obtain Ir;x=c2 rx+r xrE1y2 2r2 dy=p 2c2 ppp 2(x r+1) pp 2(x r1)E1(z2)dzp 2c2 p
2p; (3.20) which leads to (3.13). Finally, the bound in (3.14) easily follows from those in (3.12) and (3.13) and the elementary inequality  1 +2p 2p1=2 1 1 4 1 + (2p 2)1=2
4 1 4; 2(0;1]: Proof of Theorem 3.3. First, we consider the case when A+=A+ c;andA=A c;(i.e. when m0 A=m0 c;) for some c >0. We will continue to show that the solution provided by Theo- rem 2.9 is exactly mc;, forcsmall. Indeed, bearing in mind the estimates in Proposition 3.4, we consider cp 2p; so that inf x2Rm0 3;c;(x)1 2: (3.21) In view of (3.11), (3.21) and Remark 2.10, we set "0:= 4cpp;  :=1 2;  :=4 8K4C=1 27K4C; (3.22) whereC;K1are the constants given by Theorem 2.9. In this manner, from (3.11), (3.21) and (3.22), we have inf Rm0 31 +and [m0]BMO"0; and the condition (2.42) is fulfilled if "06 256K4C; or equivalently, if c~c, with ~c:=p 216K4Cp: 26Observe that in particular ~c<p 2p. For fixed 0<c< ~c, we can apply Theorem 2.9 to deduce the existence and uniqueness of a solution mof (LLG) satisfying inf x2R t>0m3(x;t)1 +2 1 +K2(+ 2)2and [m]X4K+29Kcpp:(3.23) Now by Proposition 3.4, for fixed 0<c~c, we have the following estimates for mc; [mc;]X4c=1 4and inf x2R t>0m3;c;(x;t)1 2; so in particular mc;satisfies (3.23). Thus the uniqueness of solution implies that m=mc;, provided that c~c. Defining the constants L1,L2andby L1= 4K; L 2=29Kppand=1 +2 1 +K2(+ 2)2;(3.24) the theorem is proved in the case A=A c;. For the general case, we would like to understand which angles can be reached by varying the parametercin the range (0;~c]. To this end, for fixed 0< c~c, let#c;be the angle between A+ c;andA c;. From Lemma A.6, #c;arccos 1c2+ 32c3p 2 ;for allc2 0;2p 32i : Now, it is easy to see that the function F(c) = arccos 1c2+ 32c3p 2 is strictly increasing on the interval [0;2p 48]so that F(c)>F(0) = 0;for allc2 0;2p 48i : (3.25) Letc= min(~c;2p 48)and consider the map T:c!#c;on[0;c]. By Lemma A.6, Tis continuous on [0;c],T(0) = lim c!0+T(c) = 0and, bearing in mind (3.25), T(c) =#c;>0. Thus, from the intermediate value theorem we infer that for any #2(0;#c;), there exists c2(0;c)such that #=T(c) =#c;: We can now complete the proof for any A+,A2S2. Let#be the angle between A+and A. From the previous lines, we know that there exists #:=#c;such that if #2(0;#), there exists c2(0;c)such that#=#c;. For this value of c, consider the initial value problem associated with m0 c;and the constants defined in (3.24). We have already seen the existence of a unique solution mc;of the LLG equation associated with this initial condition satisfying (3.6). LetR2SO(3)be the rotation on R3such that A+=RA+ c;andA=RA c;. Then m:=Rmc;solves (LLG ) with initial condition m0 A. Finally, recalling the above definition ofL1,L2and, using the invariance of the norms under rotations and the fact that mc;is the unique solution satisfying (3.23), it follows that mis the unique solution satisfying the conditions in the statement of the theorem. We are now in position to give the proof of Theorem 1.2, the second of our main results in this paper. In fact, we will see that Theorem 1.2 easily follows from Theorem 3.3 and the well-posedness for the LLG equation stated in Theorem 2.9. 27Proof of Theorem 1.2. Let#,,L1andL2betheconstantsdefinedintheproofofTheorem3.3. Given A+andAsuch that 0<#<#, Theorem 3.3 asserts the existence of 0<c<p 2p(3.26) andR2SO(3)such thatRmc;is the unique solution of (LLG ) with initial condition m0 A satisfying (3.6), and in particular m0 A=Rm0 c;. By hypothesis m0satisfies km0m0 AkL1cp 2p: (3.27) Hence, defining m0 R=R1m0, recalling that [f]BMO2kfkL1and using the invariance of the norms under rotations, we deduce from (3.27) that km0 RkL1km0 c;kL1+cp 2pand [m0 R]BMO[m0 c;]BMO +cpp: Then, by Proposition 3.4, km0 3;RkL12cppand [m0 R]BMO4cpp: (3.28) From (3.26) and (3.28), it follows that m0 3;R(x)1=2;for allx2R: Therefore, as in the proof of Theorem 3.3, we can apply Theorem 2.9 with the values of "0, andgiven in (3.22) to deduce the existence of a unique (smooth) solution mRof (LLG) with initial condition m0 Rsatisfying inf x2R t>0m3;R(x;t)1 +2 1 +K2(+ 2)2=and [mR]X4K+29Kcpp=L1+L2c: Since we have taken the values for "0,andas in the proof Theorem 3.3, Theorem 2.9 also implies that kmRmc;kX480Kkm0 Rm0 c;kL1: The conclusion of the theorem follows defining m=RmRandL3= 480K, and using once again the invariance of the norm under rotations. 3.2.2 Multiplicity of solutions. Proof of Theorem 1.3 As proved in [17], when = 1, the self-similar solutions are explicitly given by mc;1(x;t) = (cos(cErf(x=p t));sin(cErf(x=p t));0);for all (x;t)2RR+;(3.29) for everyc>0, where Erf(
)is the non-normalized error function Erf(s) =s 0e2=4d: In particular, ~A c;1= (cos(cp);sin(cp);0) 28#c;1  c Figure 2: The angle #c;as a function of cfor= 1. and the angle between A+ c;1andA c;1is given by #c;1= arccos(cos(2 cp)): (3.30) Formula (3.30) and Figure 2 show that there are infinite values of cthat allow to reach any angle in [0;]. Therefore, using the invariance of (LLG ) under rotations, in the case when = 1, one can easily prove the existence of multiple solutions associated with a given initial data of the form m0 Afor any given vectors A2S2(see argument in the proof included below). In the case that is close enough to 1, we can use a continuity argument to prove that we still have multiple solutions. More precisely, Theorem 1.3 asserts that for any given initial data of the form m0 Awith angle between A+andAin the interval (0;), ifis sufficiently close to one, then there exist at leastk-distinct solutions of (LLG ) associated with the same initial condition, for any given k2N. The rest of this section is devoted to the proof of Theorem 1.3. Proof of Theorem 1.3. Letk2N,A2S2and#2(0;)be the angle between A+andA. Using the invariance of (LLG ) under rotations, it suffices to prove the existence of k2(0;1) such that for every 2[k;1]there exist 0< c1<


< cksuch that the angle #cj;between A+ cj;andA cj;, satisfies #cj;=#;for allj2f1;:::;kg: (3.31) In what follows, and since we want to show the existence of at least k-distinct solutions, we will assume without loss of generality that kis large enough. First observe that, since A c;= (A+ 1;c;;A+ 2;c;;A+ 3;c;), we have the explicit formula cos(#c;) = 2(A+ 1;c;)21; and using Lemma A.8 in the Appendix, we get jcos(#c;)cos(#c;1)j=j2((A+ 1;c;)2(A+ 1;c;1)2)j4jA+ 1;c;A+ 1;c;1j4h(c)p 1;(3.32) for all2[1=2;1], withh:R+!R+an increasing function satisfying lims!1h(s) =1. Forj2N, we setaj= (2j+ 1)p=2andbj= (2j+ 2)p=2, so that (3.30) and (3.32) yield cos(#aj;)1 + 4h(aj)p 1and cos(#bj;)14h(bj)p 1;82[1=2;1]: (3.33) 29Definel= cos(#)and k= max 11l 8h(bk)2 ;11 +l 8h(bk)2 : Notice that, since #2(0;), we have1< l < 1and thusk<1. Also, since hdiverges to 1, we can assume without loss of generality that k2[1=2;1), and from the definition of kwe have 0<p1k<min1l 8h(bk);1 +l 8h(bk) : Therefore, from (3.33) and h(aj)< h(bj)h(bk)(sincehis a strictly increasing function), we get cos(#aj;)1 +l 2and1 +l 2cos(#bj;);8j2f1;:::;kg;82[k;1]; and thus cos(#aj;)<l< cos(#bj;);8j2f1;:::;kg;82[k;1]; (3.34) sincel2(1;1). Let us fix2[k;1]andj2f1;:::;kg. By Lemma A.8, c!cos(#c;)is a continuous function on c. Therefore (3.34) and the intermediate value theorem yield the existence of cj2 [aj;bj]such that cos(#cj;) =l= cos(#); or equivalently, such that #cj;=#. Finally, for each j2f1;:::;kg, letRj2SO(3)be such that m0 A=Rjm0 cj;, and define mj bymj=Rjmcj;. Then mjsolves (LLG ) with initial data m0 Aand the control of @xmjin (1.9) follows from the definition of mjin terms of mcj;and the analogous property established in Theorem A.5 for the self-similar solution mcj;(see (A.12)). In the case when = 1and#2[0;], formula (3.30) shows that sequence fcjgj1in (1.10) satisfies#cj;1=#for allj2N. The result follows by considering the sequence of solutions fmjgj1described above. Remark 3.5. Notice that the proof given above also shows how close kneeds to be to 1 in terms of the (fixed) angle #2(0;), and in particular k!1ask!1, andk!1as#!0 (i.e. whenl!1). Remark 3.6. For= 1andc>0, the function u(x;t) =P(mc;1) = exp icErf(x=p t)
is a solution of (DNLS) with initial condition u0=eicpR++eicpR: Therefore there is also a multiplicity phenomenon for the equation (DNLS). 3.3 A singular solution for a nonlocal Schrödinger equation We have used the stereographic projection to establish a well-posedness result for (LLG ). Melcher [31] showed a global well-posedness result, provided that krm0kLN";m0Q2H1(RN)\W1;N(RN); > 0; N3; 30for some Q2S2and">0small. Later, Lin, Lan and Wang [30] improved this result and proved global well-posedness under the conditions krm0kM2;2";m0Q2L2(RN); > 0; N2; for some Q2S2and">0small.6In the context of Theorem 1.1 and using the characterization ofBMO1in Theorem A.1, the second condition in (1.3) says that krm0kBMO1is small. In view of the embeddings LN(RN)M2;2(RN)BMO1(RN); forN2, we deduce that Theorem 1.1 includes initial conditions with less regularity, as long as their essential range is not S2. The argument in [30, 31] is based on the method of moving frames that produces a covariant complex Ginzburg–Landau equation. One of the aims of this subsection is to compare their approach in the context of the self-similar solutions mc;, and in particular to draw attention to a possible difficulty in using it to study these solutions. In the sequel we consider the one-dimensional case N= 1and2[0;1]. Then the moving frames technique can be recast as a Hasimoto transformation as follows. Assume that mis the tangent vector of a curve in R3, i.e.m=@xX, for some curve X(x;t)2R3parametrized by the arc-length. It can be shown (see [12]) that if mevolves under (LLG ), then the torsion and the curvature cofXsatisfy @t= c@xc +@x@xxcc2 c + c2+@x@x(c) +@xc c ; @tc =(@x(c)@xc) + @xcc2
: Hence, defining the Hasimoto transformation [19] (also called filament function) v(x;t) = c(x;t)ei
x 0(;t)d; (3.35) we verify that vsolves the following dissipative Schrödinger (or complex Ginzburg–Landau) equation i@tv+ (i)@xxv+v 2 jvj2+ 2x 0Im(v@xv)A(t) = 0; (3.36) where=p 12and A(t) =  c2+2(@xxcc2) c + 2@x(c) +@xc c (0;t): The curvature and torsion associated with the self-similar solutions mc;are (see [17]): cc;(x;t) =cp tex2 4tandc;(x;t) =x 2p t: (3.37) Therefore in this case A(t) =c2 t(3.38) 6We recall that v2M2;2(RN)ifv2L2 loc(RN)and kvkM2;2:= sup x2RN r>01 r(N2)=2kvkL2(Br(x))<1: 31and the Hasimoto transformation of mc;is vc;(x;t) =cp te(+i)x2 4t: In particular vc;is a solution of (3.36) with A(t)as in (3.38), for all 2[0;1]andc > 0. Moreover, the Fourier transform of this function (w.r.t. the space variable) is bvc;(;t) = 2cp (+i)e(+i)2t; so thatvc;is a solution of (3.36) with a Dirac delta as initial condition: vc;(
;0) = 2cp (+i): Heredenotes the delta distribution at the point x= 0andpzdenotes the square root of a complex number zsuch that Im(pz)>0. In the limit cases = 0and= 1, the first three terms in equation (3.36) lead to a cubic Schrödinger equation and to a linear heat equation, respectively. The Cauchy problem with a Dirac delta for these kind of equations associated with a power type non-linearity has been studied by several authors (see e.g. [4] and the reference therein). We recall two classical results. Theorem 3.7 ([9]).Letp2andu2Lp loc(RR+)be a solution in the sense of distributions of @tu@xxu+jujpu= 0onRR+: (3.39) Assume that lim t!0+ Ru(x;t)'(x)dx= 0;for all'2C0(Rnf0g); (3.40) whereC0(Rnf0g)denotes the space of continuous functions with compact support in Rnf0g. Thenu2C2;1(R[0;1))andu(x;0) = 0for allx2R. In particular there is no solution of (3.39)such that lim t!0+ Ru(x;t)'(x)dx='(0);for all'2C0(RN): In[9]itisalsoprovedthatif 1<p< 2, equation(3.39)hasaglobalsolutionwithaDiracdelta as initial condition, as in the case of the linear parabolic equation. Concerning the Schrödinger equation, we have the following ill-posedness result due to Kenig, Ponce and Vega [21]. Theorem 3.8 ([21]).Letp2. Either there is no solution in the sense of distributions of i@tu+@xxu+jujpu= 0onRR+; (3.41) with lim t!0+u(
;t) =inS0(R); in the class u;jujpu2L1(R+;S0(R)), or there is more than one. After performing an appropriate change of variables, equation (3.36) leads to the following equation i@tu+ (i)@xxu+u 2 juj2+ 2x 0Im(u@yu)dy = 0: (3.42) Since2[0;1], the above equation can be seen as an intermediate model between (3.39) and (3.41). Therefore one could expect that when 2(0;1], the solutions of (3.42) share similar properties to those established in Theorem 3.7 for the equation (3.39). We will continue to observe that this is not necessarily the case. To this end, we need the following: 32Proposition 3.9. For all2[0;1]and for all c2Cnf0gthe function wc;:RR+!Cgiven by wc;(x;t) =cp texpijcj2 2ln(t) + (i)x2 4t is a solution of (3.42). In addition, i) If2[0;1), thenwc;(
;t)does not converge in S0(R)ast!0+. ii) If2(0;1], then lim t!0+ Rwc;(x;t)'(x)dx= 0;for all'2C0(Rnf0g): (3.43) Proof.A straightforward computation shows that wc;satisfies (3.42). The proof that wc;(
;t) does not converge in S0(R)ast!0+ifc6= 0is the same as in [21]. Indeed, for '2S(R), by Parseval’s theorem, Rwc;(x;t)'(x)dx=1 2 Rbwc;(;t)b'()d =ceijcj2ln(t)=2 pp +i Re(+i)2tb'()d: By the dominated convergence theorem, the last integral converges: lim t!0+ Re(+i)2tb'()d= Rb'()d= 2'(0): Since6= 0,eijcj2ln(t)=2does not admit a limit at 0inS0(R). We conclude that wc;(
;t)does not converge in S0(R)ast!0+. It remains to prove (3.43). Since now '2C0(Rnf0g), we cannot proceed as before. However, using the change of variables x=p ty, we have lim t!0+ Rwc;(x;t)'(x)dx=ceijcj2ln(t)=2 Re(+i)y2=4'(p ty)dy: Therefore, since >0and'(0) = 0, the dominated convergence theorem implies that lim t!0+ Re(+i)y2=4'(p ty)dy='(0) Re(+i)y2=4dy= 0: Sincejeijcj2ln(t)=2j= 1, we obtain (3.43). The results in Proposition 3.9 lead to the following remarks: 1. Observe that if 2(0;1),wc;provides a solution to the dissipative equation (3.42). Moreover, form part (ii)in Proposition 3.9, wc;satisfies the condition (3.40). However, notice that wc;cannot be extended to C2;1(R[0;1))due to the presence of a logarithmic oscillation. This is in contrast with the properties for solutions of the cubic heat equation (3.39) established in Theorem 3.7. 2. In the case = 0, equation (3.42) corresponds to (3.41) with p= 2, i.e. to the equation cubic NLS equation that is invariant under the Galilean transformation. The proof of the ill-posedness result given in Theorem 3.8 relies on this invariance and part (i)of Proposi- tion 3.9 with = 0. Although when  > 0, equation (3.42) is no longer invariant under the Galilean transformation, part (i)of Proposition 3.9 could be an indicator that that the Cauchy problem (3.42) with a delta as initial condition is still ill-posed. This question rests open for the moment and it seems that the use of (3.36) (or (3.42)) can be more difficult to formulate a Cauchy theory for (LLG ) including self-similar solutions. 334 Appendix The characterization of BMO1 1(RN)as sum of derivatives of functions in BMO was proved by Koch and Tataru in [23]. A straightforward generalization of their proof leads to the following characterization of BMO1 (RN). Theorem A.1. Let2(0;1]andf2S0(RN). Thenf2BMO1 (RN)if and only if there exist f1;:::;fN2BMO(RN)such thatf=PN j=1@jfj. In addition, if such a decomposing holds, then kfkBMO1 .NX j=1[fj]BMO: The next results provide the equivalence between the weak solutions and the Duhamel for- mulation. We first need to introduce for T >0the spaceL1 uloc(RN(0;T))defined as the space of measurable functions on RN(0;T)such that the norm kfkuloc;T:= sup x02RN B(x0;1)T 0jf(y;t)jdtdy is finite. We refer the reader to Lemarié–Rieusset’s book [27] for more details about these kinds of spaces. In particular, we recall the following result corresponding to Lemma 11.3 in [27] in the case= 1. It is straightforward to check that the same proof still applies if 2(0;1). Lemma A.2. Let2(0;1],T2(0;1)andw2L1 uloc(RN(0;T)). Then the function W(x;t) :=t 0S(ts)w(x;s)ds is well defined and belongs to L1 uloc(RN(0;T)). Moreover, i@tW+ (i)
W=winD0(RNR+); and the application [0;T]!R t7! kW(
;t)kL1(B1(x0)) is continuous for any x02R, withkW(
;t)kL1(B1(x0))!0, ast!0+, uniformly in x0. Following the ideas in [27], we can establish now the equivalence between the notions of solutions as well as the regularity. Theorem A.3. Let2(0;1]andu2X(RNR+;C). Then the following assertions are equivalent: i) The function usatisfies iut+ (i)
u= 2(i)u(ru)2 1 +juj2inD0(RNR+): (A.1) ii) There exists u02S0(RN)such thatusatisfies u(t) =S(t)u02(i)t 0S(ts)u(ru)2 1 +juj2ds: 34Moreover, if (ii) holds, then u2C1(RNR+)and k(u(t)u0)'kL1(RN)!0;ast!0+; (A.2) for any'2S(RN). Proof.In view of Lemma A.2, we need to prove that the function g(u) =2(i)u(ru)2 1 +juj2 belongs toL1 uloc(RN(0;T)), for allT >0. Indeed, by (2.3) we have kg(u)kuloc;Tkjruj2kuloc;T: (A.3) IfT1, then kjruj2kuloc;Tsup x02RN Q1(x0)jru(y;t)j2dtdykuk2 X: (A.4) IfT1, using that jruj[u]Xp t;for anyt>0; we get kjruj2kT;ulocsup x02RN Q1(x0)jru(y;t)j2dtdy + sup x02RNT 1 B1(x0)jruj2dydt kuk2 X+ [u]2 XjB1(0)jT 11 tdt kuk2 X(1 +jB1(0)jln(T)):(A.5) In conclusion, we deduce from (A.3), (A.4) and (A.5) that g(u)2L1 uloc(RN(0;T))and then it follows from Lemma A.2 that (ii) implies (i). The other implication can be established as in [27, Theorem 11.2]. Moreover, we deduce that the function W(x;t) :=T(g(u))(x;t) =t 0S(ts)g(u)ds satisfieskW(
;t)kL1(B1(x0))!0, ast!0+, uniformly in x02RN. Let us take '2S(RN)and a constantC'>0such thatj'(x)jC'(2 +jxj)N1. Then  RNj'(y)W(y;t)jdyX k2ZN B1(k)C' (2 +jxj)N+1jW(y;t)jdy sup x02RNkW(
;t)kL1(B1(x0))X k2ZNC' (1 +jkj)N+1; so thatk'W(
;t)kL1(RN)!0ast!0+, i.e. k(u(t)S(t)u0)'kL1(RN)!0;ast!0+: (A.6) On the other hand, since u02L1(RN), kS(t)u0u0kL1(Br(0))!0;ast!0+; (A.7) 35for anyr>0(see e.g. [3, Corollary 2.4]). Given >0, we fixr>0such that 2ku0k1k'kL1(Bcr(0)): Using (A.7), we obtain lim t!0+k(S(t)u0u0)'kL1(Br(0))= 0: Then, passing to limit in the inequality k(S(t)u0u0)'kL1(RN)k(S(t)u0u0)'kL1(Br(0))+ 2ku0kL1(RN)k'kL1(Bcr(0));(A.8) we obtain lim sup t!0+k(S(t)u0u0)'kL1(RN): (A.9) Therefore lim t!0+k(S(t)u0u0)'kL1(RN)= 0: Combining with (A.6), we conclude the proof of (A.2). It remains to prove that uis smooth for t >0. Sinceu2X(RNR+;C), we get that u;ru2L1 loc(RNR+). Theng(u)2L2 loc(RNR+)so theLp-regularity theory for parabolic equations implies that a function usatisfying (A.1) belongs to u2H2;1 loc(RNR+)(see [28, 24] and [33, Remark 48.3] for notations and more details). Since the space Hk\L1is stable under multiplication (see e.g. [20, Chapter 6]), we can use a bootstrap argument to conclude that u2C1(RNR+). Remark A.4. Several authors have studied further properties of the solutions found by Koch and Tataru for the Navier–Stokes equations. For instance, analyticity, decay rates of the higher- order derivatives in space and time have been investigated by Miura and Sawada [32], Germain, Pavlović and Staffilani [14], among others. A similar analysis for the solution uof (DNLS) is beyond the scope of this paper, but it can probably be performed using the same arguments given in [32, 14]. We end this appendix with some properties of the self-similar found in [17]. Theorem A.5 ([17]).LetN= 1. For every 2[0;1]andc >0, there exists a profile fc;2 C1(R;S2)such that mc;(x;t) =fc;xp t ;for all (x;t)2RR+; is a smooth solution of (LLG)onRR+. Moreover, (i) There exist unitary vectors A c;= (A j;c;)3 j=12S2such that the following pointwise con- vergence holds when tgoes to zero: lim t!0+mc;(x;t) =8 < :A+ c;;ifx>0; A c;;ifx<0;(A.10) andA c;= (A+ 1;c;;A+ 2;c;;A+ 3;c;). (ii) There exists a constant C(c;;p ), depending only on c,andpsuch that for all t>0 kmc;(
;t)A+ c;(0;1)(
)A c;(1;0)(
)kLp(R)C(c;;p )t1 2p; (A.11) for allp2(1;1). In addition, if >0,(A.11)also holds for p= 1. 36(iii) Fort>0andx2R, the derivative in space satisfies j@xmc;(x;t)j=cp tex2 4t: (A.12) (iv) Let2[0;1]. Then A+ c;!(1;0;0)asc!0+. Lemma A.6. Letc >0,2(0;1],A+ c;;A c;be the unit vectors given in Theorem A.5 and #c;the angle between A+ c;andA c;. Then, for fixed 2(0;1],#c;is a continuous function inc. Also, for 0<c<2p=32, #c;arccos 1c2+32c3p 2 : (A.13) Remark A.7. If2(0;1]andc2(0;2p=32), then 1c2+32c3p 22(0;1), so that its arccosis well-defined. Proof.The continuity was proved in [17]. To show the estimate (A.13), we use Theorem 1.3 in [17], to get A+ 2;c;cp (1 +)p 2c2 4+c2 p 2 1 +c2 8+cp (1 +) 2p 2! +c2 2p 22 : Since2(0;1], we have for any c2(0;1],  4+ p 2 1 +c2 8+cp (1 +) 2p 2! +c22 821 2 4+p 2 1 + 8+p 2 +2 8 8 2: We deduce that for all ;c2(0;1], A+ 2;c;cp (1 +)p 28c2 2cpp 28c2 2: In particular A+ 2;c;0ifc2p=(8p 2). Thus (A+ 2;c;)2c2 216c3pp 22+64c4 4c2 216c3p 2; so that cos(#c;) = 12((A+ 2;c;)2+ (A+ 3;c;)2)1c2+32c3p 2; which implies (A.13). The following lemma is a slightly refinement of Theorem 1.4 in [17]. Lemma A.8 ([17]).Letc >0,2[0;1]andA+ c;be the unit vector given in Theorem A.5. ThenA+ c;is a continuous function of in[0;1]and jA+ c;A+ c;1jh(c)p 1;for all2[1=2;1]; (A.14) whereh:R+!R+is a strictly increasing function satisfying lim s!1h(s) =1: 37Proof.Inviewof[17, Theorem1.4], weonlyneedtoprovethattheconstant C(c)inthestatement of the Theorem 1.4 (notice that c0in [17] corresponds to cin our notation) is polynomial in c with nonnegative coefficients. Looking at the proof of [17, Theorem 1.4], we see that the constant C(c)behaves like the constant in inequality (3.108) in [17]. In view of (3.17), the estimate (3.23) in [17] can be written as jf(s)jp 2andjf0(s)jc 2es2=4; and then (3.18) can be recast as jgj c 4+c2p 2 8!s es2=4+s2es2=2 : Then, it can be easily checked that the function his a polynomial with nonnegative coefficients. Acknowledgments. A.deLairewaspartiallysupportedbytheLabexCEMPI(ANR-11-LABX- 0007-01) and the MathAmSud program. S. Gutierrez was partially supported by the EPSRC grant EP/J01155X/1 and the ERCEA Advanced Grant 2014 669689 - HADE. References [1] M.AbramowitzandI.A.Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. [2] F.AlougesandA.Soyeur. OnglobalweaksolutionsforLandau-Lifshitzequations: existence and nonuniqueness. Nonlinear Anal. , 18(11):1071–1084, 1992. [3] J. M. Arrieta, A. Rodriguez-Bernal, J. W. Cholewa, and T. Dlotko. Linear parabolic equa- tions in locally uniform spaces. Math. Models Methods Appl. Sci. , 14(2):253–293, 2004. [4] V. Banica and L. Vega. On the Dirac delta as initial condition for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire , 25(4):697–711, 2008. [5] V. Banica and L. Vega. On the stability of a singular vortex dynamics. Comm. Math. Phys. , 286(2):593–627, 2009. [6] V. Banica and L. Vega. Scattering for 1D cubic NLS and singular vortex dynamics. J. Eur. Math. Soc. (JEMS) , 14(1):209–253, 2012. [7] V. Banica and L. Vega. Stability of the self-similar dynamics of a vortex filament. Arch. Ration. Mech. Anal. , 210(3):673–712, 2013. [8] P. Biernat and P. Bizoń. Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres. Nonlinearity , 24(8):2211–2228, 2011. [9] H. Brezis and A. Friedman. Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl. (9) , 62(1):73–97, 1983. 38[10] H. Brezis and L. Nirenberg. Degree theory and BMO. I. Compact manifolds without bound- aries.Selecta Math. (N.S.) , 1(2):197–263, 1995. [11] J.-M. Coron. Nonuniqueness for the heat flow of harmonic maps. Ann. Inst. H. Poincaré Anal. Non Linéaire , 7(4):335–344, 1990. [12] M. Daniel and M. Lakshmanan. Perturbation of solitons in the classical continuum isotropic Heisenberg spin system. Physica A: Statistical Mechanics and its Applications , 120(1):125– 152, 1983. [13] S. Ding and C. Wang. Finite time singularity of the Landau-Lifshitz-Gilbert equation. Int. Math. Res. Not. IMRN , (4):Art. ID rnm012, 25, 2007. [14] P. Germain, N. Pavlović, and G. Staffilani. Regularity of solutions to the Navier-Stokes equations evolving from small data in BMO1.Int. Math. Res. Not. IMRN , (21):Art. ID rnm087, 35, 2007. [15] P. Germain and M. Rupflin. Selfsimilar expanders of the harmonic map flow. Ann. Inst. H. Poincaré Anal. Non Linéaire , 28(5):743–773, 2011. [16] T. L. Gilbert. A lagrangian formulation of the gyromagnetic equation of the magnetization field.Phys. Rev. , 100:1243, 1955. [17] S. Gutiérrez and A. de Laire. Self-similar solutions of the one-dimensional Landau–Lifshitz– Gilbert equation. Nonlinearity , 28(5):1307, 2015. [18] S. Gutiérrez, J. Rivas, and L. Vega. Formation of singularities and self-similar vortex motion under the localized induction approximation. Comm. Partial Differential Equations , 28(5- 6):927–968, 2003. [19] H. Hasimoto. A soliton on a vortex filament. J. Fluid Mech , 51(3):477–485, 1972. [20] L. Hörmander. Lectures on nonlinear hyperbolic differential equations , volume 26 of Math- ématiques & Applications (Berlin) [Mathematics & Applications] . Springer-Verlag, Berlin, 1997. [21] C. E. Kenig, G. Ponce, and L. Vega. On the ill-posedness of some canonical dispersive equations. Duke Math. J. , 106(3):617–633, 2001. [22] H. Koch and T. Lamm. Geometric flows with rough initial data. Asian J. Math. , 16(2):209– 235, 2012. [23] H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Adv. Math. , 157(1):22–35, 2001. [24] O. Ladyzhenskaya, V. Solonnikov, and N. Ural’tseva. Linear and quasi-linear equations of parabolic type . Amer. Math. Soc., Transl. Math. Monographs. Providence, R.I., 1968. [25] M. Lakshmanan and K. Nakamura. Landau-Lifshitz equation of ferromagnetism: Exact treatment of the Gilbert damping. Phys. Rev. Lett. , 53:2497–2499, 1984. [26] L. Landau and E. Lifshitz. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion , 8:153–169, 1935. [27] P. G. Lemarié-Rieusset. Recent developments in the Navier-Stokes problem , volume 431 ofChapman & Hall/CRC Research Notes in Mathematics . Chapman & Hall/CRC, Boca Raton, FL, 2002. 39[28] G. M. Lieberman. Second order parabolic differential equations . World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [29] J. Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete Contin. Dyn. Syst. , 33(2):739–755, 2013. [30] J. Lin, B. Lai, and C. Wang. Global well-posedness of the Landau-Lifshitz-Gilbert equation for initial data in Morrey spaces. Calc. Var. Partial Differential Equations , 54(1):665–692, 2015. [31] C. Melcher. Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equa- tion in higher dimensions. Indiana Univ. Math. J. , 61(3):1175–1200, 2012. [32] H. Miura and O. Sawada. On the regularizing rate estimates of Koch-Tataru’s solution to the Navier-Stokes equations. Asymptot. Anal. , 49(1-2):1–15, 2006. [33] P. Quittner and P. Souplet. Superlinear parabolic problems . Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states. [34] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory inte- grals, volume 43 of Princeton Mathematical Series . Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [35] C. Wang. Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal. , 200(1):1–19, 2011. [36] D. Wei. Micromagnetics and Recording Materials . SpringerBriefs in Applied Sciences and Technology. Springer Berlin Heidelberg, 2012. 40

2017-01-11

We prove a global well-posedness result for the Landau-Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any dimension. In the one-dimensional case, we characterize the self-similar solutions associated with an initial data given by some ($\mathbb{S}^2$-valued) step function and establish their stability. We also show the existence of multiple solutions if the damping is strong enough. Our arguments rely on the study of a dissipative quasilinear Schr\"odinger obtained via the stereographic projection and techniques introduced by Koch and Tataru.

The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions

1701.03083v2

arXiv:1806.04782v3 [cond-mat.other] 9 Dec 2020Dynamical and current-induced Dzyaloshinskii-Moriya int eraction: Role for damping, gyromagnetism, and current-induced torques in noncolline ar magnets Frank Freimuth1,2,∗Stefan Bl¨ ugel1, and Yuriy Mokrousov1,2 1Peter Gr¨ unberg Institut and Institute for Advanced Simula tion, Forschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y and 2Institute of Physics, Johannes Gutenberg University Mainz , 55099 Mainz, Germany Both applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya in- teraction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respec- tively. We report a theory of CIDMI and DDMI. The inverse of CI DMI consists in charge pumping byatime-dependentgradient ofmagnetization ∂2M(r,t)/∂r∂t, while theinverseofDDMIdescribes the torque generated by ∂2M(r,t)/∂r∂t. In noncollinear magnets CIDMI and DDMI depend on the local magnetization direction. The resulting spatial g radients correspond to torques that need to be included into the theories of Gilbert damping, gyromag netism, and current-induced torques (CITs) in order to satisfy the Onsager reciprocity relation s. CIDMI is related to the modification of orbital magnetism induced by magnetization dynamics, whic h we call dynamical orbital magnetism (DOM), and spatial gradients of DOM contribute to charge pum ping. We present applications of this formalism to the CITs and to the torque-torque correlat ion in textured Rashba ferromagnets. I. INTRODUCTION Since the Dzyaloshinskii-Moriya interaction (DMI) controls the magnetic texture of domain walls and skyrmions, methods to tune this chiral interaction by external means have exciting prospects. Application of gatevoltage[1–3]orlaserpulses[4]arepromisingwaysto modify DMI. Additionally, theory predicts that in mag- netic trilayer structures the DMI in the top magnetic layer can be controlled by the magnetization direction in the bottom magnetic layer [5]. Moreover, methods to generatespin currentsmaybe usedto induceDMI, which is predicted by the relations between the two [6, 7]. Re- cent experiments show that also electric currents modify DMI in metallic magnets, which leads to large changes in the domain-wallvelocity [8, 9]. However,a rigoroustheo- retical formalism for the investigation of current-induced DMI (CIDMI) in metallic magnets has been lacking so far, and the development of such a formalism is one goal of this paper. Recently, a Berry phase theory of DMI [6, 10, 11] has been developed, which formally resembles the mod- ern theory of orbital magnetization [12–14]. Orbital magnetism is modified by the application of an electric field, which is known as the orbital magnetoelectric re- sponse [15]. In the case of insulators it is straightfor- ward to derive the expressions for the magnetoelectric response directly. However, in metals it is much easier to derive expressions instead for the inverse of the mag- netoelectric response, i.e., for the generation of electric currents by time-dependent magnetic fields [16]. The in- verse current-induced DMI (ICIDMI) consists in charge pumping by time-dependent gradients of magnetization. Due to the analogies between orbital magnetism and the BerryphasetheoryofDMIonemayexpectthatinmetals it is convenient to obtain expressions for ICIDMI, which canthen be used todescribe the CIDMI byexploitingthereciprocity between CIDMI and ICIDMI. We will show in this paper that this is indeed the case. In noncentrosymmetric ferromagnets spin-orbit inter- action (SOI) generates torques on the magnetization – the so-called spin-orbit torques (SOTs) – when an elec- tric current is applied [17]. The Berry phase theory of DMI [6, 10, 11] establishes a relation to SOTs. The for- mal analogies between orbital magnetism and DMI have been shown to be a very useful guiding principle in the development of the theory of SOTs driven by heat cur- rents [18]. In particular, it is fruitful to consider the DMI coefficients as a spiralization, which is formally analo- gous to magnetization. In the theory of thermoelectric effects in magnetic systems the curl of magnetization de- scribes a bound current, which cannot be measured in transport experiments and needs to be subtracted from the Kubo linear response in order to obtain the mea- surable current [19–21]. Similarly, in the theory of the thermal spin-orbit torque spatial gradients of the DMI spiralization, which result from the temperature gradi- ent together with the temperature dependence of DMI, need to be subtracted in order to obtain the measurable torqueandto satisfyaMott-likerelation[10, 18]. In non- collinear magnets the question arises whether gradients of the spiralization that are due to the magnetic texture correspond to torques like those from thermal gradients. We will show that indeed the spatial gradients of CIDMI need to be included into the theory of current-induced torques (CITs) in noncollinear magnets in order to sat- isfy the Onsager reciprocity relations [22]. When the system is driven out of equilibrium by mag- netization dynamics rather than electric current one may expect DMI to be modified as well. The inverse effect of this dynamical DMI (DDMI) consists in the generation of torques by time-dependent magnetization gradients. In noncollinear magnets the DDMI spiralization varies in space. We will show that the resulting gradient cor-2 responds to a torque that needs to be considered in the theory of Gilbert damping and gyromagnetism in non- collinear magnets. This paper is structured as follows. In section IIA we give an overview of CIT in noncollinear magnets and introduce the notation. In section IIB we describe the formalism used to calculate the response of electric cur- rent to time-dependent magnetization gradients. In sec- tion IIC we show that current-induced DMI (CIDMI) and electric current driven by time-dependent magneti- zation gradients are reciprocal effects. This allows us to obtain an expression for CIDMI based on the formal- ism of section IIB. In section IID we discuss that time- dependent magnetization gradients generate additionally torques on the magnetization and show that the inverse effect consists in the modification of DMI by magnetiza- tion dynamics, which we calldynamical DMI (DDMI). In section IIE we demonstrate that magnetization dynam- ics induces orbital magnetism, which we call dynamical orbitalmagnetism (DOM) and showthat DOM is related to CIDMI. In section IIF we explain how the spatial gra- dients of CIDMI and DOM contribute to the direct and to the inverse CIT, respectively. In section IIG we dis- cuss how the spatial gradients of DDMI contribute to the torque-torque correlation. In section IIH we complete the formalism used to calculate the CIT in noncollinear magnets by adding the chiral contribution of the torque- velocity correlation. In section III we finalize the theory of the inverse CIT by adding the chiral contribution of the velocity-torque correlation. In section IIJ we fin- ish the computational formalism of gyromagnetism and damping by adding the chiral contribution of the torque- torque correlation and the response of the torque to the time-dependent magnetization gradients. In section III we discuss the symmetry properties of the response to time-dependent magnetizationgradients. In section IVA we present the results for the chiral contributions to the directand the inverseCITin the Rashbamodel andshow that both the perturbation by the time-dependent mag- netization gradient and the spatial gradients of CIDMI and DOM need to be included to ensure that they are reciprocal. In section IVB we present the results for the chiralcontributiontothe torque-torquecorrelationinthe Rashba model and show that both the perturbation by the time-dependent magnetization gradient and the spa- tialgradientsofDDMI need tobe included toensurethat it satisfies the Onsager symmetry relations. This paper ends with a summary in section V.II. FORMALISM A. Direct and inverse current-induced torques in noncollinear magnets Even in collinearmagnets the application of an electric fieldEgenerates a torque TCIT1on the magnetization when inversion symmetry is broken [17, 23]: TCIT1 i=/summationdisplay jtij(ˆM)Ej, (1) wheretij(ˆM) is the torkance tensor, which depends on the magnetization direction ˆM. This torque is called spin-orbit torque (SOT), but we denote it here CIT1, because it is one contribution to the current-induced torques (CITs) in noncollinear magnets. Inversely, mag- netization dynamics pumps a charge current JICIT1ac- cording to [24] JICIT1 i=/summationdisplay jtji(−ˆM)ˆej·/bracketleftBigg ˆM×∂ˆM ∂t/bracketrightBigg ,(2) whereˆejis a unit vector that points into the j-th spa- tial direction. Generally, JICIT1can be explained by the inverse spin-orbit torque [24] or the magnonic charge pumping [25]. We denote it here by ICIT1, because it is one contribution to the inverse CIT in noncollinear magnets. In the special case of magnetic bilayers one im- portantmechanism responsiblefor JICIT1arisesfrom the combination of spin pumping and the inverse spin Hall effect [26, 27]. In noncollinear magnets there is a second contribution to the CIT, which is proportional to the spatial deriva- tives of magnetization [28]: TCIT2 i=/summationdisplay jklχCIT2 ijklEjˆek·/bracketleftBigg ˆM×∂ˆM ∂rl/bracketrightBigg .(3) The description of noncollinearity by the derivatives ∂ˆM/∂rlisonlyapplicablewhenthe magnetizationdirec- tion changes slowly in space like in magnetic skyrmions with large radius and in wide magnetic domain walls. In order to treat noncollinear magnets such as Mn 3Sn [29], where the magnetization direction varies strongly on the scale of one unit cell, Eq. (3) needs to be modified, which is beyond the scope of the present paper. The adia- batic and the non-adiabatic [30] spin transfer torques are two important contributions to χCIT2 ijkl, but the in- terplay between broken inversion symmetry, SOI, and noncollinearity can lead to a large number of additional mechanisms [22, 31]. Similarly, the current pumped by magnetization dynamics contains a contribution that is proportional to the spatial derivatives of magnetiza-3 tion [22, 32, 33]: JICIT2 i=/summationdisplay jklχICIT2 ijklˆej·/bracketleftBigg ˆM×∂ˆM ∂t/bracketrightBigg ˆek·/bracketleftBigg ˆM×∂ˆM ∂rl/bracketrightBigg . (4) TCIT2 iandJICIT2 ican be considered as chiral contribu- tionsto the CIT and to the ICIT, respectively, because they distinguish between left- and right-handed spin spi- rals. Due to the reciprocity between direct and inverse CIT [22, 24] the coefficients χICIT2 ijklandχCIT2 jiklare related according to χICIT2 ijkl(ˆM) =χCIT2 jikl(−ˆM). (5) B. Response of electric current to time-dependent magnetization gradients In order to compute JICIT2based on the Kubo lin- ear response formalism it is necessary to split it into two contributions, JICIT2aandJICIT2b. While JICIT2a is obtained as linear response to the perturbation by atime-dependent magnetization gradient in a collinear ferromagnet, JICIT2bis obtained as linear response to the perturbation by magnetization dynamics in a non- collinear ferromagnet. Therefore, as will become clear below,JICIT2acan be expressed by a correlation func- tion of two operators, because it describes the response of the current to a time-dependent magnetization gradi- ent: A time-dependent magnetization gradient is a single perturbation, which is described by a single perturbing operator. In contrast, JICIT2binvolves the correlation of three operators, because it describes the response of the current to magnetization dynamics in the presence of perturbation by noncollinearity. These are twoperturba- tions: One perturbation by the magnetization dynamics, andasecondperturbationtodescribethenoncollinearity. In the Kubo formalism the expressions for the response onethe onehand toatime-dependent magnetizationgra- dient, which is described by a single perturbing operator, and the response on the other hand to a time-dependent magnetization in the presence of a magnetization gradi- ent, which is described by two perturbing operators, are different. Therefore, we split JICIT2into these two con- tributions, which we call JICIT2aandJICIT2b. In the remainder of this section we discuss the calculation of the contribution JICIT2a. The contribution JICIT2bis discussed in section III below. JICIT2ais determined by the second derivative of mag- netization with respect to time and space variables and can be written as JICIT2a i=/summationdisplay jkχICIT2a ijk∂2ˆMj ∂rk∂t. (6) Anonzerosecondderivative∂2ˆMj ∂rk∂tis what we referto asa time-dependent magnetization gradient . Wewillshowbe-low that in special cases∂2ˆMj ∂rk∂tcan be expressed in terms of the products∂ˆMl ∂rk∂ˆMl ∂t, which will allow us to rewrite JICIT2a iin the form of Eq. (4) in the cases relevant for the chiral ICIT. However, as will become clear below, Eq. (6) is the most general expression for the response to time-dependent magnetization gradients, and it can- not generally be rewritten in the form of Eq. (4): This is only possible when it describes a contribution to the chiral ICIT. JICIT2aoccurs in two different situations, which need to be distinguished. In one case the magnetization gra- dient varies in time like sin( ωt) everywhere in space. An example is ˆM(r,t) = ηsin(q·r)sin(ωt) 0 1 , (7) whereηis the amplitude and the derivatives at t= 0 and r= 0 are ∂ˆM(r,t) ∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0=∂ˆM(r,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0= 0 (8) and ∂2ˆM(r,t) ∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0= ηqiω 0 0 . (9) In the other case the magnetic texture varies like a propagating wave, i.e., proportional to sin( q·r−ωt). An example is given by ˆM(r,t) = ηsin(q·r−ωt) 0 1−η2 2sin2(q·r−ωt) ,(10) where the derivatives at t= 0 and r= 0 are ∂ˆM(r,t) ∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0= ηqi 0 0 , (11) ∂ˆM(r,t) ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0= −ηω 0 0 (12) and ∂2ˆM(r,t) ∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle r=t=0= 0 0 η2qiω . (13) In the latter example, Eq. (10), the second derivative, Eq. (13), is along the magnetization ˆM(r= 0,t= 0), while in the former example, Eq. (7), the second deriva- tive, Eq. (9), is perpendicular to the magnetization when r= 0 andt= 0.4 We assume that the Hamiltonian is given by H(r,t) =−/planckover2pi12 2me∆+V(r)+µBˆM(r,t)·σΩxc(r)+ +1 2ec2µBσ·[∇V(r)×v], (14) where the first term describes the kinetic energy, the sec- ond term is a scalar potential, Ωxc(r) in the third term is the exchange field, and the last term describes the spin- orbit interaction. Around t= 0 and r= 0 we can de- compose the Hamiltonian as H(r,t) =H0+δH(r,t), whereH0is obtained from H(r,t) by replacing ˆM(r,t) byˆM(r= 0,t= 0) and δH(r,t) =∂H0 ∂ˆMxηsin(q·r)sin(ωt) =µBΩxc(r)σxηsin(q·r)sin(ωt)(15) in the case of the first example, Eq. (7). In the case of the second example, Eq. (10), δH(r,t)≃∂H ∂ˆMxηsin(q·r−ωt) +∂H ∂ˆMzη2sin(q·r)sin(ωt),(16) whereforsmall randtonlythesecondterm ontheright- hand side contributes to∂2H(r,t) ∂rk∂t. We consider here only the time-dependence of the exchange field direction and ignore the time-dependence of the exchange field mag- nitude Ωxc(r) that is induced by the time-dependence of the exchange field direction. While the variation of the exchange field magnitude drives currents and torques as well, as shown in Ref. [34], the variation of the ex- change field magnitude is a small response and therefore these secondary responses are suppressed in magnitude when compared to the direct primary responses of the current and torque to the variation in the exchange field direction. We will use the perturbations Eq. (15) and Eq. (16) in order to compute the response of current and torque within the Kubo response formalism. An alterna- tive approach for the calculation of the response to time- dependent fields is variational linear-response, which has been applied to the spin susceptibility by Savrasov [35]. The perturbation by the time-dependent gradient can be written as δH=∂H ∂ˆM·∂2ˆM ∂ri∂tsin(qiri) qisin(ωt) ω,(17) which turns into Eq. (15) when Eq. (9) is inserted. When Eq. (13) is inserted it turns into the second term in Eq. (16). In Appendix A we derive the linear response to pertur- bations of the type of Eq. (17) and show that the corre- sponding coefficient χICIT2a ijkin Eq. (6) can be expressedas χICIT2a ijk=ie 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBig viRvkRROjR+viRRvkROjR+ −viRROjRvkR−viRvkROjAA +viROjAvkAA+viROjAAvkA −viRvkRROjA−viRRvkROjA +viRROjAvkA+viAvkAOjAA −viAOjAvkAA−viAOjAAvkA/bracketrightBig ,(18) whereR=GR k(E) andA=GA k(E) are shorthands for the retarded and advanced Green’s functions, respectively, andOj=∂H/∂ˆMj.e >0 is the positive elementary charge. In the case of the perturbation of the type Eq. (7) the second derivative∂2ˆM ∂ri∂tis perpendicular to M. In this case it is convenient to rewrite Eq. (6) as JICIT2a i=/summationdisplay jkχICIDMI ijkˆej·/bracketleftBigg ˆM×∂2ˆM ∂rk∂t/bracketrightBigg ,(19) where the coefficients χICIDMI ijkare given by χICIDMI ijk=ie 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBig viRvkRRTjR+viRRvkRTjR+ −viRRTjRvkR−viRvkRTjAA +viRTjAvkAA+viRTjAAvkA −viRvkRRTjA−viRRvkRTjA +viRRTjAvkA+viAvkATjAA −viATjAvkAA−viATjAAvkA/bracketrightBig ,(20) and T=ˆM×∂H ∂ˆM(21) is the torque operator. In Sec. IIC we will explain that χICIDMI ijkdescribes the inverse of current-induced DMI (ICIDMI). In the case of the perturbation of the type of Eq. (10) the second derivative∂2ˆMj ∂rk∂tmay be rewritten as product of the first derivatives∂ˆMl ∂tand∂ˆMl ∂rk. This may be seen5 as follows: ∂H ∂ˆM·∂2ˆM ∂ri∂t=∂2H ∂t∂ri= =∂ ∂t/bracketleftBigg/parenleftbigg ˆM×∂H ∂ˆM/parenrightbigg ·/parenleftBigg ˆM×∂ˆM ∂ri/parenrightBigg/bracketrightBigg = =/bracketleftBigg/parenleftBigg ∂ˆM ∂t×∂H ∂ˆM/parenrightBigg ·/parenleftBigg ˆM×∂ˆM ∂ri/parenrightBigg/bracketrightBigg = =/bracketleftBigg/parenleftBigg/parenleftBigg ˆM×∂ˆM ∂t/parenrightBigg ׈M/parenrightBigg ×∂H ∂ˆM/bracketrightBigg ·/bracketleftBigg ˆM×∂ˆM ∂ri/bracketrightBigg = =−/bracketleftBigg ˆM×∂ˆM ∂t/bracketrightBigg ·/bracketleftBigg ˆM×∂ˆM ∂ri/bracketrightBigg/bracketleftbigg ˆM·∂H ∂ˆM/bracketrightbigg = =−∂ˆM ∂t·∂ˆM ∂ri/bracketleftbigg ˆM·∂H ∂ˆM/bracketrightbigg . (22) This expression is indeed satisfied by Eq. (11), Eq. (12) and Eq. (13): ∂ˆM ∂ri·∂ˆM ∂t=−∂2ˆM ∂ri∂t·ˆM (23) atr= 0,t= 0. Consequently, Eq. (6) can be rewritten as JICIT2a i=/summationdisplay jkχICIT2a ijk∂2ˆMj ∂rk∂t= =−/summationdisplay jklχICIT2a ijk∂ˆMl ∂rk∂ˆMl ∂t[1−δjl] =/summationdisplay jklχICIT2a ijklˆej·/bracketleftBigg ˆM×∂ˆM ∂t/bracketrightBigg ˆek·/bracketleftBigg ˆM×∂ˆM ∂rl/bracketrightBigg , (24) where χICIT2a ijkl=−/summationdisplay mχICIT2a iml[1−δjm]δjk.(25) Thus, Eq. (24) and Eq. (25) can be used to express JICIT2a iin the form of Eq. (4). C. Direct and inverse CIDMI Eq. (20) describes the response of the electric current to time-dependent magnetization gradients of the type Eq. (15). The reciprocal process consists in the current- induced modification of DMI. This can be shown by ex- pressing the DMI coefficients as [10] Dij=1 V/summationdisplay nf(Ekn)/integraldisplay d3r(ψkn(r))∗Dijψkn(r) =1 V/summationdisplay nf(Ekn)/integraldisplay d3r(ψkn(r))∗Ti(r)rjψkn(r), (26)where we defined the DMI-operator Dij=Tirj. Using the Kubo formalism the current-induced modification of DMI may be written as DCIDMI ij=/summationdisplay kχCIDMI kijEk (27) with χCIDMI kij=1 Vlim ω→0/bracketleftBige /planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig ,(28) where ∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) =−i∞/integraldisplay 0dteiωt∝an}bracketle{t[Dij(t),vk(0)]−∝an}bracketri}ht(29) is the Fourier transform of a retarded function and Vis the volume of the unit cell. Since the position operator rin the DMI operator Dij=Tirjis not compatible with Bloch periodic bound- ary conditions, we do not use Eq. (28) for numerical calculations of CIDMI. However, it is convenient to use Eq. (28) in order to demonstrate the reciprocity between direct and inverse CIDMI. InverseCIDMI (ICIDMI) describes the electric current that responds to the perturbation by a time-dependent magnetization gradient according to JICIDMI k=/summationdisplay ijχICIDMI kijˆei·/bracketleftBigg ˆM×∂2ˆM ∂t∂rj/bracketrightBigg .(30) The perturbation by a time-dependent magnetization gradient may be written as δH=−/summationdisplay jm·∂2ˆM ∂t∂rjrjΩxc(r)sin(ωt) ω= =/summationdisplay jT·/bracketleftBigg ˆM×∂2ˆM ∂t∂rj/bracketrightBigg rjsin(ωt) ω =/summationdisplay ijDijˆei·/bracketleftBigg ˆM×∂2ˆM ∂t∂rj/bracketrightBigg sin(ωt) ω.(31) Consequently, the coefficient χICIDMI kijis given by χICIDMI kij=1 Vlim ω→0/bracketleftBige /planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig .(32) Using ∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,ˆM) =−∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,−ˆM) (33) we find that CIDMI and ICIDMI are related through the equations χCIDMI kij(ˆM) =−χICIDMI kij(−ˆM). (34) In order to calculate CIDMI we use Eq. (20) for ICIDMI and then use Eq. (34) to obtain CIDMI.6 The perturbation Eq. (16) describes a different kind of time-dependent magnetization gradient, for which the reciprocaleffect consists in the modification of the expec- tation value ∝an}bracketle{tσ·ˆMrj∝an}bracketri}ht. However, while the modification of∝an}bracketle{tTirj∝an}bracketri}htby an applied current can be measured [8, 9] from the change of the DMI constant Dij, the quantity ∝an}bracketle{tσ·ˆMrj∝an}bracketri}hthas not been considered so far in ferromagnets. In noncollinear magnets the quantity ∝an}bracketle{tσrj∝an}bracketri}htcan be used todefinespintoroidization[36]. Therefore,whiletheper- turbation of the type of Eq. (15) is related to CIDMI and ICIDMI, which are both accessible experimentally [8, 9], in the case of the perturbation of the type of Eq. (16) we expect that only the effect of driving current by the time-dependent magnetization gradient is easily accessi- ble experimentally, while its inverse effect is difficult to measure. D. Direct and inverse dynamical DMI Not only applied electric currents modify DMI, but also magnetization dynamics, which we call dynamical DMI (DDMI). DDMI can be expressed as DDDMI ij=/summationdisplay kχDDMI kijˆek·/bracketleftBigg ˆM×∂ˆM ∂t/bracketrightBigg .(35) In Sec. IIG we will show that the spatial gradient of DDMI contributes to damping and gyromagnetism in noncollinear magnets. The perturbation used to describe magnetization dynamics is given by [24] δH=sin(ωt) ω/parenleftBigg ˆM×∂ˆM ∂t/parenrightBigg ·T.(36) Consequently, the coefficients χDDMI kijmay be written as χDDMI kij=−1 Vlim ω→0/bracketleftbigg1 /planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg .(37) Since the position operator in Dijis not compatible with Bloch periodic boundary conditions, we do not use Eq. (37) for numerical calculations of DDMI, but instead we obtain it from its inverse effect, which consists in the generation of torques on the magnetization due to time- dependent magnetization gradients. These torques can be written as TIDDMI k=/summationdisplay ijχIDDMI kijˆei·/bracketleftBigg ˆM×∂2ˆM ∂t∂rj/bracketrightBigg ,(38) where the coefficients χIDDMI kijare χIDDMI kij=1 Vlim ω→0/bracketleftbigg1 /planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg ,(39)becausethe perturbationby the time-dependent gradient can be expressed in terms of Dijaccording to Eq. (31) and because the torque on the magnetizationis described by−T[23]. Consequently,DDMIandIDDMIarerelated by χDDMI kij(ˆM) =−χIDDMI kij(−ˆM). (40) For numerical calculations of IDDMI we use χIDDMI ijk=i 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBig TiRvkRRTjR+TiRRvkRTjR+ −TiRRTjRvkR−TiRvkRTjAA +TiRTjAvkAA+TiRTjAAvkA −TiRvkRRTjA−TiRRvkRTjA +TiRRTjAvkA+TiAvkATjAA −TiATjAvkAA−TiATjAAvkA/bracketrightBig ,(41) whichisderivedinAppendix A. InordertoobtainDDMI wecalculateIDDMIfromEq.(41)andusethereciprocity relation Eq. (40). Eq.(38)is validfortime-dependent magnetizationgra- dients that lead to perturbations of the type of Eq. (15). Perturbations of the second type, Eq. (16), will induce torques on the magnetization as well. However, the in- verse effect is difficult to measure in that case, because it corresponds to the modification of the expectation value ∝an}bracketle{tσ·ˆMrj∝an}bracketri}htby magnetization dynamics. Therefore, while in the case of Eq. (15) both direct and inverse response are expected to be measurable and correspond to ID- DMI and DDMI, respectively, we expect that in the case of Eq. (16) only the direct effect, i.e., the response of the torque to the perturbation, is easy to observe. E. Dynamical orbital magnetism (DOM) Magnetization dynamics does not only induce DMI, but also orbital magnetism, which we call dynamical or- bital magnetism (DOM). It can be written as MDOM ij=/summationdisplay kχDOM kijˆek·/bracketleftBigg ˆM×∂ˆM ∂t/bracketrightBigg ,(42) where we introduced the notation MDOM ij=e V∝an}bracketle{tvirj∝an}bracketri}htDOM, (43) which defines a generalized orbital magnetization, such that MDOM i=1 2/summationdisplay jkǫijkMDOM jk (44)7 corresponds to the usual definition of orbital magnetiza- tion. The coefficients χDOM kijare given by χDOM kij=−1 Vlim ω→0/bracketleftBige /planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvirj;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig ,(45) because the perturbation by magnetization dynamics is described by Eq. (36). We will discuss in Sec. IIF that the spatial gradient of DOM contributes to the inverse CIT. Additionally, we will show below that DOM and CIDMI are related to each other. In order to obtain an expression for DOM it is conve- nient to consider the inverse effect, i.e., the generation of atorquebythe applicationofa time-dependent magnetic fieldB(t) that actsonly onthe orbitaldegreesoffreedom of the electrons and not on their spins. This torque can be written as TIDOM k=1 2/summationdisplay ijlχIDOM kijǫijl∂Bl ∂t, (46) where χIDOM kij=−1 Vlim ω→0/bracketleftBige /planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;virj∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig ,(47) because the perturbation by the time-dependent mag- netic field is given by δH=−e 2/summationdisplay ijkǫijkvirj∂Bk ∂tsin(ωt) ω.(48) Therefore, thecoefficientsofDOMandIDOM arerelated by χDOM kij(ˆM) =−χIDOM kij(−ˆM). (49) In Appendix A we show that the coefficient χIDOM ijkcan be expressed as χIDOM ijk=−ie 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBig TiRvkRRvjR+TiRRvkRvjR+ −TiRRvjRvkR−TiRvkRvjAA +TiRvjAvkAA+TiRvjAAvkA −TiRvkRRvjA−TiRRvkRvjA +TiRRvjAvkA+TiAvkAvjAA −TiAvjAvkAA−TiAvjAAvkA/bracketrightBig .(50) Eq. (50) and Eq. (20) differ only in the positions of the two velocity operators and the torque operator be- tween the Green functions. As a consequence, IDOM are ICIDMI are related. In Table I and Table II we list the relations between IDOM and ICIDMI for the Rashba model Eq. (83). We will explain in Sec. III that IDOM andICIDMI arezeroin the Rashbamodel when themag- netization is along the zdirection. Therefore, we discussin Table I the case where the magnetization lies in the xz plane, and in Table II we discuss the case where the mag- netization lies in the yzplane. According to Table I and Table II the relation between IDOM and ICIDMI is of the formχIDOM ijk=±χICIDMI jik. This is expected, because the indexiinχIDOM ijkis connected to the torque operator, while the index jinχICIDMI ijkis connected to the torque operator. TABLEI:Relations betweentheinverseofthemagnetization - dynamics induced orbital magnetism (IDOM) and inverse current-inducedDMI (ICIDMI)in the 2d Rashbamodel when ˆMlies in the zxplane. The components of χIDOM ijk(Eq. (50)) andχICIDMI ijk(Eq. (20)) are denoted by the three indices ( ijk). ICIDMI IDOM (211) (121) (121) (211) -(221) (221) (112) (112) -(212) (122) -(122) (212) (222) (222) (231) (321) (132) (312) -(232) (322) TABLE II: Relations between IDOM and ICIDMI in the 2d Rashba model when ˆMlies in the yzplane. ICIDMI IDOM (111) (111) -(211) (121) -(121) (211) (221) (221) -(112) (112) (212) (122) (122) (212) -(131) (311) (231) (321) (132) (312) F. Contributions from CIDMI and DOM to direct and inverse CIT In electronic transport theory the continuity equation determines the current only up to a curl field [37]. The curl of magnetization corresponds to a bound current that cannot be measured in electron transport experi- ments such that J=JKubo−∇×M (51) hastobeusedtoextractthetransportcurrent Jfromthe currentJKuboobtained from the Kubo linear response.8 The subtraction of ∇×Mhas been shown to be impor- tant when calculating the thermoelectric response [37] and the anomalous Nernst effect [20]. Similarly, in the theory of the thermal spin-orbit torque [10, 18] the gra- dients of the DMI spiralization have to be subtracted in order to obtain the measurable torque: Ti=TKubo i−/summationdisplay j∂ ∂rjDij, (52) where the spatial derivative of the spiralization arises from its temperature dependence and the temperature gradient. Since CIDMI and DOM depend on the magnetization direction, they vary spatially in noncollinear magnets. Similar to Eq. (52) the spatial derivatives of the current- induced spiralization need to be included into the theory of CIT. Additionally, the gradients of DOM correspond tocurrentsthatneedtobeconsideredinthetheoryofthe inverse CIT, similar to Eq. (51). In section IV we explic- itly show that Onsager reciprocity is violated if spatial gradients of DOM and CIDMI are not subtracted from the Kubo response expressions. By trial-and-error we find that the following subtractions are necessary to ob- tain response currents and torques that satisfy this fun- damental symmetry: JICIT i=JKubo i−1 2/summationdisplay j∂ˆM ∂rj·∂MDOM ij ∂ˆM(53) and TCIT i=TKubo i−1 2/summationdisplay j∂ˆM ∂rj·∂DCIDMI ij ∂ˆM,(54) whereJICIT iis the current driven by magnetization dy- namics, and TCIT iis the current-induced torque. Interestingly, we find that also the diagonal elements MDOM iiare nonzero. This shows that the generalized def- inition Eq. (43) is necessary, because the diagonal ele- mentsMDOM iido not contribute in the usual definition ofMiaccording to Eq. (44). These differences in the symmetry properties between equilibrium and nonequi- librium orbital magnetism can be traced back to sym- metry breaking by the perturbations. Also in the case of the spiralization tensor Dijthe nonequilibrium cor- rectionδDijhas different symmetry properties than the equilibrium part (see Sec. III). The contribution of DOM to χICIT2 ijklcan be written as χICIT2c ijkl=−1 2ˆek·/bracketleftBigg ˆM×∂χDOM jil ∂ˆM/bracketrightBigg (55) and the contribution of CIDMI to χCIT2 ijklis given by χCIT2b ijkl=−1 2ˆek·/bracketleftBigg ˆM×∂χCIDMI jil ∂ˆM/bracketrightBigg .(56)G. Contributions from DDMI to gyromagnetism and damping The response to magnetization dynamics that is de- scribed by the torque-torque correlation function con- sists of torques that are related to damping and gyro- magnetism [24]. The chiral contribution to these torques can be written as TTT2 i=/summationdisplay jklχTT2 ijklˆej·/bracketleftBigg ˆM×∂ˆM ∂t/bracketrightBigg ˆek·/bracketleftBigg ˆM×∂ˆM ∂rl/bracketrightBigg , (57) where the coefficients χTT2 ijklsatisfy the Onsager relations χTT2 ijkl(ˆM) =χTT2 jikl(−ˆM). (58) SinceDDMIdependsonthemagnetizationdirection,it varies spatially in noncollinear magnets and the resulting gradients of DDMI contribute to the damping and to the gyromagnetic ratio: TTT i=TKubo i−1 2/summationdisplay j∂ˆM ∂rj·∂DDDMI ij ∂ˆM.(59) The resulting contribution of the spatial derivatives of DDMI to the coefficient χTT2 ijklis χTT2c ijkl=−1 2ˆek·/bracketleftBigg ˆM×∂χDDMI jil(ˆM) ∂ˆM/bracketrightBigg .(60) H. Current-induced torque (CIT) in noncollinear magnets The chiral contribution to CIT consists of the spatial gradient of CIDMI, χCIT2b ijklin Eq. (56), and the Kubo linear response of the torque to the applied electric field in a noncollinear magnet, χCIT2a ijkl: χCIT2 ijkl=χCIT2a ijkl+χCIT2b ijkl. (61) In orderto determine χCIT2a ijkl, we assume that the magne- tization direction ˆM(r) oscillates spatially as described by ˆM(r) = ηsin(q·r) 0 1 1/radicalBig 1+η2sin2(q·r),(62) wherewewilltakethelimit q→0attheendofthecalcu- lation. Since the spatial derivative of the magnetization direction is ∂ˆM(r) ∂ri= ηqicos(q·r) 0 0 +O(η3),(63)9 the chiralcontributiontothe CIToscillatesspatiallypro- portional to cos( q·r). In order to extract this spatially oscillating contribution we multiply with cos( q·r) and integrate over the unit cell. The resulting expression for χCIT2a ijklis χCIT2a ijkl=−2e Vηlim q→0lim ω→0/bracketleftBigg 1 ql/integraldisplay cos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) /planckover2pi1ωd3rd3r′/bracketrightBigg , (64) whereVis the volume of the unit cell, and the retarded torque-velocity correlation function ∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) needs to be evaluated in the presence of the perturbation δH=Tkηsin(q·r) (65) due to the noncollinearity (the index kin Eq. (65) needs to match the index kinχCIT2a ijkl). In Appendix B we show that χCIT2a ijklcan be written as χCIT2a ijkl=−2e /planckover2pi1Im/bracketleftBig W(surf) ijkl+W(sea) ijkl/bracketrightBig ,(66) where W(surf) ijkl=1 4π/planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf′(E)Tr/bracketleftBigg TiGR k(E)vlGR k(E)vjGA k(E)TkGA k(E) +TiGR k(E)vjGA k(E)vlGA k(E)TkGA k(E) −TiGR k(E)vjGA k(E)TkGA k(E)vlGA k(E) +/planckover2pi1 meδjlTiGR k(E)GA k(E)TkGA k(E)/bracketrightBigg(67) is a Fermi surface term ( f′(E) =df(E)/dE) and W(sea) ijkl=1 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)/bracketleftBigg −Tr[TiRvlRRvjRTkR]−Tr[TiRvlRTkRRvjR] −Tr[TiRRvlRvjRTkR]−Tr[TiRRvjRvlRTkR] +Tr[TiRRvjRTkRvlR]+Tr[TiRRTkRvjRvlR] +Tr[TiRRTkRvlRvjR]−Tr[TiRRvlRTkRvjR] −Tr[TiRvlRRTkRvjR]+Tr[TiRTkRRvjRvlR] +Tr[TiRTkRRvlRvjR]+Tr[TiRTkRvlRRvjR] −/planckover2pi1 meδjlTr[TiRRRTkR]−/planckover2pi1 meδjlTr[TiAAATkA] −/planckover2pi1 meδjlTr[TiAATkAA]/bracketrightBigg(68) is a Fermi sea term.I. Inverse CIT in noncollinear magnets The chiral contribution JICIT2(see Eq. (4)) to the charge pumping is described by the coefficients χICIT2 ijkl=χICIT2a ijkl+χICIT2b ijkl+χICIT2c ijkl,(69) whereχICIT2a ijkldescribes the response to the time- dependentmagnetizationgradient(seeEq.(18),Eq.(25), and Eq. (24)) and χICIT2c ijklresults from the spatial gra- dient of DOM (see Eq. (55)). χICIT2b ijkldescribes the re- sponseto the perturbation bymagnetizationdynamics in a noncollinear magnet. In order to derive an expression forχICIT2b ijklwe assume that the magnetization oscillates spatially as described by Eq. (62). Since the correspond- ing response oscillates spatially proportional to cos( q·r), we multiply by cos( q·r) and integrate over the unit cell in order to extract χICIT2b ijklfrom the retarded velocity- torque correlation function ∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω), which is evaluated in the presence of the perturbation Eq. (65). We obtain χICIT2b ijkl=2e Vηlim q→0lim ω→0/bracketleftBigg 1 ql/integraldisplay cos(qlrl)Im∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) /planckover2pi1ωd3rd3r′/bracketrightBigg , (70) which can be written as (see Appendix B) χICIT2b ijkl=2e /planckover2pi1Im/bracketleftBig V(surf) ijkl+V(sea) ijkl/bracketrightBig ,(71) where V(surf) ijkl=1 4π/planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf′(E)Tr/bracketleftBig viGR k(E)vlGR k(E)TjGA k(E)TkGA k(E) +viGR k(E)TjGA k(E)vlGA k(E)TkGA k(E) −viGR k(E)TjGA k(E)TkGA k(E)vlGA k(E)/bracketrightBig(72) is the Fermi surface term and V(sea) ijkl=1 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBig −Tr[viRvlRRTjRTkR]−Tr[viRvlRTkRRTjR] −Tr[viRRvlRTjRTkR]−Tr[viRRTjRvlRTkR] +Tr[viRRTjRTkRvlR]+Tr[viRRTkRTjRvlR] +Tr[viRRTkRvlRTjR]−Tr[viRRvlRTkRTjR] −Tr[viRvlRRTkRTjR]+Tr[viRTkRRTjRvlR] +Tr[viRTkRRvlRTjR]+Tr[viRTkRvlRRTjR]/bracketrightBig(73) is the Fermi sea term. In Eq. (70) we use the Kubo formula to describe the response to magnetization dynamics combined with per- turbation theory to include the effect of noncollinearity.10 Thereby, the time-dependent perturbation and the per- turbation by the magnetization gradient are separated and perturbations of the form of Eq. (15) or Eq. (16) are not automatically included. For example the flat cy- cloidal spin spiral ˆM(x,t) = sin(qx−ωt) 0 cos(qx−ωt) (74) moving inxdirection with speed ω/qand the helical spin spiral ˆM(y,t) = sin(qy−ωt) 0 cos(qy−ωt) (75) movinginydirectionwith speed ω/qbehavelikeEq.(10) whentandraresmall. Thus, these movingdomainwalls correspond to the perturbation of the type of Eq. (10) and the resulting contribution JICIT2afrom the time- dependent magnetization gradient is not described by Eq. (70) and needs to be added, which we do by adding χICIT2a ijklin Eq. (69). J. Damping and gyromagnetism in noncollinear magnets The chiral contribution Eq. (57) to the torque-torque correlation function is expressed in terms of the coeffi- cient χTT ijkl=χTT2a ijkl+χTT2b ijkl+χTT2c ijkl, (76) whereχTT2c ijklresults from the spatial gradient of DDMI (see Eq. (60)), χTT2a ijkldescribes the response to a time- dependent magnetization gradient in a collinear magnet, andχTT2b ijkldescribes the response to magnetization dy- namics in a noncollinear magnet. In order to derive an expression for χTT2b ijklwe as- sume that the magnetization oscillates spatially accord- ing to Eq. (62). We multiply the retarded torque-torque correlation function ∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) with cos(qlrl) and integrate over the unit cell in order to extract the part of the response that varies spatially proportional to cos(qlrl). We obtain: χTT2b ijkl=2 Vηlim ql→0lim ω→0/bracketleftBigg 1 ql/integraldisplay cos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) /planckover2pi1ωd3rd3r′/bracketrightBigg . (77) In Appendix B we discuss how to evaluate Eq. (77) in first order perturbation theory with respect to the per- turbation Eq.(65) and showthat χTT2b ijklcan be expressedas χTT2b ijkl=2 /planckover2pi1Im/bracketleftBig X(surf) ijkl+X(sea) ijkl/bracketrightBig ,(78) where X(surf) ijkl=1 4π/planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf′(E)Tr/bracketleftBigg TiGR k(E)vlGR k(E)TjGA k(E)TkGA k(E) +TiGR k(E)TjGA k(E)vlGA k(E)TkGA k(E) −TiGR k(E)TjGA k(E)TkGA k(E)vlGA k(E)/bracketrightBigg(79) is a Fermi surface term and X(sea) ijkl=1 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBigg −(TiRvlRRTjRTkR)−(TiRvlRTkRRTjR) −(TiRRvlRTjRTkR)−(TiRRTjRvlRTkR) +(TiRRTjRTkRvlR)+(TiRRTkRTjRvlR) +(TiRRTkRvlRTjR)−(TiRRvlRTkRTjR) −(TiRvlRRTkRTjR)+(TiRTkRRTjRvlR) +(TiRTkRRvlRTjR)+(TiRTkRvlRRTjR)/bracketrightBigg (80) is a Fermi sea term. The contribution χTT2a ijklfrom the time-dependent gra- dients is given by χTT2a ijkl=−/summationdisplay mχTT2a iml[1−δjm]δjk,(81) where χTT2a iml=i 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBig TiRvlRROmR+TiRRvlROmR+ −TiRROmRvlR−TiRvlROmAA +TiROmAvlAA+TiROmAAvlA −TiRvlRROmA−TiRRvlROmA +TiRROmAvlA+TiAvlAOmAA −TiAOmAvlAA−TiAOmAAvlA/bracketrightBig ,(82) withOm=∂H/∂ˆMm(see Appendix A). III. SYMMETRY PROPERTIES In this section we discuss the symmetry properties of CIDMI, DDMI and DOM in the case of the magnetic Rashba model Hk(r) =/planckover2pi12 2mek2+α(k׈ez)·σ+∆V 2σ·ˆM(r).(83)11 Additionally, we discuss the symmetry properties of the currents and torques induced by time-dependent magne- tization gradients of the form of Eq. (10). We consider mirror reflection Mxzat thexzplane, mirror reflection Myzat theyzplane, and c2 rotation around the zaxis. When ∆ V= 0 these operations leave Eq. (83) invariant, but when ∆ V∝ne}ationslash= 0 they modify the magnetization direction ˆMin Eq. (83), as shown in Ta- ble III. At the same time, these operations affect the torqueTandthecurrent Jdrivenbythe time-dependent magnetization gradients (see Table III). In Table IV and Table V we show how ˆM×∂ˆM/∂rkis affected by the symmetry operations. Aflat cycloidalspin spiralwith spinsrotatingin the xz plane is mapped by a c2 rotation around the zaxis onto the same spin spiral. Similarly, a flat helical spin spiral with spins rotating in the yzplane is mapped by a c2 ro- tationaroundthe zaxisontothesamespinspiral. There- fore, when ˆMpoints inzdirection, a c2 rotation around thezaxis does not change ˆM×∂ˆM/∂ri, but it flips the in-plane current Jand the in-plane components of the torque,TxandTy. Consequently, ˆM×∂2ˆM/∂ri∂tdoes not induce currents or torques, i.e., ICIDMI, CIDMI, ID- DMI and DDMI are zero, when ˆMpoints inzdirection. However, they become nonzero when the magnetization has an in-plane component (see Fig. 1). Similarly, IDOM vanishes when the magnetization points inzdirection: In that case Eq. (83) is invariant under the c2 rotation. A time-dependent magnetic field alongzdirection is invariant under the c2 rotation as well. However, TxandTychange sign under the c2 rota- tion. Consequently, symmetryforbidsIDOM inthiscase. However, when the magnetization has an in-plane com- ponent, IDOM and DOM become nonzero (see Fig. 2). That time-dependent magnetization gradients of the type of Eq. (7) do not induce in-plane currents and torqueswhen ˆMpoints inzdirectioncan alsobe seendi- rectly from Eq. (7): The c2 rotation transforms q→ −q andMx→ −Mx. Since sin( q·r) is odd in r, Eq. (7) is in- variantunder c2rotation, whilethe in-planecurrentsand torques induced by time-dependent magnetization gradi- ents change sign under c2 rotation. In contrast, Eq. (10) is not invariant under c2 rotation, because sin( q·r−ωt) is not odd in rfort>0. Consequently, time-dependent magnetization gradients of the type of Eq. (10) induce currents and torques also when ˆMpoints locally into thezdirection. These currents and torques, which are described by Eq. (24) and Eq. (82), respectively, need to be added to the chiral ICIT and the chiral torque-torque correlation. While CIDMI, DDMI, and DOM are zero when the magnetization points in zdirection, their gra- dients are not (see Fig. 1 and Fig. 2). Therefore, the gra- dients of CIDMI, DOM, and DDMI contribute to CIT, to ICIT and to the torque-torque correlation, respectively, even when ˆMpoints locally into the zdirection.TABLE III: Effect of mirror reflection Mxzat thexzplane, mirror reflection Myzat theyzplane, and c2 rotation around thezaxis. The magnetization Mand the torque Ttransform like axial vectors, while the current Jtransforms like a polar vector. MxMyMzJxJyTxTyTz Mxz−MxMy−MzJx−Jy−TxTy−Tz MyzMx−My−Mz−JxJyTx−Ty−Tz c2-Mx-MyMz-Jx−Jy−Tx−TyTz TABLE IV: Effect of symmetry operations on the magneti- zation gradients. Magnetization gradients are described b y three indices ( ijk). The first index denotes the magnetiza- tion direction at r= 0. The third index denotes the di- rection along which the magnetization changes. The second index denotes the direction of ∂ˆM/∂rkδrk. The direction of ˆM×∂ˆM/∂rkis specified by the number below the indices (ijk). (1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1) 3-2 -3 1 2-1 Mxz(-1,2,1)(-1,-3,1) (2,-1,1) (2,-3,1) (-3,-1,1) (-3,2,1) -3 -2 3 -1 2 1 Myz(1,2,1) (1,3,1)(-2,-1,1) (-2,3,1) (-3,-1,1) (-3,2,1) 3-2 -3 -1 2 1 c2(-1,2,1)(-1,-3,1) (-2,1,1) (-2,-3,1) (3,1,1) (3,2,1) -3 -2 3 1 2-1 . TABLE V: Continuation of Table IV (1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2) 3 -2 -3 1 2-1 Mxz(-1,-2,2) (-1,3,2) (2,1,2) (2,3,2)(-3,1,2)(-3,-2,2) 3 2-3 1-2 -1 Myz(1,-2,2) (1,-3,2) (-2,1,2)(-2,-3,2) (-3,1,2)(-3,-2,2) -3 2 3 1-2 -1 c2(-1,2,2) (-1,-3,2) (-2,1,2)(-2,-3,2) (3,1,2) (3,2,2) -3 -2 3 1 2-1 A. Symmetry properties of ICIDMI and IDDMI InthefollowingwediscusshowTableIII,TableIV,and Table V can be used to analyze the symmetry of ICIDMI andIDDMI.AccordingtoEq.(19)thecoefficient χICIDMI ijk describes the response of the current JICIT2a ito the time- dependent magnetization gradient ˆej·[ˆM×∂2ˆM ∂rk∂t]. Since ˆM×∂2ˆM ∂rk∂t=∂ ∂t[ˆM×∂ˆM ∂rk] fortime-dependent magnetiza- tion gradients of the type Eq. (7) the symmetry proper- ties ofχICIDMI ijkfollow from the transformation behaviour ofˆM×∂ˆM ∂rkandJunder symmetry operations. We consider the case with magnetization in xdirec- tion. The component χICIDMI 132describes the current in x direction induced by the time-dependence of a cycloidal magnetizationgradientin ydirection(withspinsrotating12 FIG. 1: ICIDMI in a noncollinear magnet. (a) Arrows illus- trate the magnetization direction. (b) Arrows illustrate t he currentJyinduced by a time-dependent magnetization gra- dient, which is described by χICIDMI 221. When ˆMpoints in z direction, χICIDMI 221andJyare zero. The sign of χICIDMI 221and ofJychanges with the sign of Mx. FIG. 2: DOM in a noncollinear magnet. (a) Arrows illustrate the magnetization direction. (b) Arrows illustrate the orb ital magnetization induced by magnetization dynamics (DOM). WhenˆMpoints in zdirection, DOM is zero. The sign of DOM changes with the sign of Mx. in thexyplane).Myzflips both ˆM×∂ˆM ∂yandJx, but it preserves ˆM.Mzxpreserves ˆM×∂ˆM ∂yandJx, but it flipsˆM. A c2 rotation around the zaxis flips ˆM×∂ˆM ∂y, ˆMandJx. Consequently, χICIDMI 132(ˆM) is allowed by symmetry and it is even in ˆM. The component χICIDMI 122 describes the current in xdirection induced by the time- dependence of a helical magnetization gradient in ydi- rection (with spins rotating in the xzplane).Myzflips ˆM×∂ˆM ∂yandJx, but it preserves ˆM.MzxflipsˆM×∂ˆM ∂y andˆM, but it preserves Jx. A c2 rotation around the z axis flipsJxandˆM, but it preserves ˆM×∂ˆM ∂y. Conse- quently,χICIDMI 122is allowed by symmetry and it is odd in ˆM. The component χICIDMI 221describes the current in y direction induced by the time-dependence of a cycloidal magnetization gradient in xdirection (with spins rotat- ing in thexzplane).Mzxpreserves ˆM×∂ˆM ∂x, but it flipsJyandˆM.Myzpreserves ˆM,Jy, andˆM×∂ˆM ∂x. The c2 rotation around the zaxis preserves ˆM×∂ˆM ∂x, but it flipsˆMandJy. Consequently, χICIDMI 221is allowed by symmetry and it is odd in ˆM. The component χICIDMI 231 describes the current in ydirection induced by the time- dependence of a cycloidal magnetization gradient in xdi- rection (with spins rotating in the xyplane).Mzxflips ˆM×∂ˆM ∂x,ˆM, andJy.Myzpreserves ˆM×∂ˆM ∂x,ˆMand Jy. The c2 rotation around the zaxis flips ˆM×∂ˆM ∂x,Jy, andˆM. Consequently, χICIDMI 231is allowed by symmetry and it is even in ˆM. These properties are summarized in Table VI. Due to the relations between CIDMI and DOM (see Table I and Table II), they can be used for DOM as well. When the magnetization lies at a general angle in the xzplane or in theyzplaneseveraladditionalcomponentsofCIDMIand DOMarenonzero(seeTableIandTableII,respectively). TABLE VI: Allowed components of χICIDMI ijkwhenˆMpoints inxdirection. + components are even in ˆM, while - compo- nents are odd in ˆM. 132 122 221 231 + - - + Similarly, one can analyze the symmetry of DDMI. Ta- ble VII lists the components of DDMI, χDDMI ijk, which are allowed by symmetry when ˆMpoints inxdirection. TABLEVII:Allowedcomponentsof χDDMI ijkwhenˆMpointsin xdirection. +componentsareevenin ˆM, while -components are odd in ˆM. 222 232 322 332 - + + - B. Response to time-dependent magnetization gradients of the second type (Eq. (10)) According to Eq. (13) the time-dependent magneti- zation gradient is along the magnetization. Therefore, in contrast to the discussion in section IIIA we can- not use ˆM×∂2ˆM ∂rk∂tin the symmetry analysis. Eq. (24) and Eq. (25) show that χICIT2a ijjldescribes the response of JICIT2a itoˆej·/bracketleftBig ˆM×∂ˆM ∂t/bracketrightBig ˆej·/bracketleftBig ˆM×∂ˆM ∂rl/bracketrightBig whileχICIT2a ijkl= 0 forj∝ne}ationslash=k. According to Eq. (23) the symmetry prop- erties of/bracketleftBig ˆM×∂ˆM ∂t/bracketrightBig ·/bracketleftBig ˆM×∂ˆM ∂rl/bracketrightBig agree to the symmetry properties of ˆM·∂2ˆM ∂rl∂t. Therefore, in order to under- stand the symmetry properties of χICIT2a ijjlwe consider the transformation of JandˆM·∂2ˆM ∂rl∂tunder symmetry operations. We consider the case where ˆMpoints inzdirection. χICIT2a 1jj1describes the current driven in xdirection, when13 the magnetization varies in xdirection. MxzflipsˆM, but preserves JxandˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,Jx, andˆM·∂2ˆM/(∂x∂t). c2 rotation flips ˆM·∂2ˆM/(∂x∂t) andJx, but preserves ˆM. Consequently, χICIT2a 1jj1is al- lowed by symmetry and it is even in ˆM. χICIT2a 2jj1describes the current flowing in ydirection, when magnetization varies in xdirection. MxzflipsˆM andJy, but preserves ˆM·∂2ˆM/(∂x∂t).MyzflipsˆM, andˆM·∂2ˆM/(∂x∂t), but preserves Jy. c2 rotation flipsˆM·∂2ˆM/(∂x∂t) andJy, but preserves ˆM. Conse- quently,χICIT2a 2jj1is allowed by symmetry and it is odd in ˆM. Similarly, one can show that χICIT2a 1jj2is odd in ˆMand thatχICIT2a 2jj2is even in ˆM. Analogously, one can investigate the symmetry prop- erties ofχTT2a ijjl. We find that χTT2a 1jj1andχTT2a 2jj2are odd inˆM, whileχTT2a 2jj1andχTT2a 1jj2are even in ˆM. IV. RESULTS In the following sections we discuss the results for the direct and inverse chiral CIT and for the chiral torque- torque correlation in the two-dimensional (2d) Rashba model Eq. (83), and in the one-dimensional (1d) Rashba model [38] Hkx(x) =/planckover2pi12 2mek2 x−αkxσy+∆V 2σ·ˆM(x).(84) Additionally, we discuss the contributions of the time- dependent magnetization gradients, and of DDMI, DOM and CIDMI to these effects. While vertex corrections to the chiral CIT and to the chiral torque-torque correlation are important in the Rashba model [38], the purpose of this work is to show the importance ofthe contributionsfrom time-dependent magnetization gradients, DDMI, DOM and CIDMI. We therefore consider only the intrinsic contributions here, i.e., we set GR k(E) =/planckover2pi1[E −Hk+iΓ]−1, (85) where Γ is a constant broadening, and we leave the study of vertex corrections for future work. The results shown in the following sections are ob- tained for the model parameters ∆ V= 1eV,α=2eV˚A, and Γ = 0 .1Ry = 1.361eV, when the magnetization points inzdirection, i.e., ˆM=ˆez. The unit of χCIT2 ijkl is charge times length in the 1d case and charge in the 2d case. Therefore, in the 1d case we discuss the chiral torkance in units of ea0, wherea0is Bohr’s radius. In the 2d case we discuss the chiral torkance in units of e. The unit ofχTT2 ijklis angular momentum in the 1d case and angular momentum per length in the 2d case. Therefore, we discussχTT2 ijklin units of /planckover2pi1in the 1d case, and in units of/planckover2pi1/a0in the 2d case.-2 -1 0 1 2 Fermi energy [eV]-0.02-0.0100.010.020.030.040.05χijklCIT2 [ea0]2121 1121 2121 (gauge-field) 1121 (gauge-field) FIG. 3: Chiral CIT in the 1d Rashba model for cycloidal gra- dients vs. Fermi energy. General perturbation theory (soli d lines) agrees to the gauge-field approach (dashed lines). A. Direct and inverse chiral CIT In Fig. 3 we show the chiral CIT as a function of the Fermi energyfor cycloidalmagnetization gradients in the 1d Rashba model. The components χCIT2 2121andχCIT2 1121are labelled by 2121 and 1121, respectively. The component 2121ofCITdescribesthe non-adiabatictorque, while the component 1121 describes the adiabatic STT (modified by SOI). In the one-dimensional Rashba model, the con- tributionsχCIT2b 2121andχCIT2b 1121(Eq. (56)) from the CIDMI are zero when ˆM=ˆez(not shown in the figure). For cy- cloidal spin spirals, it is possible to solve the 1d Rashba model by a gauge-field approach [38], which allows us to test the perturbation theory, Eq. (66). For comparison we show in Fig. 3 the results obtained from the gauge- field approach, which agree to the perturbation theory, Eq. (66). This demonstrates the validity of Eq. (66). In Fig. 4 we show the chiral ICIT in the 1d Rashba model. The components χICIT2 1221andχICIT2 1121are labelled by 1221and 1121, respectively. The contribution χICIT2a 1221 from the time-dependent gradient is of the same order of magnitude as the total χICIT2 1221. Comparison of Fig. 3 and Fig. 4 shows that CIT and ICIT satisfy the reciprocity relationsEq. (5), that χCIT2 1121is odd in ˆM, and thatχCIT2 2121 is even in ˆM, i.e.,χCIT2 2121=χICIT2 1221andχCIT2 1121=−χICIT2 1121. The contribution χICIT2a 1221from the time-dependent gradi- ents is crucial to satisfy the reciprocity relations between χCIT2 2121andχICIT2 1221. In Fig. 5 and Fig. 6 we show the CIT and the ICIT, re- spectively, for helical gradients in the 1d Rashba model. The components χCIT2 2111andχCIT2 1111are labelled 2111 and 1111, respectively, in Fig. 5, while χICIT2 1211andχICIT2 1111 are labelled 1211 and 1111, respectively, in Fig. 6. The contributions χCIT2b 2111andχCIT2b 1111from CIDMI are of the14 -2 -1 0 1 2 Fermi energy [eV]-0.0200.020.04χijklICIT2 [ea0]1221 1121 χ1221ICIT2a FIG. 4: Chiral ICIT in the 1d Rashba model for cycloidal gradients vs. Fermi energy. Dashed line: Contribution from the time-dependent gradient. same order of magnitude as the total χCIT2 2111andχCIT2 1111. Similarly, the contributions χICIT2c 1211andχICIT2c 1111from DOM are of the same order of magnitude as the to- talχICIT2 1211andχICIT2 1111. Additionally, the contribution χICIT2a 1111from the time-dependent gradient is substantial. ComparisonofFig.5andFig.6showsthatCITandICIT satisfy the reciprocity relation Eq. (5), that χCIT2 2111is odd inˆM, and thatχCIT2 1111is even in ˆM, i.e.,χCIT2 1111=χICIT2 1111 andχCIT2 2111=−χICIT2 1211. These reciprocity relations be- tween CIT and ICIT are only satisfied when CIDMI, DOM, and the response to time-dependent magnetiza- tion gradients are included. Additionally, the compar- ison between Fig. 5 and Fig. 6 shows that the contri- butions of CIDMI to CIT ( χCIT2b 1111andχCIT2b 2111) are re- lated to the contributions of DOM to ICIT ( χICIT2c 1111and χICIT2c 1211). These relations between DOM and ICIT are expected from Table I. In Fig. 7 and Fig. 8 we show the CIT and the ICIT, respectively, for cycloidal gradients in the 2d Rashba model. In this case there are contributions from CIDMI and DOM in contrast to the 1d case with cycloidal gra- dients (Fig. 3). Comparison between Fig. 7 and Fig. 8 shows that χCIT2 1121andχCIT2 2221are odd in ˆM, thatχCIT2 1221 andχCIT2 2121are even in ˆM, and that CIT and ICIT sat- isfy the reciprocity relation Eq. (5) when the gradients of CIDMI and DOM are included, i.e., χCIT2 1121=−χICIT2 1121, χCIT2 2221=−χICIT2 2221,χCIT2 1221=χICIT2 2121, andχCIT2 2121=χICIT2 1221. χCIT2 1121describesthe adiabatic STT with SOI, while χCIT2 2121 describes the non-adiabatic STT. Experimentally, it has been found that CITs occur also when the electric field is applied parallel to domain-walls (i.e., perpendicular to theq-vector of spin spirals) [39]. In our calculations, the components χCIT2 2221andχCIT2 1221describe such a case, where the applied electric field points in ydirection, while the-2 -1 0 1 2 Fermi energy [eV]-0.04-0.0200.020.040.06χijklCIT2 [ea0]1111 2111 χ1111CIT2b χ2111CIT2b FIG. 5: Chiral CIT for helical gradients in the 1d Rashba model vs. Fermi energy. Dashed lines: Contributions from CIDMI. -2 -1 0 1 2 Fermi energy [eV]-0.0200.020.040.06χijklICIT2 [ea0]1111 1211 χ1111ICIT2a χ1111ICIT2c χ1211ICIT2c FIG. 6: Chiral ICIT for helical gradients in the 1d Rashba model vs. Fermi energy. Dashed lines: Contributions from DOM. Dashed-dotted line: Contribution from the time- dependent magnetization gradient. magnetization direction varies with the xcoordinate. In Fig. 9 and Fig. 10 we show the chiral CIT and ICIT, respectively, for helical gradients in the 2d Rashba model. The component χCIT2 2111describes the adiabatic STT with SOI and the component χCIT2 1111describes the non-adiabatic STT. The components χCIT2 2211andχCIT2 1211 describe the case when the applied electric field points inydirection, i.e., perpendicular to the direction along which the magnetization direction varies. Comparison between Fig. 9 and Fig. 10 shows that χCIT2 1111andχCIT2 2211 are even in ˆM, thatχCIT2 1211andχCIT2 2111are odd in ˆMand that CIT andICIT satisfythe reciprocityrelationEq.(5) whenthegradientsofCIDMIandDOMareincluded, i.e.,15 -2 -1 0 1 2 Fermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]1121 2221 1221 2121 χ2221CIT2b χ1221CIT2b χ2121CIT2b FIG. 7: Chiral CIT for cycloidal gradients in the 2d Rashba model vs. Fermi energy. Dashed lines: Contributions from CIDMI. -2 -1 0 1 2 Fermi energy [eV]-0.00200.0020.0040.006χijklICIT2 [e]1121 1221 2121 2221 χ2221ICIT2a χ1221ICIT2a χ2121ICIT2c χ1221ICIT2c χ2221ICIT2c FIG. 8: Chiral ICIT for cycloidal gradients in the 2d Rashba model vs. Fermi energy. Dashed lines: Contributions from DOM. Dashed-dotted lines: Contributions from the time- dependent gradients. χCIT2 1111=χICIT2 1111,χCIT2 2211=χICIT2 2211,χCIT2 1211=−χICIT2 2111, and χCIT2 2111=−χICIT2 1211. B. Chiral torque-torque correlation In Fig. 11 we show the chiral contribution to the torque-torque correlation in the 1d Rashba model for cycloidal gradients. We compare the perturbation the- ory Eq. (78) plus Eq. (82) to the gauge-field approach from Ref. [38]. This comparison shows that perturba- tion theory provides the correct answer only when the contribution χTT2a ijkl(Eq. (82)) from the time-dependent-2 -1 0 1 2 Fermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]2211 1111 1211 2111 χ2111CIT2b χ1211CIT2b χ2211CIT2b χ1111CIT2b FIG. 9: Chiral CIT for helical gradients in the 2d Rashba model vs. Fermi energy. Dashed lines: Contributions from CIDMI. -2 -1 0 1 2 Fermi energy [eV]-0.004-0.00200.0020.0040.006χijklICIT2 [e]1111 1211 2111 2211 χ1111ICIT2a χ2221ICIT2a χ1111ICIT2c χ2111ICIT2c χ1211ICIT2c χ2211ICIT2c FIG. 10: Chiral ICIT for helical gradients in the 2d Rashba model vs. Fermi energy. Dashed lines: Contributions from DOM. Dashed-dotted lines: Contributions from the time- dependent gradient. gradients is taken into account. The contributions χTT2a 1221 andχTT2a 2221fromthe time-dependent gradientsarecompa- rable in magnitude to the total values. In the 1d Rashba model the DDMI-contribution in Eq. (60) is zero for cy- cloidal gradients (not shown in the figure). The compo- nentsχTT2 2121andχTT2 1221describe the chiral gyromagnetism while the components χTT2 1121andχTT2 2221describe the chi- ral damping [38, 40, 41]. The components χTT2 2121and χTT2 1221are odd in ˆMand they satisfy the Onsagerrelation Eq. (58), i.e., χTT2 2121=−χTT2 1221. In Fig. 12 we show the chiral contributions to the torque-torque correlation in the 1d Rashba model for helical gradients. In contrast to the cycloidal gradients16 -2 -1 0 1 2 Fermi energy [eV]-0.00500.0050.01χijklTT2 [h_]2121 1221 2221 1121 χ1221TT2a χ2221TT2a 2121 (gf) 1221 (gf) 1121 (gf) 2221 (gf) FIG. 11: Chiral contribution to the torque-torque correla- tion for cycloidal gradients in the 1d Rashba model vs. Fermi energy. Perturbation theory (solid lines) agrees to the gau ge- field (gf) approach (dotted lines). Dashed lines: Contribut ion from the time-dependent gradient. (Fig. 11) there are contributions from the spatial gra- dients of DDMI (Eq. (60)) in this case. The Onsager relation Eq. (58) for the components χTT2 2111andχTT2 1211is satisfied only when these contributions from DDMI are taken into account, which are of the same order of mag- nitude as the total values. The components χTT2 2111and χTT2 1211are even in ˆMand describe chiral damping, while the components χTT2 1111andχTT2 2211are odd in ˆMand de- scribe chiral gyromagnetism. As a consequence of the Onsager relation Eq. (58) we obtain χTT2 1111=χTT2 2211= 0 for the total components: Eq. (58) shows that diagonal components of the torque-torque correlation function are zero unless they are even in ˆM. However, χTT2a 1111,χTT2c 1111, andχTT2b 1111=−χTT2a 1111−χTT2c 1111are individually nonzero. Interestingly, the off-diagonal components of the torque- torquecorrelationdescribechiraldampingforhelicalgra- dients, while for cycloidal gradients the off-diagonal ele- ments describe chiral gyromagnetism and the diagonal elements describe chiral damping. In Fig. 13 we show the chiral contributions to the torque-torque correlation in the 2d Rashba model for cy- cloidal gradients. In contrast to the 1d Rashba model with cycloidal gradients (Fig. 11) the contributions from DDMIχTT2c ijkl(Eq.(60))arenonzerointhiscase. Without thesecontributionsfromDDMI theOnsagerrelation(58) χTT2 2121=−χTT2 1221is violated. The DDMI contribution is of the same order of magnitude as the total values. The components χTT2 2121andχTT2 1221are odd in ˆMand describe chiral gyromagnetism, while the components χTT2 1121and χTT2 2221are even in ˆMand describe chiral damping. In Fig. 14 we show the chiral contributions to the torque-torque correlation in the 2d Rashba model for he- lical gradients. The components χTT2 1211andχTT2 2111are even-2 -1 0 1 2 Fermi energy [eV]-0.00500.0050.01χijklTT2 [h_]1111 2111 1211 2211 χ1111TT2c χ2111TT2c χ1211TT2c χ2211TT2c χ1111TT2a χ2111TT2a FIG. 12: Chiral contribution to the torque-torque correla- tion for helical gradients in the 1d Rashba model vs. Fermi energy. Dashed lines: Contributions from DDMI. Dashed- dotted lines: Contributions from the time-dependent gradi - ents. -2 -1 0 1 2 Fermi energy [eV]-0.000500.00050.001χijklTT2 [h_/a0]1121 2121 1221 2221 χ1221TT2a χ2221TT2a χ2121TT2c χ1221TT2c FIG. 13: Chiral contribution to the torque-torque correla- tion for cycloidal gradients in the 2d Rashba model vs. Fermi energy. Dashed lines: Contributions from DDMI. Dashed- dotted lines: Contributions from the time-dependent gradi - ents. inˆMand describe chiral damping, while the compo- nentsχTT2 1111andχTT2 2211are odd in ˆMand describe chiral gyromagnetism. The Onsager relation Eq. (58) requires χTT2 1111=χTT2 2211= 0 andχTT2 2111=χTT2 1211. Without the contributions from DDMI these Onsager relations are vi- olated.17 -2 -1 0 1 2 Fermi energy [eV]-0.000500.00050.001χijklTT2 [h_ /a0]1111 2111 1211 2211 χ1111TT2a χ2111TT2a χ1111TT2c χ1211TT2c χ2211TT2c FIG. 14: Chiral contribution to the torque-torque correla- tion for helical gradients in the 2d Rashba model vs. Fermi energy. Dashed lines: Contributions from DDMI. Dashed- dotted lines: Contributions from the time-dependent gradi - ents. V. SUMMARY Finding ways to tune the Dzyaloshinskii-Moriya inter- action (DMI) by external means, such as an applied elec- triccurrent,holdsmuchpromiseforapplicationsinwhich DMI determines the magnetic texture of domain walls or skyrmions. In order to derive an expression for current- induced Dzyaloshinskii-Moriya interaction (CIDMI) we first identify its inverse effect: When magnetic textures vary as a function of time, electric currents are driven by various mechanisms, which can be distinguished accord- ingtotheirdifferentdependenceonthetime-derivativeof magnetization, ∂ˆM(r,t)/∂t, and on the spatial deriva- tive∂ˆM(r,t)/∂r: One group of effects is proportional to∂ˆM(r,t)/∂t, a second group of effects is propor- tional to the product ∂ˆM(r,t)/∂t ∂ˆM(r,t)/∂r, and a third group is proportional to the second derivative ∂2ˆM(r,t)/∂r∂t. We show that the response of the elec- tric current to the time-dependent magnetization gradi- ent∂2ˆM(r,t)/∂r∂tcontais the inverse of CIDMI. We establish the reciprocity relation between inverse and di- rectCIDMI and therebyobtainan expressionforCIDMI. We find that CIDMI is related to the modification of orbital magnetism induced by magnetization dynamics, which we call dynamical orbital magnetism (DOM). We show that torques are generated by time-dependent gra- dients of magnetization as well. The inverse effect con- sists in the modification of DMI by magnetization dy- namics, which we call dynamical DMI (DDMI). Additionally, we develop a formalism to calculate the chiral contributions to the direct and inverse current- induced torques (CITs) and to the torque-torque correla-tion in noncollinear magnets. We show that the response to time-dependent magnetization gradients contributes substantially to these effects and that the Onsager reci- procityrelationsareviolated when it is not takeninto ac- count. InnoncollinearmagnetsCIDMI,DDMIandDOM depend on the local magnetization direction. We show that the resulting spatial gradients of CIDMI, DDMI and DOM have to be subtracted from the CIT, from the torque-torque correlation, and from the inverse CIT, respectively. We apply our formalism to study CITs and the torque- torque correlation in textured Rashba ferromagnets. We find that the contribution of CIDMI to the chiral CIT is oftheorderofmagnitudeofthe totaleffect. Similarly, we find that the contribution of DDMI to the chiral torque- torque correlation is of the order of magnitude of the total effect. Acknowledgments WeacknowledgefinancialsupportfromLeibnizCollab- orative Excellence project OptiSPIN −Optical Control ofNanoscaleSpin Textures. Weacknowledgefundingun- der SPP 2137 “Skyrmionics” of the DFG. We gratefully acknowledge financial support from the European Re- search Council (ERC) under the European Union’s Hori- zon 2020 research and innovation program (Grant No. 856538, project ”3D MAGiC”). The work was also sup- ported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) −TRR 173 −268565370 (project A11). We gratefully acknowledge the J¨ ulich Supercomputing Centre and RWTH Aachen University for providing computational resources under project No. jiff40. Appendix A: Response to time-dependent gradients In this appendix we derive Eq. (18), Eq. (20), Eq. (41), and Eq. (82), which describe the response to time- dependent magnetization gradients, and Eq. (50), which describesthe responsetotime-dependentmagneticfields. We consider perturbations of the form δH(r,t) =Bb1 qωsin(q·r)sin(ωt).(A1) Whenweset B=∂H ∂ˆMkandb=∂2ˆMk ∂ri∂t, Eq.(A1)turnsinto Eq. (17), while when we set B=−eviandb=1 2ǫijk∂Bk ∂t we obtain Eq. (48). We need to derive an expression for the response δA(r,t) of an observable Ato this pertur- bation, which varies in time like cos( ωt) and in space like cos(q·r), because∂2ˆM(r,t) ∂ri∂t∝cos(q·r)cos(ωt). There- fore, weusethe Kubolinearresponseformalismtoobtain18 the coefficient χin δA(r,t) =χcos(q·r)cos(ωt), (A2) which is given by χ=i /planckover2pi1qωV/bracketleftBig ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) −∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(−/planckover2pi1ω)/bracketrightBig ,(A3) where∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) is the retarded function at frequency ωandVis the volume of the unit cell. The operator Bsin(q·r) can be written as Bsin(q·r) =1 2i/summationdisplay knm/bracketleftBig B(1) knmc† k+nck−m−B(2) knmc† k−nck+m/bracketrightBig , (A4) wherek+=k+q/2,k−=k−q/2,c† k+nis the cre- ation operator of an electron in state |uk+n∝an}bracketri}ht,ck−mis the annihilation operator of an electron in state |uk−m∝an}bracketri}ht, B(1) knm=1 2∝an}bracketle{tuk+n|[Bk++Bk−]|uk−m∝an}bracketri}ht(A5) and B(2) knm=1 2∝an}bracketle{tuk−n|[Bk++Bk−]|uk+m∝an}bracketri}ht.(A6) Similarly, Acos(q·r) =1 2/summationdisplay knm/bracketleftBig A(1) knmc† k+nck−m+A(2) knmc† k−nck+m/bracketrightBig , (A7) where A(1) knm=1 2∝an}bracketle{tuk+n|/bracketleftbig Ak++Ak−/bracketrightbig |uk−m∝an}bracketri}ht(A8) and A(2) knm=1 2∝an}bracketle{tuk−n|/bracketleftbig Ak++Ak−/bracketrightbig |uk+m∝an}bracketri}ht.(A9) It is convenient to obtain the retarded response func- tion in Eq. (A3) from the correspondingMatsubarafunc- tion in imaginary time τ 1 V∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(τ) = =1 4i/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n′m′/bracketleftBig A(1) knmB(2) kn′m′Z(1) knmn′m′(τ) −A(2) knmB(1) kn′m′Z(2) knmn′m′(τ)/bracketrightBig , (A10) whered= 1,2 or 3 is the dimension, Z(1) knmn′m′(τ) =∝an}bracketle{tTτc† k+n(τ)ck−m(τ)c† k−n′(0)ck+m′(0)∝an}bracketri}ht =−GM m′n(k+,−τ)GM mn′(k−,τ), (A11)Z(2) knmn′m′(τ) =∝an}bracketle{tTτc† k−n(τ)ck+m(τ)c† k+n′(0)ck−m′(0)∝an}bracketri}ht =−GM m′n(k−,−τ)GM mn′(k+,τ), (A12) and GM mn′(k+,τ) =−∝an}bracketle{tTτck+m(τ)c† k+n′(0)∝an}bracketri}ht(A13) is the single-particle Matsubara function. The Fourier transform of Eq. (A10) is given by 1 V∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) = =i 4/planckover2pi1β/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n′m′/summationdisplay p/bracketleftBig A(1) knmB(2) kn′m′GM m′n(k+,iEp)GM mn′(k−,iEp+iEN) −A(2) knmB(1) kn′m′GM m′n(k−,iEp)GM mn′(k+,iEp+iEN)/bracketrightBig , (A14) whereEN= 2πN/βandEp= (2p+ 1)π/βare bosonic andfermionicMatsubaraenergypoints, respectively, and β= 1/(kBT) is the inverse temperature. In order to carry out the Matsubara summation over Epwe make use of 1 β/summationdisplay pGM mn′(iEp+iEN)GM m′n(iEp) = =i 2π/integraldisplay dE′f(E′)GM mn′(E′+iEN)GM m′n(E′+iδ) +i 2π/integraldisplay dE′f(E′)GM mn′(E′+iδ)GM m′n(E′−iEN) −i 2π/integraldisplay dE′f(E′)GM mn′(E′+iEN)GM m′n(E′−iδ) −i 2π/integraldisplay dE′f(E′)GM mn′(E′−iδ)GM m′n(E′−iEN),(A15) whereδis a positive infinitesimal. The retarded function ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(ω) is obtained from the Mat- subara function ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) by the analytic continuation iEN→/planckover2pi1ωto real frequencies. The right-hand side of Eq. (A15) has the following analytic continuation to real frequencies: i 2π/integraldisplay dE′f(E′)GR mn′(E′+/planckover2pi1ω)GR m′n(E′) +i 2π/integraldisplay dE′f(E′)GR mn′(E′)GA m′n(E′−/planckover2pi1ω) −i 2π/integraldisplay dE′f(E′)GR mn′(E′+/planckover2pi1ω)GA m′n(E′) −i 2π/integraldisplay dE′f(E′)GA mn′(E′)GA m′n(E′−/planckover2pi1ω).(A16) Therefore, we obtain χ=−i 8π/planckover2pi12qω/integraldisplayddk (2π)d[Zk(q,ω)−Zk(−q,ω) −Zk(q,−ω)+Zk(−q,−ω)],(A17)19 where Zk(q,ω) = =/integraldisplay dE′f(E′)Tr/bracketleftBig AkGR k−(E′+/planckover2pi1ω)BkGR k+(E′)/bracketrightBig +/integraldisplay dE′f(E′)Tr/bracketleftBig AkGR k−(E′)BkGA k+(E′−/planckover2pi1ω)/bracketrightBig −/integraldisplay dE′f(E′)Tr/bracketleftBig AkGR k−(E′+/planckover2pi1ω)BkGA k+(E′)/bracketrightBig −/integraldisplay dE′f(E′)Tr/bracketleftBig AkGA k−(E′)BkGA k+(E′−/planckover2pi1ω)/bracketrightBig .(A18) We consider the limit lim q→0limω→0χ. In this limit Eq. (A17) may be rewritten as χ=−i 2π/planckover2pi12/integraldisplayddk (2π)d∂2Zk(q,ω) ∂q∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle q=ω=0.(A19) The frequency derivative of Zk(q,ω) is given by 1 /planckover2pi1∂Zk ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0=/integraldisplay dE′f(E′)Tr/bracketleftBigg Ak∂GR k−(E′) ∂E′BkGR k+(E′)/bracketrightBigg −/integraldisplay dE′f(E′)Tr/bracketleftBigg AkGR k−(E′)Bk∂GA k+(E′) ∂E′/bracketrightBigg −/integraldisplay dE′f(E′)Tr/bracketleftBigg Ak∂GR k−(E′) ∂E′BkGA k+(E′)/bracketrightBigg +/integraldisplay dE′f(E′)Tr/bracketleftBigg AkGA k−(E′)Bk∂GA k+(E′) ∂E′/bracketrightBigg . (A20) Using∂GR(E)/∂E=−GR(E)GR(E)//planckover2pi1we obtain ∂Zk ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0=−/integraldisplay dE′f(E′)Tr/bracketleftBig AkGR k−GR k−BkGR k+/bracketrightBig +/integraldisplay dE′f(E′)Tr/bracketleftBig AkGR k−BkGA k+GA k+/bracketrightBig +/integraldisplay dE′f(E′)Tr/bracketleftBig AkGR k−GR k−BkGA k+/bracketrightBig −/integraldisplay dE′f(E′)Tr/bracketleftBig AkGA k−BkGA k+GA k+/bracketrightBig . (A21) Making use of lim q→0∂GR k+ ∂q=1 2GR kv·q qGR k (A22)we finally obtain χ=−i 2π/planckover2pi12/integraldisplayddk (2π)dlim q→0lim ω→0∂2Z(q,ω) ∂q∂ω= =−i 4π/planckover2pi12q q·/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBig AkRvRRBkR+AkRRvRBkR −AkRRBkRvR−AkRvRBkAA +AkRBkAvAA+AkRBkAAvA −AkRvRRBkA−AkRRvRBkA +AkRRBkAvA +AkAvABkAA−AkABkAvAA −AkABkAAvA/bracketrightBig ,(A23) where we use the abbreviations R=GR k(E) andA= GA k(E). When we substitute B=∂H ∂ˆMj,A=−evi, and q=qkˆek, we obtain Eq. (18). When we substitute B= Tj,A=−evi, andq=qkˆek, we obtain Eq. (20). When we substitute A=−Ti,B=Tj, andq=qkˆek, we obtain Eq. (41). When we substitute B=−evj,A=−Ti, andq=qkˆek, we obtain Eq. (50). When we substitute B=∂H ∂ˆMj,A=−Ti, andq=qkˆek, we obtain Eq. (82). Appendix B: Perturbation theory for the chiral contributions to CIT and to the torque-torque correlation In this appendix we derive expressionsfor the retarded function ∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) (B1) within first-orderperturbation theory with respect to the perturbation δH=Bηsin(q·r), (B2) which may arise e.g. from the spatial oscillation of the magnetization direction. As usual, it is convenient to ob- tain the retarded response function from the correspond- ing Matsubara function ∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ) =−∝an}bracketle{tTτcos(q·r)A(τ)C(0)∝an}bracketri}ht. (B3) The starting point for the perturbative expansion is the equation −∝an}bracketle{tTτcos(q·r)A(τ1)C(0)∝an}bracketri}ht= =−Tr/bracketleftbig e−βHTτcos(q·r)A(τ1)C(0)/bracketrightbig Tr[e−βH]= =−Tr/braceleftbig e−βH0Tτ[Ucos(q·r)A(τ1)C(0)]/bracerightbig Tr[e−βH0U],(B4)20 whereH0is the unperturbed Hamiltonian and we con- sider the first order in the perturbation δH: U(1)=−1 /planckover2pi1/integraldisplay/planckover2pi1β 0dτ1Tτ{eτ1H0//planckover2pi1δHe−τ1H0//planckover2pi1}.(B5) The essentialdifference between Eq. (A3) and Eq. (B4) is that in Eq. (A3) the operator Benters together with the factor sin( q·r)sin(ωt) (see Eq. (A1)), while in Eq. (B4) only the factor sin( q·r) is connected to Bin Eq. (B2), while the factor sin( ωt) is coupled to the additional op- eratorC. We use Eq. (A4) and Eq. (A7) in order to express Acos(q·r) andBsin(q·r) in terms of annihilation and creation operators. In terms of the correlators Z(3) knmn′m′n′′m′′(τ,τ1) = ∝an}bracketle{tTτc† k−n(τ)ck+m(τ)c† k+n′(τ1)ck−m′(τ1)c† k−n′′ck−m′′∝an}bracketri}ht (B6) and Z(4) knmn′m′n′′m′′(τ,τ1) = ∝an}bracketle{tTτc† k−n(τ)ck+m(τ)c† k+n′(τ1)ck−m′(τ1)c† k+n′′ck+m′′∝an}bracketri}ht (B7) and Z(5) knmn′m′n′′m′′(τ,τ1) = ∝an}bracketle{tTτc† k+n(τ)ck−m(τ)c† k−n′(τ1)ck+m′(τ1)c† k+n′′ck+m′′∝an}bracketri}ht (B8) and Z(6) knmn′m′n′′m′′(τ,τ1) = ∝an}bracketle{tTτc† k+n(τ)ck−m(τ)c† k−n′(τ1)ck+m′(τ1)c† k−n′′ck−m′′∝an}bracketri}ht (B9) Eq. (B4) can be written as ∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1) = =ηV 4i/planckover2pi1/integraldisplayddk (2π)d/integraldisplay/planckover2pi1β 0dτ/summationdisplay nm/summationdisplay n′m′/summationdisplay n′′m′′/bracketleftBigg −B(2) knmA(1) kn′m′Ck−n′′m′′Z(3) knmn′m′n′′m′′(τ,τ1) −B(2) knmA(1) kn′m′Ck+n′′m′′Z(4) knmn′m′n′′m′′(τ,τ1) +B(1) knmA(2) kn′m′Ck+n′′m′′Z(5) knmn′m′n′′m′′(τ,τ1) +B(1) knmA(2) kn′m′Ck−n′′m′′Z(6) knmn′m′n′′m′′(τ,τ1)/bracketrightBigg(B10) within first-order perturbation theory, where we de- finedCk−n′′m′′=∝an}bracketle{tuk−n′′|C|uk−m′′∝an}bracketri}htandCk+n′′m′′= ∝an}bracketle{tuk+n′′|C|uk+m′′∝an}bracketri}ht. Note that Z(5)can be obtained from Z(3)by replac- ingk−byk+andk+byk−. Similarly, Z(6)can be obtained from Z(4)by replacing k−byk+andk+by k−. Therefore, we write down only the equations forZ(3)andZ(4)in the following. Using Wick’s theorem we find Z(3) knmn′m′n′′m′′(τ,τ1) = =−GM m′n(k−,τ1−τ)GM mn′(k+,τ−τ1)GM m′′n′′(k−,0) +GM mn′(k+,τ−τ1)GM m′′n(k−,−τ)GM m′n′′(k−,τ1) (B11) and Z(4) knmn′m′n′′m′′(τ,τ1) = =−GM mn′(k+,τ−τ1)GM m′n(k−,τ1−τ)GM m′′n′′(k+,0) +GM mn′′(k+,τ)GM m′n(k−,τ1−τ)GM m′′n′(k+,−τ1). (B12) The Fourier transform ∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) = =/integraldisplay/planckover2pi1β 0dτ1ei /planckover2pi1ENτ1∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1)(B13) of Eq. (B10) can be written as ∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) = =ηV 4i/planckover2pi1/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n′m′/summationdisplay n′′m′′/bracketleftBigg −B(2) knmA(1) kn′m′Ck−n′′m′′Z(3a) knmn′m′n′′m′′(iEN) −B(2) knmA(1) kn′m′Ck+n′′m′′Z(4a) knmn′m′n′′m′′(iEN) +B(1) knmA(2) kn′m′Ck+n′′m′′Z(5a) knmn′m′n′′m′′(iEN) +B(1) knmA(2) kn′m′Ck−n′′m′′Z(6a) knmn′m′n′′m′′(iEN)/bracketrightBigg(B14) in terms of the integrals Z(3a) knmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β 0dτ/integraldisplay/planckover2pi1β 0dτ1ei /planckover2pi1ENτ1× ×GM mn′(k+,τ−τ1)GM m′′n(k−,−τ)GM m′n′′(k−,τ1) = =1 /planckover2pi1β/summationdisplay pGM k+mn′(iEp)GM k−m′′n(iEp)GM k−m′n′′(iEp+iEN) (B15) and Z(4a) knmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β 0dτ/integraldisplay/planckover2pi1β 0dτ1ei /planckover2pi1ENτ1× ×GM mn′′(k+,τ)GM m′n(k−,τ1−τ)GM m′′n′(k+,−τ1) = =1 /planckover2pi1β/summationdisplay pGM k+mn′′(iEp)GM k−m′n(iEp)GM k+m′′n′(iEp−iEN), (B16) whereEN= 2πN/βis a bosonic Matsubara energy point and we used GM(τ) =1 /planckover2pi1β∞/summationdisplay p=−∞e−iEpτ//planckover2pi1GM(iEp),(B17)21 whereEp= (2p+1)π/βis a fermionic Matsubara point. Again Z(5a)is obtained from Z(3a)by replacing k−by k+andk+byk−andZ(6a)is obtained from Z(4a)in the same way. Summation overMatsubarapoints Epin Eq.(B15) and in Eq. (B16) and analytic continuation iEN→/planckover2pi1ωyields 2πi/planckover2pi1Z(3a) knmn′m′n′′m′′(/planckover2pi1ω) = −/integraldisplay dEf(E)GR k+mn′(E)GR k−m′′n(E)GR k−m′n′′(E+/planckover2pi1ω) +/integraldisplay dEf(E)GA k+mn′(E)GA k−m′′n(E)GR k−m′n′′(E+/planckover2pi1ω) −/integraldisplay dEf(E)GA k+mn′(E−/planckover2pi1ω)GA k−m′′n(E−/planckover2pi1ω)GR k−m′n′′(E) +/integraldisplay dEf(E)GA k+mn′(E−/planckover2pi1ω)GA k−m′′n(E−/planckover2pi1ω)GA k−m′n′′(E) (B18) and 2πi/planckover2pi1Z(4a) knmn′m′n′′m′′(/planckover2pi1ω) = −/integraldisplay dEf(E)GR k+mn′′(E)GR k−m′n(E)GA k+m′′n′(E−/planckover2pi1ω) +/integraldisplay dEf(E)GA k+mn′′(E)GA k−m′n(E)GA k+m′′n′(E−/planckover2pi1ω) −/integraldisplay dEf(E)GR k+mn′′(E+/planckover2pi1ω)GR k−m′n(E+/planckover2pi1ω)GR k+m′′n′(E) +/integraldisplay dEf(E)GR k+mn′′(E+/planckover2pi1ω)GR k−m′n(E+/planckover2pi1ω)GA k+m′′n′(E). (B19) In the next step we take the limit ω→0 (see Eq. (64), Eq. (70), and Eq. (77)): −1 Vlim ω→0Im∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) /planckover2pi1ω= =η 4/planckover2pi1Im/bracketleftBig Y(3)+Y(4)−Y(5)−Y(6)/bracketrightBig ,(B20)where we defined Y(3)=1 i/planckover2pi1/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n′m′/summationdisplay n′′m′′B(2) knmA(1) kn′m′Ck−n′′m′′× ×∂Z(3a) knmn′m′n′′m′′(/planckover2pi1ω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, Y(4)=1 i/planckover2pi1/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n′m′/summationdisplay n′′m′′B(2) knmA(1) kn′m′Ck+n′′m′′× ×∂Z(4a) knmn′m′n′′m′′(/planckover2pi1ω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, Y(5)=1 i/planckover2pi1/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n′m′/summationdisplay n′′m′′B(1) knmA(2) kn′m′Ck+n′′m′′× ×∂Z(5a) knmn′m′n′′m′′(/planckover2pi1ω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, Y(6)=1 i/planckover2pi1/integraldisplayddk (2π)d/summationdisplay nm/summationdisplay n′m′/summationdisplay n′′m′′B(1) knmA(2) kn′m′Ck−n′′m′′× ×∂Z(6a) knmn′m′n′′m′′(/planckover2pi1ω) ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ω=0, (B21) which can be expressed as Y(3)=Y(3a)+Y(3b)and Y(4)=Y(4a)+Y(4b), where 2π/planckover2pi1Y(3a)=1 /planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf(E)× ×Tr/bracketleftBigg AkGR k−(E)Ck−GA k−(E)BkGA k+(E)GA k+(E) +AkGR k−(E)GR k−(E)Ck−GA k−(E)BkGA k+(E) +AkGR k−(E)Ck−GA k−(E)GA k−(E)BkGA k+(E)/bracketrightBigg =/integraldisplayddk (2π)d/integraldisplay dEf′(E)× ×Tr/bracketleftBig AkGR k−(E)Ck−GA k−(E)BkGA k+(E)/bracketrightBig (B22) and 2π/planckover2pi1Y(3b)=−1 /planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf(E)× ×Tr/bracketleftBigg AkGA k−(E)Ck−GA k−(E)BkGA k+(E)GA k+(E) +AkGR k−(E)GR k−(E)Ck−GR k−(E)BkGR k+(E) +AkGA k−(E)Ck−GA k−(E)GA k−(E)BkGA k+(E)/bracketrightBigg .(B23)22 Similarly, 2π/planckover2pi1Y(4a)=1 /planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf(E)× ×Tr/bracketleftBigg AkGR k−(E)BkGR k+(E)Ck+GA k+(E)GA k+(E) −AkGR k−(E)GR k−(E)BkGR k+(E)Ck+GA k+(E) −AkGR k−(E)BkGR k+(E)GR k+(E)Ck+GA k+(E)/bracketrightBigg =/integraldisplayddk (2π)d/integraldisplay dEf′(E)× ×Tr/bracketleftBig AkGR k−(E)BkGR k+(E)Ck+GA k+(E)/bracketrightBig (B24) and 2π/planckover2pi1Y(4b)=−1 /planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf(E)× ×Tr/bracketleftBigg AkGA k−(E)BkGA k+(E)Ck+GA k+(E)GA k+(E) +AkGR k−(E)GR k−(E)BkGR k+(E)Ck+GR k+(E) +AkGR k−(E)BkGR k+(E)GR k+(E)Ck+GR k+(E)/bracketrightBigg .(B25) We call Y(3a)andY(4a)Fermi surface terms and Y(3b) andY(4b)Fermi sea terms. Again Y(5)is obtained from Y(3)by replacing k−byk+andk+byk−andY(6)is obtained from Y(4)in the same way. Finally, we take the limit q→0: Λ =−2 /planckover2pi1VηIm lim q→0lim ω→0∂ ∂ω∂ ∂qi∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) =1 2/planckover2pi1lim q→0∂ ∂qiIm/bracketleftBig Y(3)+Y(4)−Y(5)−Y(6)/bracketrightBig =1 2/planckover2pi1Im/bracketleftBig X(3)+X(4)−X(5)−X(6)/bracketrightBig , (B26) where we defined X(j)=∂ ∂qi/vextendsingle/vextendsingle/vextendsingle/vextendsingle q=0Y(j)(B27) forj= 3,4,5,6. Since Y(4)andY(6)are related by the interchange of k−andk+it follows that X(6)= −X(4). Similarly, since Y(3)andY(5)arerelated by the interchange of k−andk+it follows that X(5)=−X(3). Consequently, we need Λ =1 /planckover2pi1Im/bracketleftBig X(3a)+X(3b)+X(4a)+X(4b)/bracketrightBig ,(B28) where X(3a)andX(4a)are the Fermi surface terms and X(3b)andX(4b)are the Fermi sea terms. The Fermisurface terms are given by X(3a)=−1 4π/planckover2pi1/integraldisplayddk (2π)d/integraldisplay dEf′(E)Tr/bracketleftBigg AkGR k(E)vkGR k(E)CkGA k(E)BkGA k(E) +AkGR k(E)CkGA k(E)vkGA k(E)BkGA k(E) −AkGR k(E)CkGA k(E)BkGA k(E)vkGA k(E) +AkGR k(E)∂Ck ∂kGA k(E)BkGA k(E)/bracketrightBigg(B29) and X(4a)=−/bracketleftBig X(3a)/bracketrightBig∗ . (B30) The Fermi sea terms are given by X(3b)=−1 4π/planckover2pi12/integraldisplayddk (2π)d/integraldisplay dEf(E)Tr/bracketleftBigg −(ARvRRCRBR)+(AACAABAvA) −(ARRvRCRBR)−(ARRCRvRBR) +(ARRCRBRvR)−(AAvACABAA) −(AACAvABAA)+(AACABAvAA) +(AACABAAvA)−(AAvACAABA) −(AACAvAABA)−(AACAAvABA) −(ARR∂C ∂kRBR)−(AA∂C ∂kAABA) −(AA∂C ∂kABAA)/bracketrightBigg(B31) and X(4b)=−/bracketleftBig X(3b)/bracketrightBig∗ . (B32) In Eq. (B31) we use the abbreviations R=GR k(E),A= GA k(E),A=Ak,B=Bk,C=Ck. It is important to note that Ck−andCk+depend on qthrough k−= k−q/2 andk+=k+q/2 . Theqderivative therefore generates the additional terms with ∂Ck/∂kin Eq. 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2018-06-12

Both applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya interaction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respectively. We report a theory of CIDMI and DDMI. The inverse of CIDMI consists in charge pumping by a time-dependent gradient of magnetization $\partial^2 M(r,t)/\partial r\partial t$, while the inverse of DDMI describes the torque generated by $\partial^2 M(r,t)/\partial r\partial t$. In noncollinear magnets CIDMI and DDMI depend on the local magnetization direction. The resulting spatial gradients correspond to torques that need to be included into the theories of Gilbert damping, gyromagnetism, and current-induced torques (CITs) in order to satisfy the Onsager reciprocity relations. CIDMI is related to the modification of orbital magnetism induced by magnetization dynamics, which we call dynamical orbital magnetism (DOM), and spatial gradients of DOM contribute to charge pumping. We present applications of this formalism to the CITs and to the torque-torque correlation in textured Rashba ferromagnets.

Dynamical and current-induced Dzyaloshinskii-Moriya interaction: Role for damping, gyromagnetism, and current-induced torques in noncollinear magnets

1806.04782v3

1 Helicity-dependent optical control of the magnetization state emerging from the Landau-Lifshitz-Gilbert equation Benjamin Assouline, Amir Capua* Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel *e-mail: amir.capua@mail.huji.ac.il Abstract: It is well known that the Gilbert relaxation time of a magnetic moment scales inversely with the magnitude of the externally applied field, 𝑯, and the Gilbert damping, 𝜶. Therefore, in ultrashort optical pulses, where 𝑯 can temporarily be extremely large, the Gilbert relaxation time can momentarily be extremely short, reaching even picosecond timescales. Here we show that for typical ultrashort pulses, the optical control of the magnetization emerges by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. Surprisingly, when circularly polarized optical pulses are introduced to the LLG equation, an optically induced helicity-dependent torque results. We find that the strength of the interaction is determined by 𝜼=𝜶𝜸𝑯/𝒇𝒐𝒑𝒕, where 𝒇𝒐𝒑𝒕 and 𝜸 are the optical frequency and gyromagnetic ratio. Our results illustrate the generality of the LLG equation to the optical limit and the pivotal role of the Gilbert damping in the general interaction between optical magnetic fields and spins in solids. 2 The ability to control the magnetization order parameter using ultrashort circularly polarized (CP) optical pulses has attracted a great deal of attention since the early experiments of the all-optical helicity dependent switching (AO-HDS) [1-4]. This interaction was found intriguing since it appears to have all the necessary ingredients to be explained by a coherent transfer of angular momentum, yet it occurs at photon energies of 1 − 2 𝑒𝑉, very far from the typical resonant transitions in metals. The technological applications and fundamental scientific aspects steered much debate and discussion [5,6], and the experiments that followed found dependencies on a variety of parameters including material composition [7-9], magnetic structure [10-12], and laser parameters [1,3,13], that were often experiment-specific [4]. Consequently, a multitude of mechanisms that entangle photons [14,15], spins [16,17], and phonons [18,19] have been discovered. References [4,20] provide a state of the art review of the theoretical and experimental works of the field. Ferromagnetic resonance (FMR) experiments are usually carried out at the 𝐺𝐻𝑧 range. In contrast, optical fields oscillate much faster, at ~ 400−800 𝑇𝐻𝑧. Therefore, it seems unlikely that such fast-oscillating fields may interact with magnetic moments. However, the amplitude of the magnetic field in ultrashort optical pulses can, temporarily, be very large such that the magnetization may respond extremely fast. For example, in typical experiments having 40 𝑓𝑠−1 𝑝𝑠 pulses at 800 𝑛𝑚, with energy of 0.5 𝑚𝐽 that are focused to a spot size of ~0.5 𝑚𝑚$, the peak magnetic flux density can be as high as ~ 5 𝑇, for which the corresponding Gilbert relaxation time reduces to tens of picoseconds in typical ferromagnets. Here we show that ultrashort optical pulses may control the magnetization state by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. We find that the strength of the interaction is determined by 𝜂=𝛼𝛾𝐻/𝑓%&', where 𝑓%&' and 𝛼 are the angular optical frequency and the Gilbert damping, respectively, and 𝛾 is the gyromagnetic ratio. Moreover, we show that for circularly polarized (CP) pulses, the polarity of the optically induced torque is determined by the optical helicity. From a quantitative analysis, we find that a sizable effective out-of-plane field is generated which is comparable to that measured experimentally in ferromagnet/heavy-metal (FM/HM) material systems. 3 The LLG equation is typically not applied in the optical limit, and hence requires an alternative mathematical framework whose principles we adopt from the Bloch equations for semiconductor lasers [21,22]. We exploit the analogy between the magnetization state and the Bloch vector of a two-level system (TLS) [23,24] by transforming the LLG equation under a time-varying magnetic field excitation to the dynamical Maxwell-Bloch (MB) equations in the presence of an electrical carrier injection. In this transformation, the reversal of the magnetization is described in terms of population transfer between the states. The paper is organized as follows: We begin by transforming the LLG equation to the density matrix equations of a TLS. We then identify the mathematical form of a time-dependent magnetic field in the LLG equation, 𝐻AA⃗&()&↓↑, that is mapped to a time-independent carrier injection rate into the TLS. Such excitation induces a population transfer that varies linearly in time and accordingly to a magnetization switching profile that is also linear in time. The mathematical 𝐻AA⃗&()&↓↑ field emerges naturally as a temporal impulse-like excitation. We then show that when 𝛼 is sizable, 𝐻AA⃗&()&↓↑ acquires a CP component whose handedness is determined by the direction of the switching. By substituting 𝐻AA⃗&()&↓↑ for an experimentally realistic picosecond CP Gaussian optical magnetic pulse, we show that it can also exert a net torque on the magnetization. In this case as well, the helicity determines the polarity of the torque. Finally, we present a quantitative analysis that is based on experimental data. The LLG equation describing the dynamics of the magnetization, 𝑀AA⃗, where the losses are introduced in the Landau–Lifshitz form is given by [25]: 𝑑𝑀AA⃗𝑑𝑡= −𝛾1+𝛼$𝑀AA⃗×𝐻AA⃗−𝛾𝛼1+𝛼$1𝑀,𝑀AA⃗×𝑀AA⃗×𝐻AA⃗.(1) Here 𝑀, and 𝐻AA⃗ are the magnetization saturation and the time dependent externally applied magnetic field, respectively. We define 𝐻AA⃗-.. by: 𝐻AA⃗-..≜K𝐻AA⃗− 𝛼𝑀,𝐻AA⃗×𝑀AA⃗ L,(2) and in addition, 𝜅≜/012!O𝐻-.. 4−𝑗𝐻-.. 5Q/2 and 𝜅6 ≜/012!𝐻-.. 7, where 𝜅 and 𝜅6 can be regarded as effective AC and DC magnetic fields acting on 𝑀AA⃗, respectively. We 4 transform 𝑀AA⃗ to the density matrix elements of the Bloch state in the TLS picture and compare it to the Bloch equations describing a semiconductor laser that is electrically pumped [26]: ⎩⎪⎨⎪⎧𝜌̇00=𝛬0−𝛾0𝜌00+𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇$$=𝛬$−𝛾$𝜌$$−𝑗2[(𝜌0$−𝜌$0)(𝑉0$+𝑉$0)−(𝜌0$+𝜌$0)(𝑉0$−𝑉$0)]𝜌̇0$= −(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗(𝜌00−𝜌$$)𝑉0$ . (3) In this reference model, 𝛬0 and 𝛬$ are injection rates of carriers to the ground and excited states of the TLS, respectively. They are assumed to be time independent and represent a constant injection of carriers from an undepleted reservoir [27]. 𝛾0 and 𝛾$ are the relaxation rates of the ground and excited states, and 𝛾;<= is the decoherence rate due to an inhomogeneous broadening. 𝑉0$ is the interaction term and 𝜔89: is the resonance frequency of the TLS. Figure 1(a) illustrates schematically the analogy between the magnetization dynamics and the electrically pumped TLS. We find the connection between the LLG equation expressed in the density matrix form and the model of the electrically pumped TLS: ]𝛬0−𝛾0𝜌00+[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]= −𝑗𝜅𝜌$0+𝑐.𝑐.𝛬$−𝛾$𝜌$$−[𝑀5ℜ{𝑉0$}+𝑀4ℑ{𝑉0$}]= 𝑗𝜅𝜌$0+𝑐.𝑐.−(𝑗𝜔89:+𝛾;<=)𝜌0$+𝑗𝑀7𝑉0$= −𝑗𝜅6𝜌0$ +𝑗𝜅𝑀7.(4) The pumping of the excited and ground states by the constant 𝛬0 and 𝛬$ rates implies that the reversal of the magnetization along the ∓ 𝑧̂ direction is linear in time. Using Eq. (4) we find 𝜅, and hence a field 𝐻AA⃗, that produces such 𝛬0 and 𝛬$. We define this field as 𝐻AA⃗&()&↓↑: 𝐻AA⃗&()&↓↑= ±𝛬&𝑀,$−𝑀7$f𝑀5− 𝑀40g.(5) 𝐻AA⃗&()&↓↑ depends on the temporal state of 𝑀AA⃗ while 𝛬&=𝛾𝛬0/(1+𝛼$) is the effective field strength parameter. 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ induce a linear transition of 𝑀AA⃗ towards the –𝑧̂ and +𝑧̂ direction, respectively. 5 Figure 1(b) presents the outcome of the application of 𝐻AA⃗&()&↓↑ by numerically integrating the LLG equation. The Figure illustrates 𝐻AA⃗(𝑡), 𝑀7(𝑡), and the 𝑧̂ torque, O−𝑀AA⃗×𝐻AA⃗Q7, for alternating 𝐻AA⃗&()&↓ and 𝐻AA⃗&()&↑ that switch 𝑀AA⃗ between ∓𝑀,𝑧̂. The magnitude of 𝛬& determines the switching time, 𝛥𝜏↓↑, chosen here to describe a femtosecond regime. Equation (4) yields 𝛥𝜏↓↑=(1+𝛼$)𝑀,/( 𝛾𝛬&)≈𝑀,/𝛾𝛬& in which 𝑀7 is driven from 𝑀7=0 to 𝑀7≅±𝑀, (for derivation, see Supplemental Material Note 1). It is seen that O−𝑀AA⃗×𝐻AA⃗Q7 is constant when 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑ are applied so that the switching profile of 𝑀7 is linear in time. It is also seen that 𝐻AA⃗&()&↓↑ requires that m𝐻AA⃗m diverge as 𝑀7 approaches ±𝑀,, which is not experimentally feasible. To account for a more realistic excitation, in Fig. 1(c) we simulated a pulse whose trailing edge was taken as a reflection in time of 𝐻AA⃗&()&↓↑, and that is shorter by an order of magnitude as compared to the leading edge. In this case 𝑀AA⃗ remains in its final state when 𝐻AA⃗ is eventually turned off. The polarization state of 𝐻AA⃗&()&↓↑ is determined from the polarization state of the transverse components of 𝑀AA⃗. Next, we show that for larger 𝛼, 𝑀5(𝑡) becomes appreciable such that 𝐻AA⃗&()&↓↑ acquires an additional CP component. This result emerges naturally from the Bloch picture: we recall that the transverse components of 𝑀AA⃗ are expressed by the off-diagonal density matrix element. According to Eq. (3), 𝜌0$ oscillates at 𝜔89: and decays at the rate 𝛾;<=, whereas the sign of 𝜔89: determines the handedness of the transverse components of 𝑀AA⃗. Namely, the ratio between 𝜔89: and 𝛾;<= determines the magnitude of the circular component in the n𝑀4(𝑡),𝑀5(𝑡)o trajectory. Under the application of 𝐻AA⃗&()&↓↑, Eq. (4) yields 𝜔89:=±𝛾𝛬&𝛼𝑀,/[(𝑀,$−𝑀7$)(1+𝛼$)] and 𝛾;<==∓𝛾𝛬&𝑀7/[(𝑀,$−𝑀7$)(1+𝛼$)] readily showing that |𝜔89:/𝛾;<=|=𝛼𝑀,/𝑀7 increases with 𝛼, so that 𝐻AA⃗&()&↓↑ acquires an additional CP component (see Supplemental Note 2 for full derivation). Figure 2 illustrates these results. Panel (a) presents the components of 𝑀AA⃗(𝑡) for the same simulation in Fig. 1(b). It is seen that 𝑀5(𝑡) is negligible and thus 𝐻AA⃗&()&↓↑ remains linearly polarized. When 𝛼 is increased, an elliptical trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane emerges, while the constant transition rate of 𝑀7 persists as illustrated in Fig. 2(b). In this case, 6 𝐻AA⃗&()&↓↑ acquires a right-CP (RCP) or left-CP (LCP) component depending on the choice of 𝐻AA⃗&()&↓ or 𝐻AA⃗&()&↑. The coupling between the handedness and reversal direction in a femtosecond excitation is reminiscent of the switching reported in AO-HDS experiments and emerges naturally in our model. These results call to examine the interaction of the CP magnetic field of a short optical pulse with 𝑀AA⃗. Figure 3(a) presents the calculation for experimental conditions [4]. The results are shown for an 800 𝑛𝑚 optical magnetic field of an RCP Gaussian optical pulse 𝐻AA⃗%&'(𝑡). The pulse has a duration determined by 𝜏&, an angular frequency 𝜔%&', and a peak amplitude 𝐻&->? that is reached at 𝑡=𝑡&->?. In our simulations 𝜏&=3 𝑝𝑠 and 𝑡&->?=10 𝑝𝑠. The pulse energy was ~ 5 𝑚𝐽 and assumed to be focused to a spot size of ~ 100 𝜇𝑚$, for which 𝐻&->?=8⋅10@ 𝐴/𝑚. Here we take 𝛼=0.035 [28,29]. For such conditions, the Gilbert relaxation time corresponding to 𝐻&->? is 𝜏2=02/A"#$%≈16 𝑝𝑠 [30]. It is readily seen that for such 𝜏2 the magnetization responds within the duration of the optical pulse indicating that the interaction between the optical pulse and 𝑀AA⃗ becomes possible by the LLG equation. Following the interaction, 𝑀7=−5×10BC⋅𝑀:, namely a sizable net longitudinal torque results. In agreement with the prediction of the TLS model, pulses of the opposite helicity induce an opposite transition as shown in Fig. 3(b). The results are compared to the measured data discussed in Supplemental Material Note 3. To this end we simulate the same conditions of the measurements including optical intensity and sample parameters. Accordingly, we find from our calculations an effective field which is of the same order of magnitude as measured. For a given pulse duration, we define the interaction strength parameter 𝜂=2𝜋𝛼𝛾𝐻&->?/𝜔%&', which expresses the ratio between 𝜏2 and the optical cycle and is 2.5⋅10BC in Fig. 3(a). The principles of the interaction can be better understood at the limit where 𝜂→1 and for which the interaction can be described analytically. To this end, we set 𝜂=1. The higher optical magnetic fields required for this limit are achievable using conventional amplified femtosecond lasers, for example by focusing a ~ 5 𝑚𝐽 pulse into a spot size of ~ 1 𝜇𝑚$. Figure 3(c) illustrates the results for an RCP 𝐻AA⃗%&' pulse of a duration of 20 𝑓𝑠 determined by the full width at half-maximum of the 7 intensity. The Figure reveals the different stages of the interaction. During the leading edge, for 𝑡<~ 40 𝑓𝑠, the relative phase between 𝐻AA⃗%&' and 𝑀AA⃗ seems arbitrary. As 𝑡&->? is reached, the Gilbert relaxation time becomes as short as the optical cycle allowing 𝑀AA⃗ to follow 𝐻AA⃗%&' until it is entirely locked to 𝐻AA⃗%&'. In this case, 𝑀AA⃗ undergoes a right-circular trajectory about 𝑧̂. The switching of 𝑀AA⃗ takes place at the final stage of the interaction: During the trailing edge of the pulse, the amplitude of 𝐻AA⃗%&' reduces and 𝜏2 extends, thereby releasing the locking between 𝑀AA⃗ and 𝐻AA⃗%&'. In this case, the switching profile of 𝑀7 is monotonic linear-like in time, closely resembling the transition stemming from a constant carrier injection rate in the Bloch picture. The optically induced transition can be described analytically following the calculation presented in Supplemental Note 4, from which we find the transition rate: 𝛤/𝑀,=∓32√2𝑙𝑛K43L1𝜏&}~𝑙𝑛K𝐻&->?0.27𝐻'=L−~𝑙𝑛𝐻&->?𝐻'=/√2,(6) where 𝐻'==D&"'$E/2 is the value of 𝐻&->? at 𝜂=1. The rate 𝛤/𝑀, is plotted as well in Fig. 3(c) and reproduces the numerical calculation. 𝛤 depends on the ratio between 𝐻&->? and 𝐻'= and is only weekly dependent on 𝐻&->?. Namely, when 𝐻&->?≫𝐻'=, the circular trajectory of 𝑀AA⃗ in the 𝑥−𝑦 plane persists longer after 𝑡&->?, but as the amplitude of the pulse decays below 𝐻'=/√2, 𝑀AA⃗ is driven out of the 𝑥−𝑦 plane and the reversal takes place (see Supplemental Material Note 5). This analysis also holds for LCP pulses, which result in an opposite reversal of 𝑀AA⃗, as shown in Fig. 3(d). To summarize, in this work we demonstrated that the control of the magnetization by an optical field arises from first principles by introducing the magnetic part of the optical radiation to the LLG equation. This was seen from the comparison between the case where 𝜂≪1 and the case of 𝜂=1. Using the TLS model, we demonstrated the coupling between the optical helicity state and the polarity of the longitudinal torque. A quantitative analysis of the optically induced torque revealed that it can be comparable to that observed in experiments. 8 Figure 1 Fig. 1. (a) Left panel: Illustration of 𝑴AAA⃗ on the Bloch sphere. Right panel: Illustration of the electrically pumped TLS. (b) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ of Eq. (5). The Figure illustrates the temporal plots of 𝑴𝒛/𝑴𝒔, 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑,𝒚 and O−𝑴AAA⃗×𝑯AAA⃗Q𝒛 normalized to unity. (c) Interaction with 𝑯AAA⃗𝒑𝒖𝒎𝒑↓↑ and a more realistic trailing edge, for the same conditions in (b). Full lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and dashed lines correspond to 𝑯AAA⃗𝒑𝒖𝒎𝒑↑. 9 Figure 2 Fig. 2. Temporal evolution of the components of 𝑴AAA⃗ under the influence of alternating 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑ for (a) small and (b) large damping. Black dashed lines indicate the alternation between 𝑯AAA⃗𝒑𝒖𝒎𝒑↓ and 𝑯AAA⃗𝒑𝒖𝒎𝒑↑. 10 11 Fig. 3. (a) Magnetization reversal induced by an RCP Gaussian pulse for 𝜼=𝟐.𝟓⋅𝟏𝟎B𝟒. 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2023-06-07

It is well known that the Gilbert relaxation time of a magnetic moment scales inversely with the magnitude of the externally applied field, H, and the Gilbert damping, {\alpha}. Therefore, in ultrashort optical pulses, where H can temporarily be extremely large, the Gilbert relaxation time can momentarily be extremely short, reaching even picosecond timescales. Here we show that for typical ultrashort pulses, the optical control of the magnetization emerges by merely considering the optical magnetic field in the Landau-Lifshitz-Gilbert (LLG) equation. Surprisingly, when circularly polarized optical pulses are introduced to the LLG equation, an optically induced helicity-dependent torque results. We find that the strength of the interaction is determined by {\eta}={\alpha}{\gamma}H/f_opt, where f_opt and {\gamma} are the optical frequency and gyromagnetic ratio. Our results illustrate the generality of the LLG equation to the optical limit and the pivotal role of the Gilbert damping in the general interaction between optical magnetic fields and spins in solids.

Helicity-dependent optical control of the magnetization state emerging from the Landau-Lifshitz-Gilbert equation

2306.04617v2

arXiv:1011.5868v1 [cond-mat.mes-hall] 26 Nov 2010Draft Dependence of nonlocal Gilbert damping on the ferromagneti c layer type in FM/Cu/Pt heterostructures A. Ghosh, J.F. Sierra, S. Auffret, U. Ebels1and W.E. Bailey2 1)SPINTEC, UMR(8191) CEA / CNRS / UJF / Grenoble INP ; INAC, 17 rue des Martyrs, 38054 Grenoble Cedex, France 2)Dept. of Applied Physics & Applied Mathematics, Columbia Un iversity, New York NY 10027, USA (Dated: 7 September 2021) We have measured the size effect in nonlocal Gilbert relaxation rate in FM(tFM) / Cu (5nm) [/ Pt (2nm)] / Al(2nm) heterostructures, FM = {Ni81Fe19, Co60Fe20B20, pure Co}. Common behavior is observed for three FM layers, where the addit ional relaxation obeys botha strict inverse power law dependence ∆ G=Ktn,n=−1.04± 0.06 and a similar magnitude K= 224±40 Mhz·nm. As the tested FM layers span an order of magnitude in spin diffusion length λSDL, the results are in support of spin diffusion, rather than nonlocal resistivity, as the origin of the e ffect. 1Theprimarymaterialsparameter which describes thetemporal res ponseofmagnetization Mto applied fields His the Gilbert damping parameter α, or relaxation rate G=|γ|Msα. Understanding of the Gilbert relaxation, particularly in structures of reduced dimension, is an essential question for optimizing the high speed / Ghz response o f nanoscale magnetic devices. Experiments over the last decade have established that the Gilbert relaxation of ferro- magnetic ultrathin films exhibits a size effect, some component of whic h is nonlocal. Both α(tFM) =α0+α′(tFM) andG(tFM) =G0+G′(tFM) increase severalfold with decreasing FM film thickness tFM, from near-bulk values α0,G0fortFM>∼20 nm. Moreover, the damp- ing size effect can have a nonlocal contribution responsive to layers or scattering centers removed, through a nonmagnetic (NM) layer, from the precessing FM. Contributed Gilbert relaxation has been seen from other FM layers1as well as from heavy-element scattering layers such as Pt.2 The nonlocal damping size effect is strongly reminiscent of the electr ical resistivity in ferromagnetic ultrathin films. Electrical resistivity ρis size-dependent by a similar factor overasimilarrangeof tFM; theresistivity ρ(tFM)issimilarlynonlocal,dependentuponlayers not in direct contact.3–5. It isprima facie plausible that the nonlocal damping and nonlocal electrical resistivity share a common origin in momentum scattering ( with relaxation time τM) by overlayers. If the nonlocal damping arises from nonlocal scat teringτ−1 M, however, there should be a marked dependence upon FM layer type. Damping in materials with short spin diffusion length λSDLis thought to be proportional to τ−1 M(ref.6); the claim for ”resistivity-like” damping hasbeenmadeexplicitly forNi 81Fe19byIngvarsson7et al. ForFM with along λSDL, onthe other hand, relaxation Giseither nearly constant withtemperature or ”conductivity-like,” scaling as τM. Interpretation of the nonlocal damping size effect has centered in stead on a spin current model8advanced by Tserkovnyak et al9. An explicit prediction of this model is that the magnitude of the nonlocal Gilbert relaxation rate ∆ Gis only weakly dependent upon the FM layer type. The effect has been calculated10as ∆G=|γ|2¯h/4π/parenleftBig g↑↓ eff/S/parenrightBig t−1 FM (1) , where the effective spin mixing conductance g↑↓ eff/Sis given in units of channels per area. Ab-initio calculationspredictaveryweakmaterialsdependencefortheinter facialparameters 2g↑↓/S, with±10% difference in systems as different as Fe/Au and Co/Cu, and neglig ible dependence on interfacial mixing.11 Individual measurements exist of the spin mixing conductance, thr ough the damping, in FM systems Ni 81Fe1912, Co13, and CoFeB14. However, these experiments do not share a common methodology, which makes a numerical comparison of the r esults problematic, especially given that Gilbert damping estimates are to some extent mo del-dependent15. In our experiments, we have taken care to isolate the nonlocal dampin g contribution due to Pt overlayers only, controlling for growth effects, interfacial interm ixing, and inhomogeneous losses. The only variable in our comparison of nonlocal damping ∆ G(tFM), to the extent possible, has been the identity of the FM layer. Gilbert damping αhas been measured through ferromagnetic resonance (FMR) fro m ω/2π= 2-24 Ghz using a broadband coplanar waveguide (CPW) with broad c enter conduc- tor width w= 400µm, using field modulation and lock-in detection of the transmitted signa l to enhance sensitivity. The Gilbert damping has been separated fro minhomogeneous broad- ening inthe filmsmeasured using the well-known relation∆ Hpp(ω) = ∆H0+/parenleftBig 2/√ 3/parenrightBig αω/|γ|. We have fit spectra to Lorenzian derivatives with Dysonian compone nts at each frequency, for each film, to extract the linewidth ∆ Hppand resonance field Hres;αhas been extracted using linear fits to ∆ H(ω). For the films, six series of heterostructures were deposited of th e form Si/ SiO 2/ X/ FM(tFM)/ Cu(3nm)[ /Pt(3nm)]/ Al(3nm), FM = {Ni81Fe19(”Py”), Co 60Fe20B20 (”CoFeB”), pure Co }, andtFM= 2.5, 3.5, 6.0, 10.0, 17.5, 30.0 nm, for 36 heterostruc- tures included in the study. For each ferromagnetic layer type FM, one thickness series tFM was deposited with the Pt overlayer and one thickness series tFMwas deposited without the Pt overlayer. This makes it possible to record the additional damping ∆α(tFM) introduced by the Pt overlayer alone, independent of size effects present in th e FM/Cu layers deposited below. In the case of pure Co, a X=Ta(5nm)/Cu(5nm) underlayer w as necessary to sta- bilize low-linewidth films, otherwise, depositions were carried out direc tly upon the in-situ ion-cleaned substrate. Field-for-resonance data are presented in Figure 1. The main pane l showsω(H/bardbl B) data for Ni 81Fe19(tFM). Note that there is a size effect in ω(H/bardbl B): the thinner films have a substantially lower resonance frequency. For tFM= 2.5 nm, the resonance frequency is depressed by ∼5 Ghz from ∼20 Ghz resonance HB≃4 kOe. The behavior is fitted through 3the Kittel relation (lines) ω(H/bardbl B) =|γ|/radicalbigg/parenleftBig H/bardbl B+HK/parenrightBig/parenleftBig 4πMeff s+H/bardbl B+HK/parenrightBig , and the inset shows a summary of extracted 4 πMeff s(tFM) data for the three different FM layers. Samples with (open symbols) and without (closed symbols) Pt overlayers sho w negligible differences. Linear fits according to 4 πMeff s(tFM) = 4πMs−(2Ks/Ms)t−1 FMallow the extraction of bulk magnetization 4 πMsand surface anisotropy Ks; we find 4 πMPy s= 10.7 kG, 4 πMCoFeB s= 11.8 kG, 4 πMCo s= 18.3 kG, and KPy s= 0.69 erg/cm2,KCoFeB s= 0.69 erg/cm2,KCo s= 1.04 erg/cm2. The value of gL/2 =|γ|/(e/mc),|γ|= 2π·(2.799 Mhz/Oe) ·(gL/2) is found from the Kittel fits subject to this choice, yielding gPy L= 2.09,gCoFeB L= 2.07,gCo L= 2.15. The 4πMsandgLvalues, taken to be size-independent, are in good agreement with b ulk values. FMR linewidth as a function of frequency ∆ Hpp(ω) is plotted in Figure 2. The data for Py show a near-proportionality, with negligble inhomogeneous co mponent ∆ H0≤4 Oe even for the the thinnest layers, facilitating the extraction of intr insic damping parameter α. The size effect in in α(tFM) accounts for an increase by a factor of ∼3, fromαPy 0= 0.0067 (GPy 0= 105 Mhz) for the thickest films ( tFM= 30.0 nm) to α= 0.021 for the thinnest films ( tFM= 2.5 nm). The inset shows the line shapes for films with and without Pt, illustrating the broadening without significant frequency shift o r significant change in peak asymmetry. A similar analysis has been carried through for CoFeB and Co (not pict ured). Larger inhomogeneous linewidths are observed for pure Co, but homogene ous linewidth still ex- ceeds inhomogeneous linewidth by a factor of three over the frequ ency range studied, and inhomogeneous linewidths agree within experimental error for the t hinnest films with and without Pt overlayers. We extract for these films αCoFeB 0= 0.0065 ( GCoFeB 0= 111 Mhz) andαCo 0= 0.0085( GCo 0= 234 Mhz). The latter value is in very good agreement with the average of easy- and hard-axis values for epitaxial FCC Co films mea sured up to 90 Ghz, GCo 0= 225 Mhz.16 We isolate the effect of Pt overlayers on the damping size effect in Figu re 3. Values ofαhave been fitted for each deposited heterostructure: each FM t ype, at each tFM, for films with and without Pt overlayers. We take the difference ∆ α(tFM) for identical FM(tFM)/Cu(5nm)/Al(3nm) depositions with and without the insertion of Pt (3nm) after the Cu deposition. Data, as shown on the logarithmic plot in the main pa nel, are found 4to obey a power law ∆ α(tFM) =Ktn, withn= -1.04±0.06. This is excellent agreement with an inverse thickness dependence ∆ α(tFM) =KFM/tFM, where the prefactor clearly depends on the FM layer, highest for Py and lowest for Co. Note tha t efforts to extract ∆α(tFM) =Ktnwithout the FM( tFM)/Cu baselines would meet with significant errors; numerical fits to α(tFM) =KtFMnfor the FM( tFM)/Cu/Pt structures yield exponents n≃1.4. Expressing now the additional Gilbert relaxation as ∆ G(tFM) =|γ|Ms∆α(tFM) = |γFM|MFM sKFM/tFM, we plot ∆ G·tFMin Figure 4. We find ∆ G·tPy= 192±40 Mhz, ∆G·tCoFeB= 265±40 Mhz, and ∆ G·tCo= 216±40 Mhz. The similarity of values for ∆G·tFMis in good agreement with predictions of the spin pumping model in Equa tion 1, given that interfacial spin mixing parameters are nearly equal in diffe rent systems. The similarity of the ∆ G·tFMvalues for the different FM layers is, however, at odds with expectations from the ”resistivity-like” mechanism. In Figure 4 ,inset, we show the dependence of ∆ G·tFMupon the tabulated λSDLof these layers from Ref17. It can be seen thatλCo SDLis roughly an order of magnitude longer than it is for the other two FM layers, Py and CoFeB, but the contribution of Pt overlayers to damping is ve ry close to their average. Since under the resistivity mechanism, only Py and CoFeB s hould be susceptible to a resistivity contribution in ∆ α(tFM), the results imply that the contribution of Pt to the nonlocal damping size effect has a separate origin. Finally, we compare the magnitude of the nonlocal damping size effect with that pre- dicted by the spin pumping model in Ref.10. According to ∆ G·tFM=|γ|2¯h/4π= 25.69 Mhz ·nm3(gL/2)2/parenleftBig g↑↓ eff/S/parenrightBig , our experimental ∆ G·tFMandgLdata yield effective spin mixing conductances g↑↓ eff/S[Py/Cu/Pt ] = 6.8 nm−2,g↑↓ eff/S[Co/Cu/Pt ] = 7.3 nm−2, andg↑↓ eff/S[CoFeB/Cu/Pt ] = 9.6 nm−2. The Sharvin-corrected form, in the realistic limit ofλN SDL≫tN11is (g↑↓ eff/S)−1= (g↑↓ F/N/S)−1−1 2(g↑↓ N,S/S)−1+ 2e2h−1ρ tN+ (˜g↑↓ N1/N2/S)−1. Using conductances 14.1nm−2(Co/Cu), 15.0nm−2(Cu), 211nm−2(bulkρCu,tN= 3nm), 35 nm−2(Cu/Pt) would predict a theoretical g↑↓ eff,th./S[Co/Cu/Pt ] = 14.1 nm−2. Reconciling theory and experiment would require an order of magnitude larger ρCu≃20µΩ·cm, likely not physical. To summarize, a common methodology, controlling for damping size eff ects and intermix- ing in single films, has allowed us to compare the nonlocal damping size eff ect in different FM layers. We observe, for Cu/Pt overlayers, the same power law in thickness t−1.04±0.06, 5the same materials independence, but roughly half the magnitude th at predicted by the spin pumping theory of Tserkovnyak10. The rough independence on FM spin diffusion length, shown here for the first time, argues against a resistivity-based in terpretation for the effect. We would like to acknowledge the US NSF-ECCS-0925829, the Bourse Accueil Pro n◦ 2715 of the Rhˆ one-Alpes Region, the French National Research A gency (ANR) Grant ANR- 09-NANO-037, and the FP7-People-2009-IEF program no 252067 . REFERENCES 1R. Urban, G. Woltersdorf, and B. Heinrich, “Gilbert damping in single a nd multilayer ultrathin films: role of interfaces in nonlocal spin dynamics,” Physical Review Letters 87, 217204–7 (2001). 2S. Mizukami, Y. Ando, and T. Miyazaki, “Effect of spin diffusion onGilber t damping for a verythinpermalloylayer inCu/permalloy/Cu/Pt films,”Phys. Rev. B 66, 104413 (2002). 3B. Dieny, J. Nozieres, V. Speriosu, B. Gurney, and D. Wilhoit, “Chan ge in conductance is the fundamental measure of spin-valve magnetoresistance,” Ap plied Physics Letters 61, 2111–3 (1992). 4W. H. Butler, X. G. Zhang, D. M. C. Nicholson, T. C. Schulthess, and J. M. MacLaren, “Giant magnetoresistance from an electron waveguide effect in cob alt-copper multilayers,” Physical Review Letters 76, 3216–19 (1996). 5W. E. Bailey, S. X. Wang, and E. Y. Tsymbal, “Electronic scattering from Co/Cu interfaces: In situ measurement and comparison with t heory,” Phys. Rev. B 61, 1330–1335 (2000). 6V. Kambersk´ y, “On the landau-lifshitz relaxation in ferromagnetic metals,” Canadian Journal of Physics 48, 2906 (1970). 7S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczewski, P. Trouillo ud, and R. Koch, “Role of electron scattering in the magnetization relaxation of thin N i81Fe19films,” Phys. Rev, B, Condens, Matter Mater. Phys. (USA) 66, 214416 – 1 (2002/12/01). 8R. Silsbee, A. Janossy, and P. Monod, “Coupling between ferromag netic and conduction- spin-resonancemodesataferromagneticnormal-metalinterfac e,”PhysicalReviewB(Con- densed Matter) 19, 4382 – 99 (1979). 9Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, “Enhanced Gilbe rt damping in thin 6ferromagnetic films,” Phys. Rev. Lett. 88, 117601 (2002). 10Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, “Nonloca l magnetization dy- namics in ferromagnetic heterostructures,” Reviews in Modern Phy sics77, 1375 – 421 (2005). 11M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W . Bauer, “First- principles study of magnetization relaxation enhancement and spin t ransfer in thin mag- netic films,” Phys. Rev. B 71, 064420 (2005). 12S. Mizukami, Y. Ando, and T. Miyazaki, “Magnetic relaxation of norma l-metal NM/NiFe/NM films,” Journal of Magnetism and Magnetic Materials 239, 42 – 4 (2002); “The study on ferromagnetic resonance linewidth for NM/ 80NiFe/ N M, (NM=Cu, Ta, Pd and Pt) films,” Japanese Journal of Applied Physics, Part 1 (Regu lar Papers, Short Notes and Review Papers) 40, 580 – 5 (2001). 13J. Beaujour, W. Chen, A. Kent, and J. Sun, “Ferromagnetic reso nance study of poly- crystalline cobalt ultrathin films,” Journal of Applied Physics 99, 08N503 (2006); J.-M. Beaujour, J. Lee, A. Kent, K. Krycka, and C.-C. Kao, “Magnetiza tion damping in ultra- thin polycrystalline co films: evidence for nonlocal effects,” Physical Review B (Condensed Matter and Materials Physics) 74, 214405 – 1 (2006). 14H.Lee,L.Wen, M.Pathak, P.Janssen, P.LeClair,C.Alexander, C .Mewes, andT.Mewes, “Spin pumping in Co 56Fe24B20multilayer systems,” Journal of Physics D: Applied Physics 41, 215001 (5 pp.) – (2008). 15R. McMichael and P. Krivosik, “Classical model of extrinsic ferroma gnetic resonance linewidth in ultrathin films,” IEEE Transactions on Magnetics 40, 2 – 11 (2004). 16“Gilbert damping and g-factor in Fe xCo1−xalloy films,” Solid State Communications 93, 965 – 968 (1995). 17J. Bass and J. Pratt, W.P., “Spin-diffusion lengths in metals and alloys, and spin-flipping at metal/metal interfaces: an experimentalist’s critical review,” Jo urnal of Physics: Con- densed Matter 19, 41 pp. –(2007); C. Ahn, K.-H. Shin, andW. Pratt, “Magnetotran sport properties of CoFeB and Co/Ru interfaces in the current-perpen dicular-to-plane geome- try,” Applied Physics Letters 92, 102509 – 1 (2008). 7FIGURES ω/ 2π (Ghz) H (Oe)B 1 / t (nm ) FM -1 FIG. 1. Fields for resonance ω(HB) for in-plane FMR, FM=Ni 81Fe19, 2.5 nm ≤tFM≤30.0 nm; solid lines are Kittel fits. Inset:4πMeff sfor all three FM/Cu, with and without Pt overlayers. /s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48 /s49/s52/s48/s48 /s49/s53/s48/s48 /s49/s54/s48/s48 /s49/s55/s48/s48/s45/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s50/s48/s46/s48/s48/s48/s51 /s49/s50/s32/s71/s72/s122/s72 /s114/s101/s115 /s72 /s112/s112/s32 /s32/s78/s105 /s56/s49/s70/s101 /s49/s57/s40/s54/s110/s109/s41 /s32 /s32/s78/s105 /s56/s49/s70/s101 /s49/s57/s40/s54/s110/s109/s41/s45/s80/s116 /s32 /s32/s67/s111 /s54/s48/s70/s101 /s50/s48/s66 /s50/s48/s40/s54/s110/s109/s41 /s32 /s32/s67/s111 /s54/s48/s70/s101 /s50/s48/s66 /s50/s48/s40/s54/s110/s109/s41/s45/s80/s116/s39/s39/s47 /s72/s40/s97/s46/s117/s46/s41 /s70/s105/s101/s108/s100/s32/s40/s79/s101/s41 /s32/s50/s46/s53/s110/s109 /s32/s51/s46/s53/s110/s109 /s32/s54/s110/s109 /s32/s49/s48/s110/s109 /s32/s49/s55/s46/s53/s110/s109 /s32/s51/s48/s110/s109/s112/s112/s40/s79/s101/s41 /s84/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41 FIG. 2. Frequency-dependent peak-to-peak FMR linewidth ∆ Hpp(ω) for FM=Ni 81Fe19,tFM as noted, films with Pt overlayers. Inset:lineshapes and fits for films with and without Pt, FM=Ni 81Fe19, CoFeB. 8 t (nm)FM t (nm) FM α FIG. 3. Inset:αno Pt(tFM) andαPtfor Py, after linear fits to data in Figure 2. Main panel: ∆α(tFM) =αPt(tFM)−αno Pt(tFM) for Py, CoFeB, and Co. The slopes express the power law exponent n= -1.04±0.06. λPy CoFeB Co t (nm)FM ∆G·t (Mhz·nm) FM ∆G·t FM (nm) SDL FIG. 4. The additional nonlocal relaxation due to Pt overlay ers, expressed as a Gilbert relaxation rate - thickness product ∆ G·tFMfor Py, CoFeB, and Co. Inset:dependence of ∆ G·tFMon spin diffusion length λSDLas tabulated in17. 9

2010-11-26

We have measured the size effect in nonlocal Gilbert relaxation rate in FM(t$_{FM}$) / Cu (5nm) [/ Pt (2nm)] / Al(2nm) heterostructures, FM = \{ Ni$_{81}$Fe$_{19}$, Co$_{60}$Fe$_{20}$B$_{20}$, pure Co\}. Common behavior is observed for three FM layers, where the additional relaxation obeys both a strict inverse power law dependence $\Delta G =K \:t^{n}$, $n=-\textrm{1.04}\pm\textrm{0.06}$ and a similar magnitude $K=\textrm{224}\pm\textrm{40 Mhz}\cdot\textrm{nm}$. As the tested FM layers span an order of magnitude in spin diffusion length $\lambda_{SDL}$, the results are in support of spin diffusion, rather than nonlocal resistivity, as the origin of the effect.

Dependence of nonlocal Gilbert damping on the ferromagnetic layer type in FM/Cu/Pt heterostructures

1011.5868v1

Tuning Non -Gilbert -type damping in FeGa film s on MgO(001) via oblique deposition Yang Li1,2, Yan Li1,2, Qian Liu3, Zhe Yuan3, Qing -Feng Zhan4, Wei He1, Hao-Liang Liu1, Ke Xia3, We i Yu1, Xiang-Qun Zhang1, Zhao -Hua Cheng1,2,5 a) 1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 China 4State Key Laboratory of Precision Spectroscopy, School of Physics and Materials Science, East Ch ina Normal University, Shanghai 200241, China 5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China a) Corresponding author , e-mail: zhcheng@iphy.ac.cn Abstract The ability to tailor the damping factor is essential for spintronic and spin- torque application s. Here, we report an approach to manipulate the damping factor of FeGa/MgO(001) films by oblique deposition. Owing to the defects at the surface or interface in thin films , two -magnon scatterin g (TMS) acts as a non -Gilbert damping mechanism in magnetization relaxation. In this work, the contribution of TMS was characterized by in-plane angul ar dependent ferromagnetic resonance (FMR) . It is demonstrated that the intrinsic Gilbert damping is isotropic and invariant , while the extrinsic mechanism related to TMS is anisotropic and can be tuned by oblique deposition. Furthermore, the two and fourfold TMS related to the uniaxial magnetic anisotropy (UMA) and m agnetocrystalline anisotropy were discussed. Our result s open an avenue to manipulate magnetization relaxation in spintronic devices. 1 Keywords : Gilbert damping , two -magnon scattering, FMR, oblique deposition, magnetic anisotropy 2 1. Introduction In the past decades, controlling magnetization dynamics in magnetic nanostructures has been extensively studied due to its great importance for spintronic and spin- torque applications [1,2] . The magnetic relaxation is described within the framework of the Landa u-Lifshitz Gilbert (LLG) phenomenology using the Gilbert damping factor α [3]. The intrinsic Gilbert damping depends primarily on the spin- orbit coupling (SOC) [4,5] . It has been demonstrated that alloying or doping with non- magnetic transition metals provides an opportunity to tune the intrinsic damping [6,7] . Unfortunately, in this way the soft magnetic properties will reduce . In addition to the intrinsic damping, the two-magnon scatterin g (TMS) process se rves as a n important extrinsic mechanism i n magnetization relaxation in ultrathin films due to the defects at surface or interface [8,9] . This process describes the scattering between the uniform magnons and degener ate final -state spin wave modes [10]. The existence of TMS has been demonstrated in many systems of ferrites [11-13]. Since the anisotropic scattering centers , the angular dependence of the extrinsic TMS process exhibi ts a strong in -plane anisotropy [14], which allows us to adjust the overall magnetic relaxation , including both the int ensity of relaxation rate and the anisotropic behavior. Here, we report an approach to engineer the damping factor of Fe81Ga19 (FeGa ) films by oblique deposition. The FeGa alloy exhibits large magnetostriction and narrow microwave resonance linewidth [15] , which could assure it as a promising material for spintronic devices. For the geometry of off -normal deposition, it has been demonstrated to provoke shadow effects and create a periodic stripe defect matrix. This can introduce a strong uniaxial magnetization anisotropy (UMA) pe rpendicular to the projection of the atom flux [16-19]. Even though some reports have shown oblique deposition provokes a twofold TMS channel [20-22], the oblique angle dependence of the intrinsic 3 Gilbert damping and the TMS still remain in doubt. For our case, on the basis of the first-principles calculation and the in -plane angular -dependent FMR measurements, we found that the intrinsic Gilbert damping is isotropic and invariant with varying oblique deposition angles, while the extrinsic mechanism related to the two -magnon -scattering (TMS ) is anisotropic and can be tuned by oblique deposition. In addition, importantly we firstly observe a phenomenon that the cubic magnetocrystalline anisotropy determines the area including degenerate magnon modes, as well as the intensity of fourfold TMS. In general , the strong connection between the extrinsic TMS and the magnetic anisotropy , as well their direct impact on the damping constants , are system ically investigated, which offer us a useful approach to tailor the damping factor. 2. Experimental details FeGa thin films with a thickness of 20 nm were grown on MgO(001) substrates in a magnetron sputtering system with a base pressure below 3 × 10−7 Torr. Prior to deposition, t he substrates were annealed at 700 °C for 1 h in a vacuum chamber to remove surface contaminations and then held at 250 °C during deposition. The incident FeGa beam was at different obl ique angles of ψ =0°, 15°, 30°, and 45°, with respect to the surface normal , and named S1, S2, S3, and S4 in this paper , respectively. The projection of FeGa beam on the plane of the substrates was set perpendicular to the MgO[110] direction, which induces a UMA perpendicular to the projection of FeGa beam , i.e., parallel to the MgO[110] direction, due to the we ll-known self-shadowing effect. Finally , all the samples were covered with a 5 nm Ta capping layer to avoid surface oxidation [see figure 1(a)]. The epitaxial relation of FeGa(001)[ 110]||MgO (001)[ 100] was characterized by using t he X -ray in -plane Φ- scans , as described elsewhere [23]. Magnetic hy steresis loops were measured at various in-plane magnetic field orientations φ H with respect to the FeGa [100] axis using 4 magneto -optical Kerr effect (MOKE) technique at room temperature . The d ynamic magnetic properties were investigated by broadband FMR measurements based on a broadband vector network analyzer (VNA) with a transmission geometry coplanar waveguide (VNA- FMR ) [24]. This setup allows both frequency and field- sweeps measurements with external field applied parallel to the sample plane. During measurements, the sampl es were placed face down on the coplanar wavegu ide and the transmission coefficient S 21 was recorded. 3. Results and discussion Figure 1(b) displays the Kerr hysteresis loops of sample S1 and S4 recorded along with the main crystallographic directions of FeGa [100], [110], and [010] . The sample S1 exhibit s rectangular hysteresis curves with sm all coercivities for the magnetic field along [100] and [010] easy axes. In contrast, the S4 displays a hysteresis curve with two step s for the magnetic field along the [010] axis, which indicates a UMA along the FeGa[100 ] axis superimposed on the four fold magnetocrystalline anisotropy . As a result, with increasing the oblique angle, the angular dependence of normalized remnant magnetization ( Mr/Ms) gradually reveals a four fold symmetry combined with a uniaxial symmetry, as shown i n the inset of f igure 1(b). Subsequently, the magnetic anisotropic properties can be further precisely characterized by the in -plane angular -dependent FMR measurements. Figure 1(c) and 1(d) show typical FMR spectra for the real and imaginary part s of coefficient S 21 for the sample S2 . Recorded FMR spectra contain a symmetric and an antisymmetric Lorentzian peak , from which the resonant field H r with linewidth ∆𝐻𝐻 can be obtained [24,25] . Figure 2(a) shows the in -plane angular dependence of H r measured at 13 .0 GHz and can be fitted by the following expression [26,27] : 5 𝑓𝑓=𝛾𝛾𝜇𝜇0 2𝜋𝜋�𝐻𝐻𝑎𝑎𝐻𝐻𝑏𝑏 (1 ) Here,𝐻𝐻𝑎𝑎=𝐻𝐻4(3+𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M)/4+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H)+𝑀𝑀eff and 𝐻𝐻𝑏𝑏= 𝐻𝐻4𝑐𝑐𝑐𝑐𝑐𝑐4𝜑𝜑M+𝐻𝐻u𝑐𝑐𝑐𝑐𝑐𝑐2𝜑𝜑M+𝐻𝐻r𝑐𝑐𝑐𝑐𝑐𝑐(𝜑𝜑M−𝜑𝜑H), H4 and Hu represent the fourfold anisotropy field and the UMA field caused by the self -shadowing effect , respectively. 𝜑𝜑H(𝜑𝜑M) is the azimuthal angles of the applied field ( the tipped magnetization ) with respect to the [100] direction , as depicted in figure 1(a). 𝜇𝜇0𝑀𝑀eff=𝜇𝜇0𝑀𝑀𝑠𝑠−2𝐾𝐾out 𝑀𝑀𝑠𝑠, Ms is the saturation magnetization and Kout is the out -of-plane uniaxial anisotropy constant . 𝑓𝑓 is the resonance frequency , 𝛾𝛾 is the gyromagnetic ratio and here used as the accepted value for Fe films, 𝛾𝛾=185 rad GHz/T [28] . The angular dependent Hr reveals only a fourfold symmetry for the none - obliquely deposited sample , which indicates the cubic lattice texture of FeGa on MgO . With increasing the oblique angle , a uniaxial symmetry is found to be superimposed on the four fold symmetry, clearly confirming a UMA is produced by the oblique growth , which agrees with the MOKE’ results. The fitted parameter 𝜇𝜇0𝑀𝑀eff=1.90± 0.05T is found to be independent on the oblique deposition and close to 𝜇𝜇0𝑀𝑀𝑠𝑠= 1.89 ± 0.02 T estimated using VSM, which is almost same as the value of the literature [29] . This indicates negligible out- of-plane ma gnetic anisotropy in the thick FeGa films . As shown in figure 2(b), it is observed that the UMA (Ku=HuMs/2) exhibits a general increasing trend with oblique angle , which coincides with the fact the shadowing effect is stronger at larger angles of incidence [16-19]. Interestingly the oblique deposition also affects the cubic anisotropy K4 (K4=H4Ms/2). Different from the K4 increases slightly with deposition angle in Co/Cu system [16], here the value of K4 is the lowest at a n oblique angle of 15°. It is well known th at film stress significantly influences the crystallization tendency [30,31] . FeGa alloy is highly stress ed sensitive 6 due to its larger magnetostriction . Thus, t he change in K4 of FeGa films may be attributed to the anisotropy dispersion created due to the stress variations during grain growth. It should be mentioned that the best way to determine magnetic parameters is to measure the out -of-plane FMR . But the effective saturation magnetization 𝜇𝜇0𝑀𝑀eff= 1.90T of FeGa alloy leads to the perpendicular applied field beyond our instrument limit. Meanwhile, t he results obtained above are also in accord with those extracted by fitting field dependence of the resonance frequency with H//FeGa[100] shown in figure 2(c). The effective Gilbert damping 𝛼𝛼eff is extracted by linearly fitting the dependence of linewidth on frequency : 𝜇𝜇0∆H=𝜇𝜇0∆H0+2𝜋𝜋𝜋𝜋𝛼𝛼𝑒𝑒ff 𝛾𝛾, where ∆𝐻𝐻0 is the inhomogeneous broadening. For the sake of clarity , figure 3(a) only shows the frequency dependence of linewidth for the samples S1 and S2 along [110] and [100] axes. It is evident that, for the sample S1, both linear slopes of two direction s are almost same. While w ith regard to the sample S2 , the slope of the ∆H-f curve along the easy axis is approximately a factor of 2 greater than that of the hard axis. The obtained values of 𝛼𝛼eff are shown in figure 3(b). Firstly, the results clearly indicate that the effective damping exhibits anisotropy , with higher value along the easy axis . Secondly, f or the easy axis, the oblique angle dependence on the damping parameter indicates an extraordinary trend and has a peak at deposition angle 15°. However, the damping shows an increasing trend with the oblique angle for the field along the hard axis. In the following part, we will explore the effect of oblique deposition on the mechanism of the anisotropic damping and the magnetic relaxation pr ocess. So far, convincing experimental evidence is still lacking to prove the existence of anisotropic damping in bulk magnets. Chen et al. have shown the emergence of anisotropic Gilbert damping in ultrathin Fe (1.3nm)/GaAs and its anisotropy disappears 7 rapidly when the Fe thickness increases [32]. We perform the first-principles calculation of the Gilber t damping of Fe Ga alloy considering the effect induced by the lattice distortion. W e artificially make a tetragonal lattice with varying the lattice constant of the c -axis. The electronic structure of Fe -Ga alloy is calculated self - consistently using the coherent potential approx imation implemented with the tight- binding linear muffin- tin orbitals. Then the atomic potentials of Fe and Ga are randomly distributed in a 5× 5 lateral supercell, which is connected to two semi -infinite Pd leads . A thermal lattice disorder is included via displacing atoms randomly from the perfect lattice sites following a Gaussian type of distribution [ 33]. The root -mean -square displacement at room temperature is determined by the Debye model with the Debye temperature 470 K. The length of the supercell is variable and the calculated total damping is scaled linearly with this length. Thus, a linear least- squares fitting can be performed to extract the bulk damping of the Fe -Ga alloy [34]. The calculated Gilber t damping is plotted in f igure 3(c) as a funct ion of the lattice distortion (𝑐𝑐−𝑎𝑎)𝑎𝑎⁄. The Gilbert damping is nearly independent of the lattice distortion and there is no evidence of anisotropy in t he intrinsic bulk damping of Fe Ga alloy. So the extrinsic contributions are responsible for the anisotropic behavior of damping , which can be separated from the in -plane angular dependent linewidth. The recorded FMR linewidth have the following different cont ributions [11] : 𝜇𝜇0∆𝐻𝐻=𝜇𝜇0∆𝐻𝐻inh+2𝜋𝜋𝛼𝛼𝐺𝐺𝑓𝑓 𝛾𝛾𝛾𝛾+�𝜕𝜕𝐻𝐻r 𝜕𝜕𝜑𝜑H∆𝜑𝜑H�+�Γ<𝑥𝑥𝑖𝑖>𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>� <𝑥𝑥𝑖𝑖>𝑎𝑎𝑎𝑎𝑐𝑐𝑐𝑐𝑎𝑎𝑎𝑎 �(�𝜔𝜔2+(𝜔𝜔0 2)2−𝜔𝜔0 2)/(�𝜔𝜔2+(𝜔𝜔0 2)2+𝜔𝜔0 2)+Γtwofoldmaxcos4(φM- φtwofold) (2) ∆Hinh is both frequency and angle independent term due to the sample inhomogeneity . The second term is the intrinsic Gilbert damping (𝛼𝛼𝐺𝐺) contribution. 𝛾𝛾 8 is a correction factor owing to the field dragging effect caused by magnetic anisotropy [12], 𝛾𝛾 =cos (φM-φH). The 𝜑𝜑M as a function of φH for the sample S2 at fixed 13 GHz is calculated and show n in figure 4(a). Note that the draggi ng effect vanishes (𝜑𝜑M= φH) when the field is along the hard or easy axes . The third term describes the mosaicity contribution originating from the angular dispersion of the crystallographic cubic axes and yield s a broader linewidth [35]. The four th term is the TMS contribution. The Γ<𝑥𝑥𝑖𝑖> signifies the intensity of the TMS along the principal in -plane crystallographic direction <𝑥𝑥𝑖𝑖>. The 𝑓𝑓�𝜑𝜑H−𝜑𝜑<𝑥𝑥𝑖𝑖>� term indica tes the TMS contribution depending on the in- plane direction of the field rel ative to <𝑥𝑥𝑖𝑖> and commonly expressed as cos2[2(φM-φ<xi>)] [14]. In addition, 𝜔𝜔 is the angular resonant frequency and 𝜔𝜔0= 𝛾𝛾𝜇𝜇0𝑀𝑀eff. In our case, besides the fourfold TMS caused by expected lattice geometric defects, the other twofold TMS channel is induced by the dipolar fields emerging from periodic stripelike defects [20,21] . This term is parameterized by its strength Γtwofoldmax and the axis of maximal scattering rate φtwofold. As an example, t he angle- dependent linewidth measured at 13 .0 GHz for the sample S2 is shown in f igure 4(b). It clearly exhibits a strong in -plane anisotropy, and the linewidth along the [100] direction is significantly larger than that along the [110] direction . Taking only isotropic Gilbert damping into account , the dragging effect vanishes with field applied along the hard and easy axes . Meanwhile, the mosaicity term gives an angular variation of the linewidth proportional to |𝜕𝜕𝐻𝐻𝑟𝑟𝜕𝜕𝜑𝜑𝐻𝐻|⁄ , which is also zero along with the principal <100> and < 110> directions. This gives direct evidence that the rel axation is not exclusively governed only considering the intrinsic Gilbert mechanism and mosaicity term. Because the probability of defect formation along with <100> directions is higher than that along the <110> directions [12], the 9 TMS contribution is stronger along the easy axes , which is in accordance with t he fact that the linewidth s along the [100] and [110] direction s are non -equivalent. Moreover , the linewidth of [010] direction is slightly larger than that along the [100] dire ction, suggesting that another twofold TMS channel is induce d by oblique deposition. As indicated by the red solid line in figure 4(b), the linewidth can be well fitted. D ifferent parts making sense to the linewidth can therefore be sepa rated and summarized in Tab le I. As we know, the TMS predicts the curved non- linear frequency dependence of linewidth, which not appear in a small frequency range for our case (as shown in f igure 3(a)). The linewidth as function of frequency was also well fitted including the TMS - damping using the parameters in Table I (not shown here) . The larger strength of TMS along the easy axis can clearly explain the anisotropic behavior of da mping , with higher value along the easy axis shown in f igure 3(b). The obtained Gilbert damping factor of ~ 7×10-3 is isotropic and invariant with different oblique angle s. The value of damping is slightly larger than the bulk value of 5.5×10-3 [29], which may be attributed to spin pumping of the Ta capping layer. The obtained maxi ma of twofold TMS exhibits an increasing trend with the oblique angle [shown in f igure 4(c)]. According to previous works on the shadowing effect [16-19], the larger deposition angle makes the shadow ing effect stronger , and the dipolar fields within stripe like defects increase just like the UMA. This can clearly explain that the intensity of two fold TMS follows exactly the same trend with the deposition angle as the UMA . The axis of the maximal intensity of two fold TMS is paral lel to the projection of the FeGa atom flux from the fitting data. As shown in Table I, amazingly the modified growth conditions also influence the fourfold TMS, especially the strength of TMS along the <100> axis. Figure 4(c) also presents the changes of the fourfold TMS intensity as the deposition angle and shows a peak at 15° , 10 which follows a similar trend as that of 𝛼𝛼eff along [100] axis as shown in f igure 3(b). This indeed confirm s TMS -damping plays an important role in FeGa thin films. For the dispersion relation ω(k∥) in thin magnetic films , the propagation angle 𝜑𝜑𝑘𝑘∥����⃗ defined as the angle between k∥���⃗ and the projection of the saturation magnetization Ms into the sample plane is less than the critical value : 𝜑𝜑max = 𝑐𝑐𝑎𝑎𝑎𝑎−1�𝜇𝜇0𝐻𝐻r(𝜇𝜇0𝐻𝐻r+𝜇𝜇0𝑀𝑀eff) ⁄ [9,36,37] . This implies no degenerate modes are available for the angle 𝜑𝜑𝑘𝑘∥����⃗ larger than φmax. Based on this theory, we propose a hypothesis that the crystallographic anisotropy determine s the area including degenerate magnon modes , as well as the intensity of the fourfold TMS. The resonance field along <100> axis change s due to the various crystallographic anisotropy , which has a great effect on the φmax. The values of φmax of samples are shown in f igure 4(d). The data follow the same trend with the oblique angle as Γ<100>. During the grain growth, the cubic anisotropy is influenced possibly since the anisotropy dispersion due to the stress. For the lower anisotropy of sample S2 , a relatively larger amount of stress and defects present in the sample and lead to a larger four fold TMS. 4. Conclusions In conclusion, the effects of oblique deposition on the dynamic properties of FeGa thin films have been investigated systematically . The pronounced TMS as non-Gilbert damping results in an anisotropic magnetic relaxation . As the oblique angle increases, the magnitude of the twofold TMS increases due to the larger shadowing effect . Furthermore, the cubic anisotropy dominates the area including degenerate magnon modes, as well as the intensity of fourfold TMS. The reported results confirm that the modified anisotropy can influence the extrinsic relaxation pr ocess and open a n avenue to tailor magnetic relaxation in spintronic devices. 11 Acknowledgments This work is supported by the National Key Research Program of China (Grant Nos. 2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural Sciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212) and the Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDJ -SSW - JSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Norma l University is partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the Recruitment Program of Global Youth Experts and the Fundamental Research Funds for the Central Universities (Grant No. 2018EYT03). 12 References [1] Slonczewski J C 1996 J. Magn. Magn. Mater. 159 L1 [2] Žutić I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 [3] Gilbert T L2004 IEEE Trans. Magn. 40 3443 [4] He P, Ma X, Zhang J W, Zhao H B, Lüpke G, Shi Z and Zhou S M 2013 Phys. Rev. Lett. 110 077203 [5] Heinrich B, Meredith D J and Cochran J F 1979 J. Appl. Phys. 50 7726 [6] Lee A J, Brangham J T, Cheng Y, White S P, Ruane W T, Esser B D, McComb D W, Hammel P C and Yang F Y 2017 Nat. Commun. 8 234 [7] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey W E 2007 Phys. Rev. Lett. 98 117601 [8] Azzawi S, Hindmarch A and Atkinson D 2017 J. Phys. D: Appl. Phys. 50 473001 [9] Arias R and Mills D L 1999 Phys. Rev. B 60 7395 [10] Lenz K, Wende H, Kuch W, Baberschke K, Nagy K and Jánossy A 2006 Phys. Rev. B 73 144424 [11] Kurebayashi H et al 2013 Appl. Phys. Lett. 102 062415 [12] Zakeri K et al 2007 Phys. Rev. B 76 104416 [13] Lindner J, Lenz K, Kosubek E, Baberschke K, Spoddig D, Meckenstock R, Pelzl J, Frait Z and Mills D L 2003 Phys. Rev. B 68 060102(R) [14] Woltersdorf G and Heinrich B 2004 Phys. Rev. B 69 184417 [15] Parkes D E et al 2013 Sci. Rep. 3 2220 [16] Dijken S, Santo G D and Poelsema B 2001 Phys. Rev. B 63 104431 [17] Shim Y and Amar J G 2007 Phys. Rev. Lett. 98 046103 13 [18] Zhan Q F, Van Haesendonck C, Vandezande S and Temst K 2009 Appl. Phys. Lett. 94 042504 [19] Fang Y P, He W, Liu H L, Zhan Q F, Du H F, Wu Q, Yang H T, Zhang X Q and Cheng Z H 2010 Appl. Phys. Lett. 97 022507 [20] Barsukov I, Meckenstock R, Lindner J, Möller M, Hassel C, Posth O, Farle M and Wende H 2010 IEEE Trans. Magn. 46 2252 [21] Barsukov I , Landeros P, Meckenstock R, Lindner J, Spoddig D, Li Z A, Krumme B, Wende H, Mills D L and Farle M 2012 Phys. Rev. B 85 014420 [22] Mendes J B S, Vilela -Leão L H, Rezende S M and Azevedo A 2010 IEEE Trans. Magn. 46(6) 2293 [23] Zhang Y, Zhan Q F, Zuo Z H, Yang H L, Zhang X S, Yu Y, Liu Y W, Wang J , Wang B M and Li R W 2015 IEEE Trans. Magn. 51 1 [24] Kalarickal S S, Krivosik P, Wu M Z, Patton C E, Schneider M L, Kabos P, Silva T J and Nibarger J P 2006 J. Appl. Phys. 99 093909 [25] Bai L H, Gui Y S, Wirthmann A, Recksiedler E, Mecking N, Hu C-M, Chen Z H and Shen S C 2008 Appl. Phys. Lett. 92 032504 [26] Suhl H 1955 Phys. Rev. 97 555 [27] Farle M 1998 Rep. Prog. Phys. 61 755 [28] Butera A, Gómez J, Weston J L and Barnard J A 2005 J. Appl. Phys. 98 033901 [29] Kuanr B K, Camley R E, Celinski Z, McClure A and Idzerda Y 2014 J. Appl. Phys. 115 17C112 [30] Jhajhria D, Pandya D K and Chaudhary S 2018 J. Alloy Compd. 763 728 [31] Jhajhria D, Pandya D K and Chaudhary S 2016 RSC Adv. 6 94717 [32] Chen L et al 2018 Nat. Phys . 14 490 [33] Liu Y, Starikov A A, Yuan Z and Kelly P J 2011 Phys. Rev. B 84 014412 14 [34] Starikov A A, Liu Y, Yuan Z and Kelly P J 2018 Phys. Rev. B 97 214415 [35] McMichael R D, Twisselmann D J and Kunz A 2003 Phys. Rev. Lett. 90 227601 [36] Arias R and Mills D L 2000 J. Appl. Phys. 87 5455 [37] Lindner J, Barsukov I, Raeder C, Hassel C, Posth O, Meckenstock R, Landeros P and Mills D L 2009 Phys. Rev. B 80 224421 15 Figure Captions Figure 1 (color online) (a) Schematic illustration of the film deposition geometry and coordinate system (b) I n-plane hysteresis loops of samples S1 and S4 with the field along [100], [110], and [010]. The inset shows the polar plot of the normalized remanence (M r/Ms) as a functi on of the in- plane angle. FMR spectrum for the sample S2 with H along [100] and [110] axes showing the real (c ) and imaginary (d ) part s of the S 21. Figure 2 (color online) (a) H r vs. φH for FeGa films. (b) The anisotropy constants K4 and Ku vs. deposition angle. (c) f vs. Hr plots measured at H //[100], Symbols are experimental data and the solid lines are the fitted results. Figure 3 (color online) (a) ∆H as a function of f for samples S1 and S2 with field along easy and hard axis. (b) The dependence of the damping parameter on the oblique angle with field along [100] and [110] directions. (c) The calculated damping of FeGa alloy as a function of lattice distortion. Figure 4 (color online) (a) φ M and (b) ∆H as a function of φH for the sample S2 measured at 13.0 GHz. (c) Oblique angle dependences of Γ<100> and Γtwofoldmax. (d) The largest angle including degenerate magnon modes as a function of the oblique angle with the applied field along <100> direction. Table Caption Table I. The magnetic relaxation parameters of the FeGa films prepared via oblique deposition (with experimental errors in parentheses). 16 Figure 1 17 Figure 2 18 Figur e 3 Figure 4 19 TableⅠ Sample 𝜇𝜇0ΔHinh (mT) 𝛼𝛼G Δ𝜑𝜑H (deg.) Γ<100> (107Hz) Γ<110> (107Hz) Γtwofoldmax (107Hz) 𝜑𝜑twofold (deg. ) S1 0 0.007 0.62 17(3) 5.8(1.8) 0(2) 90 S2 0.7 0.007 1.2 81.4(3.7) 9.3(1.9) 7.4(3) 90 S3 0 0.007 1.0 59.2(4.5) 11.1(2) 13(3.7) 90 S4 0 0.007 1.1 33.3(6) 14.8(3.7) 26(4) 90 20

2019-11-02

The ability to tailor the damping factor is essential for spintronic and spin-torque applications. Here, we report an approach to manipulate the damping factor of FeGa/MgO(001) films by oblique deposition. Owing to the defects at the surface or interface in thin films, two-magnon scattering (TMS) acts as a non-Gilbert damping mechanism in magnetization relaxation. In this work, the contribution of TMS was characterized by in-plane angular dependent ferromagnetic resonance (FMR). It is demonstrated that the intrinsic Gilbert damping is isotropic and invariant, while the extrinsic mechanism related to TMS is anisotropic and can be tuned by oblique deposition. Furthermore, the two and fourfold TMS related to the uniaxial magnetic anisotropy (UMA) and magnetocrystalline anisotropy were discussed. Our results open an avenue to manipulate magnetization relaxation in spintronic devices.

Tuning Non-Gilbert-type damping in FeGa films on MgO(001) via oblique deposition

1911.00728v1

1 Magnetization Dynamics in Synthetic Antiferromagnets with Perpendicular Magnetic Anisotropy Dingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian-Ping Wang2, and Xiaojia Wang1,* 1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA 2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA ABSTRACT: Understanding the rich physics of magnetization dynamics in perpendicular synthetic antiferromagnets (p-SAFs) is crucial for developing next-generation spintronic devices. In this work, we systematically investigate the magnetization dynamics in p-SAFs combining time- resolved magneto -optical Kerr effect (TR -MOKE) measurements with theoretical modeling . These model analyses, based on a Landau -Lifshitz -Gilbert approach incorporating exchange coupling , provide detail s about the magnetization dynamic characteristics including the amplitude s, directions, and phases of the precession of p-SAFs under varying magnetic fields . These model - predicted characteristics are in excellent quantitative agreement with TR-MOKE measurements on an asymmetric p -SAF. We further reveal the damping mechanisms of two precession modes co-existing in the p -SAF and successfully identify individual contributions from different sources , includ ing Gilbert damping of each ferromagnetic layer , spin pumping, and inhomogeneous broadening . Such a comprehensive understanding of magnetization dynam ics in p -SAFs, obtained *Author s to whom correspondence should be addressed : huan1746@umn.edu and wang4940@umn.edu 2 by integrating high -fidelity TR -MOKE measurements and theoretical modeling, can guide the design of p-SAF-based architectures for spintronic applications . KEYWORDS: Synthetic antiferromagnets; Perpendicular magnetic anisotropy; Magnetization Dynamics; Time -resolved magneto -optical Kerr effect; Spintronics 3 1 INTRODUCTION Synthetic antiferromagnet ic (SAF) structures have attracted considerable interest for applications in spin mem ory and logic devices because of their unique magnetic configuration s [1- 3]. The SAF structures are composed of two ferromagnetic (FM) layers anti -parallelly coupled through a non -magnetic (NM) spacer, offer ing great flexibilit ies for the manipulat ion of magnetic configurations through external stimuli (e.g., electric -field and spin-orbit torque , SOT) . This permit s the design of new architecture s for spintronic applications , such as magnetic tunnel junct ion (MTJ), SOT devices, domain wall devices, sky rmion devices, among others [4-7]. The SAF structures possess many advantages for such applications , including fast switching speeds (potentially in the THz regimes), low off set fields, small switching current s (and thus low energy consumption) , high thermal stability, excellent resilience to perturbations from external magnetic fields, and large turnabilit y of magnetic properties [3,8-16]. A comprehensive study of the magnetization dynami cs of SAF structures can facilitate the understanding of the switching behavior of spintronic devices , and ultimately guide the design of novel device architectures . Different from a single FM free layer, magnetization dynamics of the SAF structures involv es two modes of precession , namely high -frequency (HF) and low -frequency (LF) modes, that result from the hybridization of magnetizations precession in the two FM layers . The relative phase and precession amplitude in two FM layers can significantly affect the spin- pumping enhancement of magnetic damping [17], and thus play an important role in determining the magnetization dynamic behaviors in SAFs. Heretofore, the exchange -coupling strength and magnetic damping constant of SAFs have been studied by ferromagnetic resonance (F MR) [18- 21] and optical metrolog y [22-25]. Most FMR -based experimental studies were limited to SAFs with in -plane magnetic anisotropy (IM A). For device applications, perpendicular magnetic 4 anisotropy (PMA) gives better scalability [3,26] . Therefore , the characteristics of magnetization dynamics of perpendicular SAF (p-SAF) structures are of much valu e to investigat e. In addition, prior studies mainly focus ed on the mutual spin pumping between two FM layers [22,27,28] . A more thorough understanding of the contribution s from various sources, including inhomogeneous broadening [29], remains elusive . In this paper, we report a comprehensive study of the magnetization dynamics of p -SAFs by integrating high -fidelity experiments and theoretical modeling to detail the characteristic parameters. These parameters describe the amplitude, phase, and direction of magnetization precess ion of both the HF and LF modes for the two exchange -coupled FM layers in a p -SAF. We conduct all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) measurements [30-33] on an asymmetric p -SAF structure with two different FM layers. The field-dependent amplitude and phase of TR -MOKE signal s can be well captured by our theoretical model, which in turn provid es comprehensive physical insights into the magnetization dynamics of p -SAF structures. Most importantly, we show that inhomogeneous broadening plays a critical role in determining the effective damping of both HF and LF modes, especially at low fields. We demonstrate the quantification of contributions from inhomogeneous broadening and mutual spin pumping (i.e., the exchange of angular momentum between two FM layers via pumped spin currents ) [21] to the effect ive damping, enabl ing accurate determination of the Gilbert damping for individual FM layers. Results of this work are beneficial for designing p-SAF-based architectures in spintronic application s. Additionally, this work also serves as a successful example demonstrating that TR- MOKE, as an all -optical met rology, is a powerful tool to capture the magnetization dynamics and reveal the rich physics of complex structures that involve multilayer coupling . 5 2 METHODOLOTY 2.1 Sample preparation and characterization One SAF structure was deposited onto thermally oxidize d silicon wafers with a 300 -nm SiO 2 layer by magnetron sputtering at room temperature (RT) in a six -target ultra -high vacuum (UHV) Shamrock sputtering system. The base pressure is below 5×10−8 Torr. The stacking structure of the SAF is: [Si/SiO 2]sub/[Ta(5)/Pd(3)] seed/[Co(0.4)/Pd(0.7)/Co(0.4)] FM1/[Ru(0.6)/Ta(0.3)] NM/ CoFeB(1) FM2/[MgO(2)/Ta(3)] capping . The numbers in parentheses denote the layer thicknesses in nanometers. After deposition, the sample was annealed at 250 ℃ for 20 minutes by a rapid - thermal-annealing process. The two FM layers are CoFeB and Co/Pd/Co layers, separated by a Ru/Ta spacer, forming an asymmetric p -SAF structure ( i.e., two FM layers having different magnetic properties). The M-Hext loops were characterized by a physical propert y measurement system (PPMS) with a vibrating -sample magnetometer (VSM) module. The resulting M-Hext loops are displayed in Fig. 1(a). Under low out -of-plane fields ( Hext < 500 Oe), the total magnetic moments in two FM layers of the SAF stack perfectly cancel out each other: M1d1 = M2d2 with Mi and di being the magnetization and thickness of each FM layer ( i = 1 for the top CoFeB layer and i = 2 for the bottom Co/Pd/Co laye r). The spin-flipping field ( Hf ≈ 500 Oe ) in the out -of-plane loop indicates the bilinear interlayer -exchange -coupling (IEC) J1 between the two FM layers : J1 = −HfMs,1d1 ≈ −0.062 erg cm-2 [34]. The values of Ms,1, Ms,2, d1, and d2 can be found in Table SI of the Supplemental Material (SM) [35]. 2.2 Theoretical foundation of magnetization dynamics for a p -SAF structure The magnetic free energy per unit area for a p -SAF structure with uniaxial PMA can be expressed as [36]: 6 𝐹=−𝐽1(𝐦1⋅𝐦2)−𝐽2(𝐦1⋅𝐦2)2 +∑2 𝑖=1𝑑𝑖𝑀s,𝑖[−1 2𝐻k,eff,𝑖(𝐧⋅𝐦𝑖)2−𝐦𝑖⋅𝐇ext] (1) where J1 and J2 are the strength of the bilinear and biquadratic IEC. mi = Mi / Ms,i are the normalized magnetization vectors for individual FM layers ( i = 1, 2). di, Ms,i, and Hk,eff, i denote, respectively, the thickness, saturation magnetization, and the effective anisotropy field of the i-th layer. n is a unit vector indicating the sur face normal direction of the film. For the convenience of derivation and discussion, the direction of mi is represented in the spherical coordinates by the polar angle θi and the azimuthal angle φi, as shown in Fig. 1 (b). The equilibrium direction of magne tization in each layer (𝜃0,𝑖,𝜑0,𝑖) under a given Hext is obtained by minimizing F in the (𝜃1,𝜑1,𝜃2,𝜑2) space. The magnetization precession is governed by the Landau -Lifshitz -Gilbert (LLG) equation considering the mutual spin pumping between two FM layers [27,37 -40]: 𝑑𝐌𝑖 𝑑𝑡=−𝛾𝑖𝐌𝑖×𝐇eff,𝑖+(𝛼0,𝑖+𝛼sp,𝑖𝑖) 𝑀s,𝑖𝐌𝒊×𝑑𝐌𝒊 𝑑𝑡−𝛼sp,𝑖𝑗 𝑀s,𝑖𝐌𝒊×(𝐦𝐣×𝑑𝐦𝒋 𝑑𝑡)×𝐌𝒊 (2) On the right -hand side of Eq. (2), the first term describes the precession with the effective field Heff,i in each layer, given by the partial derivative of the total free energy in the M space via 𝐇eff,𝑖= −∇𝐌𝑖𝐹. The second term represents the relaxation induced by Gilbert damping ( α) of the i-th layer, which includes the intrinsic ( 𝛼0,𝑖) and spin -pumping -enhanced ( 𝛼sp,𝑖𝑖) damping. For TR -MOKE measurements, 𝛼0,𝑖 and 𝛼sp,𝑖𝑖 are indistinguishable. Hence, we def ine 𝛼𝑖=𝛼0,𝑖+𝛼sp,𝑖𝑖 to include both terms. The last term in Eq. (2) considers the influence of pumped spin currents from the layer j on the magn etization dynamics of the layer i. 7 The time evolution of Mi can be obtained by solving the linearized Eq. (2). Details are provided in Note 1 of the SM [35]. The solutions to Eq. (2) in spherical coordinates are: [𝜃1(𝑡) 𝜑1(𝑡) 𝜃2(𝑡) 𝜑2(𝑡)]=[𝜃0,1 𝜑0,1 𝜃0,2 𝜑0,2]+[Δ𝜃1(𝑡) Δ𝜑1(𝑡) Δ𝜃2(𝑡) Δ𝜑2(𝑡)]=[𝜃0,1 𝜑0,1 𝜃0,2 𝜑0,2]+ [ 𝐶𝜃,1HF 𝐶𝜑,1HF 𝐶𝜃,2HF 𝐶𝜑,2HF] exp(𝑖𝜔HF𝑡)+ [ 𝐶𝜃,1LF 𝐶𝜑,1LF 𝐶𝜃,2LF 𝐶𝜑,2LF] exp(𝑖𝜔LF𝑡) (3) with Δ𝜃𝑖 and Δ𝜑𝑖 representing the deviation angles of magnetization from its equilibrium direction along the polar and azimuthal directions . The last two terms are the linear combination of two eigen -solutions, denoted by superscripts HF (high -frequency mode) and LF (low -frequency mode). ω is the complex angular frequencies of two modes, with the real and imaginary parts representing the precession angular frequency ( 𝑓/2𝜋) and relaxation rate (1/ τ), respectively. For each mode, the complex prefactor vector [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 contains detailed information about the magnetization dynamics. As illustrated in Fig. 1 (c), the moduli, |𝐶𝜃,𝑖| and |𝐶𝜑,𝑖| correspond to the half cone angles of t he precession in layer i along the polar and azimuthal directions for a given mode immediately after laser heating, as shown by Δ𝜃 and Δ𝜑 in Figs. 1 (b-c). The phase difference between Δ𝜃𝑖 and Δ𝜑𝑖, defined as Arg(Δ𝜃𝑖/Δ𝜑𝑖)=Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) with Arg representing the argument of complex numbers, determines the direction of precession. If Δ𝜃𝑖 advances Δ𝜑𝑖 by 90°, meaning Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°, the precession is counter -clockwise (CCW) in the θ-φ space (from a view against Mi).Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=−90°, on the contrary, suggests clockwise (CW) precession [ Fig. 1 (d)]. Further, the argument of 𝐶𝜃,2/𝐶𝜃,1 provides the relative phase in two FM layers. Arg(𝐶𝜃,2/𝐶𝜃,1)=0° corresponds to the precession motions in two FM layers that are in -phase (IP) in terms of θ for a given mode. While the out -of-phase (OOP) precession in terms of θ is represented by Arg(𝐶𝜃,2/𝐶𝜃,1)=180° [Fig. 1 (e)]. Given the precession 8 direction in each layer and the phase difference between the two FM layers in terms of θ, the phase difference in terms of φ can be automatically determined. FIG. 1 (a) Magnetic hysteresis ( M-Hext) loops of the p -SAF stack. The magnetization is n ormalized to the saturation magnetization ( M/Ms). (b) Schematic illustration of the half cone angles (Δ θ and Δφ) and precession direction of magnetization. The precession direction is defined from a view against the equilibrium direction ( 0, φ0) of M. The representative precession direction in the schematic is counterclockwise (CCW). (c) The relation between precession half cone angles and the prefactors. (d) The relation between precession direction and the prefactors. (e) The relative phase between two FM layers for different prefactor values. As for the effective damping 𝛼eff=1/2𝜋𝑓𝜏, in addition to the intrinsic damping ( α0,i) and the spin-pumping contribution ( αsp,ii and αsp,ji) considered in Eq. (2), inhomogeneities can also bring substantial damping enhancement [32,33,41,42] . Here, we m odel the total relaxation rate as follows: 9 1 𝜏Φ=−Im(𝜔Φ)+1 𝜏inhomoΦ (4) The superscript Φ = HF or LF, representing either the high -frequency or low -frequency precession modes. 𝜔Φ includes both the intrinsic and spin -pumping contributions. The inhomogeneous broadening is calculated as: 1 𝜏inhomoΦ=∑1 𝜋|𝜕𝑓Φ 𝜕𝐻k,eff,𝑖| 𝑖Δ𝐻k,eff,𝑖+∑1 𝜋|𝜕𝑓Φ 𝜕𝐽𝑖| 𝑖Δ𝐽𝑖 (5) where the first summation represents the contrib ution from the spatial variation of the effective anisotropy field of individual FM layers (Δ Hk,eff, i). The second summation denotes the contribution from the spatial fluctuations of the bilinear and biquadratic IEC (Δ J1 and Δ J2). According to Slonczewski’s “thickness fluctuations” theory, Δ J1 generates J2 [43,44] . Therefore, the fact that J2 = 0 for our sample suggests that ΔJ1 is sufficiently small, allowing us to neglect the inhomogeneous broadening from th e fluctuations of both the bilinear and biquadratic IEC in the following analyses . 2.3 Detection of magnetization dynamics The magnetization dynamics of the p -SAF sample is detected by TR -MOKE, which is ultrafast -laser -based metrology utilizing a pump -probe configuration. In TR -MOKE, pump laser pulses interact with the sample, initiating magnetization dynamics in magnetic layers via inducing ultrafast thermal demagnetization. The laser -induced heating brings a rapid decrease to the magnetic anisotropy fields and IEC [45,46] , which changes 𝜃0,𝑖, 𝜑0,𝑖 and initiates the precession. The magnetizati on dynamics due to pump excitation is detected by a probe beam through the magneto -optical Kerr effect. In our setup, the incident probe beam is normal to the sample surface (polar MOKE); therefore, the Kerr rotation angle ( 𝜃K) of the reflected probe beam is proportional to the z component of the magnetization [47]. More details about the experimental setup can be 10 found in Refs. [30,32] . For p -SAF, TR -MOKE signals contain two oscillating frequencies that correspond to the HF and LF modes (𝑓HF>𝑓LF). The signals are proportional to the change in 𝜃K and can be analyzed as follows: Δ𝜃K(𝑡)=𝐴+𝐵𝑒−𝑡/𝜏T+𝐶HFcos(2𝜋𝑓HF𝑡+𝛽HF)𝑒−𝑡/𝜏HF+𝐶LFcos(2𝜋𝑓LF𝑡+𝛽LF)𝑒−𝑡/𝜏LF (6) where the exponential term 𝐵𝑒−𝑡/𝜏T is related to the thermal background with 𝜏T being the time scale of heat dissipation . The rest two terms on the right -hand side are the precession terms with C, f, β, and τ denoting , respectively, the amplitude, frequency, phase, and relaxation time of the HF and LF modes. After excluding the thermal background from TR -MOKE signals, the precession is modeled with the initial conditions of step -function de creases in 𝐻k,eff,𝑖 and 𝐽𝑖, following the ultrafast laser excitation [48]. This is a reasonable approximation since the precession period (~15 -100 ps for Hext > 5 kOe) is much longer than the time scales of the laser excitation (~1.5 ps) and subsequent relaxations among electrons, magnons, and lattice (~ 1 -2 ps) [49], but much shorter than the time scale of heat dissipation -governed recovery (~400 ps). With these initial conditions , the prefactors in Eq. ( 3) can be determined (see m ore details in Note 1 of the SM [35]). For our SAF structure, 𝜃K detected by the probe beam contain s weighted contributions from both the top and bottom FM layers: 𝜃K(𝑡) 𝜃K,s=𝑤cos𝜃1(𝑡)+(1−𝑤)cos𝜃2(𝑡) (7) where 𝜃K,s represents the Kerr rotation angle when the SAF s tack is saturated along the positive out-of-plane ( z) direction. w is the weighting factor, considering the different contributions to the total MOKE signals from two FM layers. w can be obtained from static MOKE measurements [50], which gives 𝑤= 0.457 (see more details in Note 2 of the SM [35]). 11 3 RESULTS AND DISCUSSION 3.1 Field -dependent p recession frequencies and equilibrium magnetization directions TR-MOKE signals measured at varying Hext are depicted in Fig. 2 (a). The external field is tilted 15 ° away from in-plane [θH = 75°, as defined by Fig. 2 (c)] to achieve larger amplitdues of TR-MOKE signals [51]. The signals can be fitted to Eq. (6) to extract the LF and HF precession modes. The field -dependen t precession frequenc ies of both modes are summarized in Fig. 2 (b). For simplicity, when analyzing precession frequencies, magnetic damping and mutual spin pumping are neglected due to its insignificant impacts on precession frequencies. By comparing the experimental data and the prediction of ωHF/2π and ωLF/2π based on E q. (3), the effective anisotropy fields and the IEC strength are fitted as Hk,eff, 1 = 1.23 ± 0.28 kOe, Hk,eff, 2 = 6.18 ± 0.13 kOe, J1 = −0.050 ± 0.020 erg cm−2, and J2 = 0. All parameters and their determination methods are summarized in Table SI of the SM [35]. The fitted J1 is close to that obtained from the M-Hext loops (~−0.062 erg cm-2). The inset of Fig 2 (b) shows the zoom ed-in view of field -dependent precession frequencies around Hext = 8 kOe, where a n anti -crossing feature is observed: a narrow gap (~2 GHz) open s in the frequency dispersion curves of the HF and LF modes owing to the weak IEC between two FM layers. Without a ny IEC, the precession frequencies of two FM layers would cross at Hext = 8 kOe, as indicated by the green dashed line and blue dashed line in the figure. We refer to t hese two sets of crossing frequencies as the single -layer natural frequencies of two FM layers (FM 1 and FM 2) in the following discussions . 12 FIG. 2 (a) TR -MOKE signals under varying Hext when θH = 75° [as defined in panel (c)]. Circles are the experimental data and black lines are the fitting curves based on Eq. (6). (b) The precession frequencies of the HF and LF modes as functions of Hext. Circles are experimental data and solid lines are fitting curves. The inset highlights the zoomed -in view of the field -dependent frequencies around 8 kOe, where the green dashed line and blue dashed line are the single -layer (SL) precession frequencies of FM 1 and FM 2 without interlayer exchange coupling. (c) Schematic illustration of the definition of the equilibrium polar angles ( θ0,1 and θ0,2), and the direction of the external magnetic field ( θH). The illustration is equivalent to Fig. 1(b) due to symmetry. (d) θ0,1 and θ0,2 as functions of Hext. The dash -dotted line plots the difference between the two equilibrium polar angles. 13 Based on the fitted stack properties ( Hk,eff,1, Hk,eff,2, J1, and J2), the equilibrium magnetization directions in the two layers can be calculated. For SAFs with weak IEC compared with uniaxial PMA, the azimuthal angles of the magnetization in two FM layers are always the same as that of the external field at equilibrium status. Therefore, two polar angles will be sufficient to describe the equilibrium magnetization con figuration. Figure 2(c) illustrates the definition of the equilibrium polar angles of two FM layers ( θ0,1, θ0,2) and the external field ( θH). The values of θ0,1, θ0,2, and the difference between these two polar angles as functions of Hext are shown in Fig. 2(d). When Hext is low (< 1.6 kOe), magnetic anisotropy and antiferromagnetic coupling are dominant and |θ0,1 − θ0,2| is larger than 90 °. As Hext increases, both θ0,1 and θ0,2 approach θH. When Hext is high (> 15 kOe), the Zeeman energy becomes dominant and both M1 and M2 are almost aligned with Hext. 3.2 Cone angle, direction, and phase of magnetization precession revealed by modeling Besides the equilibrium configuration, using sample properties extracted from Fig. 2 (b) as input parameters, the LLG -based modeling (described in section 2.2) also provide s information o n the cone angle, direction, and phase of magnetization precession for each mode ( Fig. 1 ). The discussion in this section is limited to the case without damping an d mutual spin pumping . They will be considered in Note 4 of the SM [35], sections 3.3, and 3.4. The calculation results are shown in Fig. 3 , which are categorized into three regions. At high external fields ( Hext > 1.6 kOe, regions 2 and 3), both FM layers precess CCW [ Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°], and the polar angles of magnetization in two layers are in-phase [Arg(𝐶𝜃,2/𝐶𝜃,1)= 0°] for the HF mode and out-of-phase [Arg(𝐶𝜃,2/𝐶𝜃,1) = 180° ] for the LF mode. This is the reason for the HF mode (LF mode) also being called the acoustic mode ( optical mode) in the literature [23]. The criterion to differentiate 14 region 2 from region 3 is the FM layer that dominat es a given precessional mode (i.e., the layer with larger precession cone angles) . In region 2 (1.6 kOe < Hext < 8 kOe) , the HF mode is dominated by FM 2 because FM 2 has larger cone angles than FM 1. This is reasonable since the higher precession frequency is closer to the natural frequency of FM 2 [see Fig. 2(b)] in region 2. Similarly, in region 3, the HF mode is dominated by FM 1 with larger precession cone angles. When Hext is low (region 1), the angle between two magnetizations is larger than 90° [ Fig. 2 (d)] owing to the more dominan t AF-exchange -coupling energy as compared with the Zeeman energy . In this region, magnetization dynamics exhibits some unique features. Firstly, CW [ Arg(𝐶𝜃,𝑖/ 𝐶𝜑,𝑖)=−90°] precession emerges: for each mode, the dominant layer precesses CCW (FM 2 for the HF mode and FM 1 for the LF mode) and the subservient layer precesses CW (FM 1 for the HF mode and FM 2 for the LF mode). This is because the effective field for the subservient layer [ e.g., Heff,1 for the HF mode, see Eq. (2)] precesses CW owing to the CCW precession of the dominant layer when |𝜃0,1−𝜃0,2|>90° [Fig. 2(d)] . In other words, a low Hext that makes |𝜃0,1−𝜃0,2|> 90° is a necessary condition for the CW precession. However, it is not a sufficient condition. In general, certain degrees of symmetry breaking ( Hk,eff,1 ≠ Hk,eff,2 or the field is tilted away from the direction normal to the easy axis ) are also needed to generate CW precession. For example, for symmetric a ntiferromagnets ( Hk,eff,1 = Hk,eff,2) under fields perpendicular to the easy axis, CW precession does not appear even at low fields (Fig. 2(a) in Ref. [52]). See Note 5 of the SM [35] for more details. Secondly, as shown in Fig. 3 , the precession motions in two FM layers are always in-phase for both HF and LF modes; thus, there is no longer a clear differentiation between “acoustic mode” and “optical mo de”. Instead, the two modes can be differentiated as “right -handed” and “left -handed” based on the chirality [53]. Here, we define the chirality with respect to a reference direction taken as the projection of Hext or M2 (magnetization direction of the layer with 15 a higher Hk,eff) on the easy axis [ -z direction in Fig. 3 (c)]. Lastly, the shape of the precession cone also varies in different regions. Δ θi and Δφi are almost the same for both modes in region 3, indic ating the precession trajectories are nearly circular. While in regions 1 and 2, Δ θi and Δφi are not always equal, suggesting the precession trajectories may have high ellipticities. FIG. 3 The calculated half cone angle, direction, and phase of magnetization precession for (a) the HF mode and (b) the LF mode. In the top row, four curves represent the polar and azimuthal half cone angles of precession in two FM layers. All half cone angles are normalized with r espect to Δθ1. The middle row shows the value of Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) under different Hext. A value of 90° (−90°) represents CCW (CW) precession. The bottom row is the phase difference of the polar angles in two layers. A value of 0° (180°) corresponds to the polar angles of the magnetization in two layers are IP (OOP ) during precession. Dashed lines correspon d to the reference case where damping is zero in both layers. (c) Schematic illustrations of the cone angle, direction, and phase of 16 magnetization precession for the HF and LF modes in different regions, and their corresponding characteristics regarding ch irality and phase difference. 3.3 Amplitude and phase of TR -MOKE signals Actual magnetization dynamics is resolvable as a linear combination of the two eigenmodes (the HF and the LF modes ). By taking into account the initial conditions (i.e., laser excitation , see Note 1 of the SM [35]), we can determine the amplitude and phase of the two modes in TR -MOKE signals . Figure 4 (a) summarizes the amplitudes of both HF and LF modes [CHF and CLF in Eq. (3)] under different Hext. Noted that the y-axis represents Kerr angle ( θK) instead of the cone angle of precession. The LF mode has a local minimum near 8 kOe, where the two FM layers have similar precession cone angles but opposite phase s for the LF mode [ Fig. 3 (b)]. The amplitude s of both modes decrease with Hext in the high -field region. This is similar to the single -layer case, where the amplitudes of TR -MOKE signals decrease with Hext because the decrease in Hk,eff induced by laser heating is not able to significantly alternate the equilibrium magnetization direction when the Zeeman energy dominates [51]. The LF mode also has an amplitude peak at low fields ( Hext < 3 kOe), where the dominant layer of FM 1 changes its equilibrium direction dramatically with Hext (from ~75° to 170°) as shown in Fig. 2(d). To directly compare the amplitudes of TR -MOKE signals and the LLG -based calculations , the weighting factor w and the initial conditions are needed. The initial conditions are determined by 𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′, representing the instantan eous effective anisotropy fields and IEC strength upon laser heating. These instantaneous properties are different from their corresponding room - temperature values ( Hk,eff,1, Hk,eff,2, and J1). The accurate determination of𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′ demands the modeling of the laser heating process as well as the temperature dependence of stack properties, which are challenging. Here, we treat these three variables as adjustable parameters and 17 determine their values by fitting the field -dependent amp litudes of TR -MOKE signals , which yields 𝐻k,eff,1′𝐻k,eff,1⁄=0.90±0.01, 𝐻k,eff,2′𝐻k,eff,2⁄=0.95±0.01, and 𝐽1′𝐽1⁄=0.83±0.01. It is apparent that the field dependence of TR -MOKE signal amplitude is in excellent agreement with the theoretical modeling , as s hown in Fig. 4 (a). Figure 4 (b) shows the calculated half polar cone angles for each mode in each FM layer. In TR-MOKE signals, the optical mode (the LF mode in regions 2 and 3) tends to be partially canceled out because the two layers precess out -of-phase. Therefore, compared with Fig. 4 (a), the information in Fig. 4 (b) better reflects the actual intensity of both modes in FM 1 and FM 2. In Fig. 4(b), the precession cone angles of both modes in FM 1 (Δ𝜃1HF,Δ𝜃1LF) have local maxima at the anti-crossing field (Hext ≈ 8 kOe). On the contrary, Δ𝜃2LF and Δ𝜃2HF of FM 2 have their maxima either above or below the anti-crossing field. This is because FM 2 has larger precession amplitudes (cone angles) than FM 1 at the anti-crossing field if there is no IEC [the dotted lines of FM 1 (SL) and FM 2 (SL) in Fig. 4 (b)]. With IEC, FM 2 with larger cone angles can drive the precession motion in FM 1 significantly near the anti-crossing field, where IEC is effective. Subsequently, the precession amplitudes of FM 1 exhibit local maxima as its cone angle peaks at the anti-crossing field [solid lines in Fig. 4(b)]. Also, compared with the uncoupled case [FM 1 (SL) in Fig. 4(b)], FM 1 in the SAF structure has a much larger cone angle at the boundary between regions 1 and 2 (Hext ≈ 1.6 kOe). This corresponds to the case where FM 1 fast switch ing is driven by Hext, as shown in Fig. 2( d). The energy valley of FM 1 created by IEC and uniaxial anisotropy is canceled out by Hext. As a result, any perturbation in Hk,eff,1 or IEC can induce a large change in 𝜃1. Besides amplitude, the phase of TR -MOKE signals [ HF and LF in Eq. (6)] also provides important information about the magnetization dynamics in SAF [Fig. 4 (c)]. In Fig. 4 (c), the phase of the HF mode stays constant around π. However, the LF mode goes through a π-phase shift at 18 the transition from region 2 to region 3. Th is phase shift can be explained by the change of the dominant layer from region 2 to region 3 for the LF mode [ Fig. 3(c)]. As illustrated in Fig. 4 (d), the LF mode (optical mode in regions 2 and 3) has opposite phases in FM 1 (~0°) and FM 2 (~180°). Considering the two FM layers have comparable optical contributions to TR -MOKE signals ( w ≈ 0.5), TR -MOKE signals will reflect the phase of the dominant layer for each mode. In region 3, FM 2 has larger p recession cone angles than FM 1 for the LF mode ; therefore, LF TR-MOKE signals have the same phase as FM 2 (~180°). However, in region 2, the dominant layer shifts from FM 2 to FM 1 for the LF mode. Hence, the phase of LF TR-MOKE signals also change s by ~180° t o be consistent with the phase of FM 1 (~0°). As for the HF mode, since the two layers always have almost the same phase ( ~180°), the change of the dominant layer does not cause a shift in the phase of TR -MOKE signals. By comparing Fig. 4 (d) and Fig. 3 (a-b), one can notice that the phase difference between two FM layers could deviate from 0° or 180° when damping and mutual spin pumping is considered [Fig. 4(d)]. The deviation of phase allows energy to be transferred from one FM layer to the other during precession via exchange coupling [54]. In our sample system, FM 2 has a higher damping constant ( 𝛼1= 0.020 and 𝛼2=0.060); therefore, the net transfer of energy is from FM 1 to FM 2. More details can be found in Note 4 of the SM [35], which shows the phase of TR -MOKE signals is affected by Gilbert damping in both layers and the mutual spin pumping . By fitting the phase [Fig. 4(c)] and the damping [ Fig. 5(a) ] of TR -MOKE signals simultaneously, we obtained 𝛼sp,12 = 0.010 ± 0.004, 𝛼sp,21=0.007−0.007+0.009, 1= 0.020 ± 0.002, and 2 = 0.060 ± 0.008. Nonreciprocal spin pumping damping ( 𝛼sp,12≠𝛼sp,21) has been reported in asymmetric FM 1/NM/FM 2 trilayers and attributed to the different spin -mixing conductance ( 𝑔𝑖↑↓) at the two FM/NM interfaces [27], following 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑗↑↓/(8𝜋𝑀s,𝑖𝑑𝑖), with 𝑔𝑖 the 𝑔-factor of the i-th layer and 𝜇B the Bohr 19 magneton [55]. The above equation neglects the spin -flip scattering in NM and assumes that the spin accumulation in the NM spacer equally flows back to FM 1 and FM 2 [37]. However, the uncertainties of our 𝛼sp,𝑖𝑗 are too high to justify the nonreciprocity of 𝛼sp,𝑖𝑗 (see Note 3 of the SM [35] for detailed uncertainty analyses). In fact, if the spin backflow to FM i is proportional to 𝑔𝑖↑↓, then 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑖↑↓𝑔𝑗↑↓/[4𝜋𝑀s,𝑖𝑑𝑖(𝑔𝑖↑↓+𝑔𝑗↑↓)] (Eq. 1.14 in Ref. [56]). In this case, the different spin-mixing conductance at two FM/NM interfaces ( 𝑔1↑↓≠𝑔2↑↓) will not lead to nonreciprocal 𝛼sp,𝑖𝑗. Although differences in 𝑔𝑖 and magnetic moment per area ( 𝑀s,i𝑑𝑖) can potentially lead to nonreciprocal 𝛼sp,𝑖𝑗, the values of 𝑔𝑖 and 𝑀s,i𝑑𝑖 for the two FM layers are expected to be similar (the net magnetization of SAF is zero without external fields). Therefore, nearly reciprocal 𝛼sp,𝑖𝑗 are plausible for our SAF stack. Assu ming 𝑔𝑖↑↓ values are similar at the two FM/NM interfaces (𝑔1↑↓≈𝑔2↑↓=𝑔↑↓), this yields 𝑔↑↓ =8𝜋𝑀s,𝑖𝑑𝑖𝛼sp,𝑖𝑗/(𝑔𝑖𝜇B) = 1.2 ~ 1.7 × 1015 cm−2. 𝑔↑↓ can also be estimated from the free electron density per spin ( n) in the NM layer: 𝑔↑↓ ≈ 1.2𝑛2/3 [57]. With n = 5.2 × 1028 m−3 for Ru [58] (the value of n is similar for Ta [59]), 𝑔↑↓ is estimated to be 1.7 × 1015 cm−2, the same order as the 𝑔↑↓ value from TR -MOKE measurements, which justifies the 𝛼sp,𝑖𝑗 values derived from TR -MOKE are within a reasonable range. The values of 𝛼1 and 𝛼2 will be discussed in section 3.4. 20 FIG. 4 (a) Amplitudes of TR -MOKE signals a s functions of Hext. The circles and curves represent experimental data and modeling fitting , respectively. (b) The calculated precession half cone angles at different Hext. Red curves and black curves represent the cone angles of the HF mode and the LF mode in FM 1 (solid lines) and FM 2 (dash ed lines). Dotted lines are the precession cone angles of single -layer (SL) FM 1 and FM 2 without IEC. (c) Phases of TR -MOKE signals at varying Hext. Circles and curves are experimental data and modeling fitting (𝛼sp,12=0.010, 𝛼sp,21= 0.007, 𝛼1=0.020, 𝛼2=0.060). (d) Simulated precession phase of the HF mode (red curves) and the LF mode (black curves) in FM 1 (solid lines) and FM 2 (dash ed lines). 3.4 Magnetic damping of the HF and LF precession modes In addition to the amplitude and phase of TR -MOKE signals for the p -SAF stack, the model analyses also provide a better understanding of magnetic damping. Figure 5 (a) shows the effective damping constant ( 𝛼eff=1/2𝜋𝑓𝜏) measured at different Hext (symbols), in comparison with 21 model ing fitting (solid lines). The general Hext dependence of αeff can be well captured by the model. The fitted Gilbert damping, 1= 0.020 ± 0.002 and 2 = 0.060 ± 0.008 are close to the Gilbert damping of Ta/CoFeB(1 nm)/MgO thin films (~0.017) [41,60] and Co/Pd multilayers with a similar tCo/tPd ratio (~0.085) [61]. Other fitted parameters are Δ𝐻k,eff,1=0.26±0.02 kOe, Δ𝐻k,eff,2= 1.42±0.18 kOe, 𝛼12sp=0.010±0.004 𝛼21sp=0.007−0.007+0.009. Δ𝐽1 and Δ𝐽2 are set to be zero, as explained in Sec. 2.2. More details regarding the values and determination methods of all parameters involved in our data reduction are provided in Note 3 of the SM [35]. Dashed lines show the calculated 𝛼eff without inhomogeneous broadening. At high Hext, the difference between the solid lines and dashed lines approaches zero because the inhomogeneous broadening is suppressed. At low Hext, the solid lines are significantly higher than the dashed lines , indicating substantial inhomogeneous broadening contributions . The effective damping shows interesting features near the anti-crossing field. As shown in Fig. 5(b), due to the effective coupling between two FM layers near the anti-crossing field, the hybridization of precession in two FM layers leads to a mix of damping with contributions from both layers. The effective damping of the FM 1-dominant mode reaches a maximum within the anti-crossing region ( 7 Hext 10 kOe) and is higher than the single -layer (SL) FM 1 case. Similarly, the hybridized HF and LF modes at 8.5 kOe exhibit a lower 𝛼eff (~0.073) compared to the SL FM 2 case. eff consists of contributions from Gilbert damping ( 𝛼𝑖), mutual spin pumping (𝛼sp,𝑖𝑗, 𝑖≠𝑗), and inhomogeneous broadening ( Δ𝐻k,eff,𝑖 and Δ𝐽𝑖). To better understand the mixing damping behavior, Fig. 5 (c) shows eff after excluding the inhomogeneous contribution ( 𝛼effinhomo). Compared to the SL layer c ase (green and blue dashed lines), the HF and LF modes (red and black dashed lines) clearly suggest that IEC effectively mixes the damping in two layers around the anti - crossing field. Without the IEC, precession in FM 2 with a higher damping relaxes faster than that 22 in FM 1. However, the IEC provides a channel to transfer energy from FM 1 to FM 2, such that the two layers have the same precession relaxation rate for a given mode. Near the anti -crossing field, two layers have comparable precession cone angles; therefore, the damping values of the hybridized modes are roughly the average of two FM layers. In addition to the static IEC, dynamic spin pumping can also modify the damping of individual modes. The black and red solid lines represent the cases with mutu al spin pumping ( 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0.007). Generally, in regions 2 & 3, mutual spin pumping reduces the damping of the HF mode and increases the damping of the LF mode because the HF (LF) mode is near in -phase (out -of-phase). Overall, the static IEC still plays the essential role for the damping mix near the anti -crossing field. FIG. 5 (a) Effective damping constant under varying Hext. Circles are experimental data. Solid lines are fitting curves based on Eqs. (4 -5). Dashed lines denote eff after the removal of inhomogeneous -broadening contribution. (b) A zoomed -in figure of panel (a) between 5 kOe and 15 kOe. Blue and green circles are measured effective damping of the mode dominated by FM 1 and FM 2, respectively. Blu e and green dashed lines are the 𝛼eff of FM 1 and FM 2 single layer without IEC. (c) Effective damping after excluding the inhomogeneous contribution as a function of Hext. The HF mode (red curves) and the LF mode (black curves) are represented by solid (o r dashed) curves when the mutual spin pumping terms ( 𝛼sp,12 and 𝛼sp,21) are considered (or excluded). The d ashed green and blue lines are the SL cases for FM 1 and FM 2, respectively. 23 4 CONCLUSION We systematically investigate d the magnetization dynamics excited by ultrafast laser pulses in an asymmetric p -SAF sample both theoretically and experimentally. We obtained d etailed information regarding magnetization dynamics, including the cone angles, directions, and phases of spin precession in each layer under different Hext. In particular, the dynamic features in the low - field region (region 1) exhibiting CW precession, were revealed. The r esonance between the precession of two FM layers occurs at the boundary between regions 2 an d 3, where an anti - crossing feature is present in the frequency vs. Hext profile . The dominant FM layer for a given precession mode also switches from region 2 to region 3. The amplitude and phase of TR -MOKE signals are well captured by theoretical modeling . Importantly , we successfully quantified the individual contributions from various sources to the effective damping , which enables the determination of Gilbert damping for both FM layers. At low Hext, the contribution of inhomogeneous broadening to the effective damping is significant. Near the anti-crossing field, the effective damping of two coupled modes contains substantial contributions from both FM layers owing to the strong hybridization via IEC . Although the analyses were made for an asymme tric SAF sample, this approach can be directly applied to study magnetization dynamics and magnetic properties of general complex material systems with coupled multilayers , and thus benefits the design and optimization of spintronic materials via structural engineering. Acknowledgements This work is primarily supported by the National Science Foundation ( NSF, CBET - 2226579). D.L.Z gratefully acknowledges the funding support from the ERI program (FRANC) “Advanced MTJs for computation in and near ra ndom access memory” by DARPA, and ASCENT, one of six 24 centers in JUMP (a Semiconductor Research Corporation program, sponsored by MARCO and DARPA). J.P.W and X.J.W also appreciate the partial support from the UMN MRSEC Seed program (NSF, DMR -2011401 ). D.B.H . would like to thank the support from the UMN 2022 -2023 Doctoral Dissertation Fellowship. The authors appreciated the valuable discussion with Prof. Paul Crowell. References [1] R. Chen, Q. Cui, L. Liao, Y. Zhu, R. Zhang, H. Bai, Y. Zhou, G. Xing, F. Pan, H. Yang et al., Reducing Dzyaloshinskii -Moriya interaction and field -free spin -orbit torque switching in synthetic antiferromagnets, Nat. Commun. 12, 3113 (2021). [2] W. Legrand, D. Maccariello, F. Ajejas, S. Collin, A. Vecchiola, K. 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Magn. 48, 3288 (2012). 1 Supplement al Material for Magnetization Dynamics in Synthetic Antiferromagnets with Perpendicular Magnetic Anisotropy Dingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian -Ping Wang2, and Xiaojia Wang1,* 1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA 2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA Supplement al Note 1: Analyses of the magnetization precession in each Ferromagnetic (FM) layer For the convenience of derivation, mi is represented in the spherical coordinate s with the polar angle θi and the azimuthal angle φi, as shown in Fig. 1(b): 𝐦𝑖=(sin𝜃𝑖cos𝜑𝑖,sin𝜃𝑖sin𝜑𝑖,cos𝜃𝑖) (S1) Accordingly, t he expressi on of Eq. ( 2) in the spherical coordinate s is: { 𝜃̇1=−𝛾1 𝑑1𝑀s,1sin𝜃1∂𝐹 ∂𝜑1−𝛼1sin𝜃1𝜑̇1+𝛼sp,12sin𝜃2cos(𝜃2−𝜃1)𝜑̇2 𝜑̇1=𝛾1 𝑑1𝑀s,1sin𝜃1∂𝐹 ∂𝜃1+𝛼1 sin𝜃1𝜃̇1−𝛼sp,12 sin𝜃1𝜃̇2 𝜃̇2=−𝛾2 𝑑2𝑀s,2sin𝜃2∂𝐹 ∂𝜑2−𝛼2sin𝜃2𝜑̇2+𝛼sp,21sin𝜃1cos(𝜃1−𝜃2)𝜑̇1 𝜑̇2=𝛾2 𝑑2𝑀s,2sin𝜃2∂𝐹 ∂𝜃2+𝛼2 sin𝜃2𝜃̇2−𝛼sp,21 sin𝜃2𝜃̇1 (S2) *Author s to whom correspondence should be addressed : huan1746@umn.edu and wang4940@umn.edu 2 where, a dot over variables represents a derivative with respect to time. When Mi precesses around its equilibrium direction: {𝜃𝑖=𝜃0,𝑖+Δ𝜃𝑖 𝜑𝑖=𝜑0,𝑖+Δ𝜑𝑖 (S3) with i and i representing the deviation angles of Mi from its equilibrium direction along the polar and azimuthal directions. Assuming the deviation is small, under the first -order approximation, the first -order partial derivative of F in Eq. (S2) can be expanded as: { ∂𝐹 ∂𝜃𝑖≈∂2𝐹 ∂𝜃𝑖2Δ𝜃𝑖+∂2𝐹 ∂𝜑𝑖∂𝜃𝑖Δ𝜑𝑖+∂2𝐹 ∂𝜃𝑗∂𝜃𝑖Δ𝜃𝑗+∂2𝐹 ∂𝜑𝑗∂𝜃𝑖Δ𝜑𝑗 ∂𝐹 ∂𝜑𝑖≈∂2𝐹 ∂𝜃𝑖𝜕𝜑𝑖Δ𝜃𝑖+∂2𝐹 ∂𝜑𝑖2Δ𝜑𝑖+∂2𝐹 ∂𝜃𝑗∂𝜑𝑖Δ𝜃𝑗+∂2𝐹 ∂𝜑𝑗∂𝜑𝑖Δ𝜑𝑗 (S4) By substituting Eq. ( S4), Equation ( S2) is linearized as [1]: [ Δ𝜃̇1 Δ𝜑̇1 Δ𝜃̇2 Δ𝜑̇2] =𝐊[Δ𝜃1 Δ𝜑1 Δ𝜃2 Δ𝜑2] (S5) where, K is a 4×4 matrix, con sisting of the properties of individual FM layers and the second - order derivatives of F in terms of 𝜃1,𝜑1,𝜃2,and𝜑2. Equation (S5) has four eigen -solutions, in the form of 𝐶exp(𝑖𝜔𝑡), corresponding to four precession frequencies: ±𝜔HF and ±𝜔LF. A pair of eigen -solutions with the same absolute precession frequency are physically equivalent. Therefore, only two eigen -solutions need to be considered: {Δ𝜃𝑖=𝐶𝜃,𝑖HFexp(𝑖𝜔HF𝑡) Δ𝜑𝑖=𝐶𝜑,𝑖HFexp(𝑖𝜔HF𝑡) and {Δ𝜃𝑖=𝐶𝜃,𝑖LFexp(𝑖𝜔LF𝑡) Δ𝜑𝑖=𝐶𝜑,𝑖LFexp(𝑖𝜔LF𝑡) (S6) After r earrange ment , the full solutions in the spherical coordinates are expressed as below (also Eq. (3) in the main paper). 3 [𝜃1(𝑡) 𝜑1(𝑡) 𝜃2(𝑡) 𝜑2(𝑡)]= [ 𝜃0,1 𝜑0,1 𝜃0,2 𝜑0,2] +[Δ𝜃1(𝑡) Δ𝜑1(𝑡) Δ𝜃2(𝑡) Δ𝜑2(𝑡)]=[𝜃0,1 𝜑0,1 𝜃0,2 𝜑0,2]+ [ 𝐶𝜃,1HF 𝐶𝜑,1HF 𝐶𝜃,2HF 𝐶𝜑,2HF] exp(𝑖𝜔HF𝑡)+ [ 𝐶𝜃,1LF 𝐶𝜑,1LF 𝐶𝜃,2LF 𝐶𝜑,2LF] exp(𝑖𝜔LF𝑡) (S7) The prefactors of these eigen -solutions provide information about magnetization dynamics of both the HF and LF modes. Directly from solving Eq. (S2), one can obtain the relative ratios of these prefactors , which are [𝐶𝜑1HF/𝐶𝜃1HF,𝐶𝜃2HF/𝐶𝜃1HF,𝐶𝜑2HF/𝐶𝜃1HF] and [𝐶𝜑1LF /𝐶𝜃1LF ,𝐶𝜃2LF /𝐶𝜃1LF ,𝐶𝜑2LF /𝐶𝜃1LF ]. These ratios provide precession information of each mode, as presented in Fig. 3. Obtaining the absolute values of [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 for each mode requires the initial conditions of precession , which i s necessary for fitting the actual precession amplitudes in TR - MOKE signals . In TR -MOKE measurements, magnetization precession is initiated by laser heating, which reduces the magnetic anisotropy of each FM layer and the interlayer exchange coupling streng th between two FM layers [2]. Considering the laser heating process is ultrafast compared with magnetization precession while the following cooling due to heat dissipation is much slower than magnetization dynamics, we approximately model the temporal profiles of effective anisotropy fields and exchange coupling as step functions. Owing to the sudden change in magnetic properties induced by laser heating , magnetization in each layer will establish a new equilibrium direction (𝜃0,𝑖′,𝜑0,𝑖′). In other words, M i deviates from its new eq uilibrium direction by Δ𝜃𝑖=𝜃0,𝑖−𝜃0,𝑖′, Δ𝜑𝑖=𝜑0,𝑖−𝜑0,𝑖′. Substituting 𝑡=0 to Eq. ( S7), one can get the initial conditions for magnetization dynamics: Δ𝜃𝑖(𝑡=0)=𝐶𝜃,𝑖HF+𝐶𝜃,𝑖LF=𝜃0,𝑖−𝜃0,𝑖′ Δ𝜑𝑖(𝑡=0)=𝐶𝜑,𝑖HF+𝐶𝜑,𝑖LF=𝜑0,𝑖−𝜑0,𝑖′=0 (S8) Once the initial conditions are set, the absolute values of all prefactors can be obtained . 4 Supplementa l Note 2: Estimation of each layer’s contribution to total TR -MOKE signals The contribution from each FM layer is estimated by static MOKE measurement. According to Ref. [3], the resu lt from this method matches well with that from the optical calculation. The sample is perpendicularly saturated before the static MOKE measurement. Then the out -of-plane M-Hext loop ( Fig. S1) is measured by static MOKE. As shown in the figure, two different antiferromagnetic (AF) configurations have different normalized MOKE signals, indicating the different contribution s to the total signals by two layers. The weighting factor is calculated by: −𝑤+(1−𝑤)=0.085 (S9) which gives 𝑤=0.457. Considering the relatively small layer thicknesses [FM 1: CoFeB(1), spacer: Ru(0.6)/Ta(0.3), and FM 2: Co(0.4)/Pd(0.7)/Co(0.4)], it is reasonable that FM 1 and FM 2 make comparable contributions to the total TR -MOKE signals ( i.e., w ≈ 0.5). FIG. S1 Static MOKE hysteresis loop. Magnetic fields are applied along the out -of-plane direction. 5 Supplemental Note 3: Summary of the parameters and uncertainties for data reduction Given that a number of variables are involved in the analysis, TABLE SI summarizes the major variables discussed in the manuscript, along with their values and determinatio n methods. TABLE SI. Summary of the values and determination methods of parameters used in the data reduction. The reported uncertainties are one -sigma uncertainties from the mathematical model fitting to the TR -MOKE measurement data. Parameters Values Determination Methods Hf ~500 Oe VSM Ms,1 1240 emu cm−3 VSM Ms,2 827 emu cm−3 VSM d1 1 nm Sample structure d2 1.5 nm Sample structure Hk,eff,1 1.23 ± 0.28 kOe Fitted from f vs. Hext [Fig. 2(b)] Hk,eff,2 6.18 ± 0.13 kOe Fitted from f vs. Hext [Fig. 2(b)] γ1 17.79 ± 0.04 rad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)] γ2 17.85 ± 0.04 rad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)] J1 −0.050 ± 0.020 erg cm−2 Fitted from f vs. Hext [Fig. 2(b)] J2 0 Fitted from f vs. Hext [Fig. 2(b)] w 0.457 Static MOKE 𝐻k,eff,1′/𝐻k,eff,1 0.90 ± 0.01 Fitted from Amp vs. Hext [Fig. 4(a)] 𝐻k,eff,2′/𝐻k,eff,2 0.95 ± 0.01 Fitted from Amp vs. Hext [Fig. 4(a)] 𝐽1′/𝐽1 0.83 ± 0.01 Fitted from Amp vs. Hext [Fig. 4(a)] 𝛼1 0.020 ± 0.002 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] 𝛼2 0.060 ± 0.008 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] Δ𝐻k,eff,1 0.26 ± 0.02 kOe Fitted from eff vs. Hext [Fig. 5(a)] Δ𝐻k,eff,2 1.42 ± 0.18 kOe Fitted from eff vs. Hext [Fig. 5(a)] 𝛼sp,12 0.010 ± 0.004 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] 𝛼sp,21 0.007−0.007+0.009 Fitted from eff vs. Hext [Fig. 5(a)] and vs. Hext [Fig. 4(c)] 6 Supplemental Note 4: Impacts of 𝜶𝟏, 𝜶𝟐, and mutual spin pumping on the phase Without damping, the phase difference in the precession polar angles of two FM layers [Arg(𝐶𝜃2/𝐶𝜃1)] is always 0° or 180°, as shown in Fig. 3 of the main article. However, this does not necessarily hold if either the damping or mutual spin pumping is considered. The changes in the phase difference due to damping are depicted in Fig. S2. When 1 = 2, the phase difference between two layers stays at 0° or 180° [ Fig. S2(a)], identical to the lossless case ( 1 = 2 = 0) in Fig. 3. As a result, the initial phase of TR -MOKE signals ( ) also stays at 0° or 180° [ Fig. S2(b)]. However, when 𝛼1≠𝛼2, Arg(𝐶𝜃2/𝐶𝜃1) deviates from 0° or 180° especially at high fields ( Hext > 5 kOe) [ Fig. S2(c,e)]. The layer with a higher damping [FM 1 in (c) or FM 2 in (e)] tends to have a more advanced phase at high fields (regions 2 and 3). For example, in Fig. S2(e), 0° < Arg( 𝐶𝜃2/𝐶𝜃1) < 180° for both HF and LF modes in regions 2 and 3. The deviation from the perfect in -phase (0°) or out -of-phase (180°) condition allows the IEC to transfer energy from the low -damping layer to the high -damping layer, such that the precession in both layers can damp at the same rate [4]. As a result, the initial phase of the TR -MOKE signals also changes, which opens a negative or positive gap at high fields (> 10 kOe) for both modes, as shown in Fig. S2(d,f). This enables us to determine the difference between 1 and 2 by analyzing the in itial phase of TR -MOKE signals. 7 FIG. S 2 Impact of 𝛼1 and 𝛼2 on the phase without mutual spin pumping. (a,c,e) The phase difference between the polar angles in two layers for HF and LF modes. (b,d,f) The calculated initial phase of TR -MOKE signals for each mode with 1 = 2 = 0.02 (a,b), 1 = 0.06 and 2 = 0.02 (c,d), and 1 = 0.02 and 2 = 0.06 (e,f). The mutual spin pumping is set as 𝛼sp,12=𝛼sp,21 = 0 for all three cases. The rest of the parameters used in this calculation can be found in TABLE SI. The impact of mutual spin pumping on the precession phase is illustrated in Fig. S3, where three different cases of either the one -way (𝛼sp,12 or 𝛼sp,21) or two -way (both 𝛼sp,12 and 𝛼sp,21) spin pumping are considered. A reference case without the consideration of mutual spin pumping (1 = 0.02, 2 = 0.06, and 𝛼sp,12= 𝛼sp,21 = 0) is also plotted (dashed curves) for the ease of comparison. In general, it can be seen that mutual spin pumping could also change the phase difference in the precession polar angles of two layers, and thus the initial phase o f TR -MOKE signals noticeably. This can be explained by the damping modification resulting from spin pumping. In regions 2 and 3, Eq. (2) can be approximately rearranged as: 8 𝑑𝐦𝑖 𝑑𝑡≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼𝑖𝐦𝑖×𝑑𝐦𝑖 𝑑𝑡−𝐶𝑗 𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)𝐦𝑖×𝑑𝐦𝑖 𝑑𝑡 ≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+[𝛼𝑖−𝐶𝑗 𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)]𝐦𝑖×𝑑𝐦𝑖 𝑑𝑡 =−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼̅𝑖𝐦𝑖×𝑑𝐦𝑖 𝑑𝑡 (S10) where 𝐶𝑗/𝐶𝑖 represents the ratio of the cone angles in the j-th FM layer to the i-th FM layer. 𝐶𝑗/𝐶𝑖 is positive for the in -phase mode and negative for the out -of-phase mode. θ0,1 and θ0,2 are the equilibrium polar angle s of M1 and M2, as defined in Fig. 2(c). Therefore, the mutual spin -pumping term either enhances or reduces the damping depending on the mode. 𝛼̅𝑖 = 𝛼𝑖− 𝐶𝑗 𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2) represents the effective Gilbert damping in the i-th FM layer after considering the mutual spin -pumping effect. This modification to damping is more significant when the i-th layer is subservient with a smaller cone angle ( e.g., FM 2 for the HF mode in region 3), while the j-th layer is dominant with a mu ch larger precession cone angle ( e.g., FM 1 for the LF mode in region 3), leading to a large ratio of |𝐶𝑗/𝐶𝑖|. In Fig. S 3(a), only the spin current injected from FM 1 to FM 2 is considered. According to the above analysis, 𝛼sp,21 can only bring noticeable modifications to the damping of FM 2 when FM 1 is the dominant layer. Based on Fig. 3 in the main article, the LF mode in region 2 and HF mode in region 3 satisfy this condition (FM 1 dominant and FM 2 subservient). As shown in Fig. S 3(a), the phase difference noticeably deviates from the reference case without mutual spin pumping (dashed curves) in region 2 for the LF mode (black curves) and in region 3 for the HF mode (red curves). For the LF mode in region 2, the precession motions in two layers are nearly o ut-of-phase (negative C1/C2); therefore, the spin pumping from FM 1 enhances the damping in FM 2. Since 1 (0.02) is less than 2 (0.06), the spin pumping from FM 1 to FM 2 further increases | 𝛼̅1 − 𝛼̅2| between the two layers. Consequently, the phase difference shifts further away from 180°. While 9 for the HF mode in region 3, 𝛼sp,21 reduces the damping of FM 2 because C1/C2 is positive resulting from the near in -phase feature of this mode. Hence, | 𝛼̅1 − 𝛼̅2| becomes smaller and the phase difference gets closer to 0°. In Fig. S 3(c), only 𝛼sp,12 is considered, which requires FM 2 as the dominant layer (the HF mode in region 2 and LF mode in region 3) for noticeable changes in | 𝛼̅1 − 𝛼̅2|. For the HF mode in region 2, spin pumping from FM 2 reduces 𝛼̅1 given that the precession motions in two layers are nearly in phase (positive C2/C1). Therefore, | 𝛼̅1 − 𝛼̅2| increases and the phase difference in Fig. S 3(c) shifts further away from 0° in region 2. However, for the LF mode in regions 3, the nearly out -of-phase precession in two FM layers (negative C1/C2) increases 𝛼̅1 and reduces | 𝛼̅1 − 𝛼̅2|. As a result, the phase difference in Fig. S 3(c) shifts toward 180°. When both 𝛼sp,12 and 𝛼sp,21 are considered [ Fig. S 3(e)], a combined effect is expected for the phase difference with noticeable changes for both the HF and LF modes in regions 2 and 3. The impacts of mutual spin pumping on the phase difference between the HF and LF modes are reflected by the initial phase of TR -MOKE signals [ in Fig. S 3(b,d,f)]. Compared with the reference case without mutual spin pumping (dashed curves), the introduction of mutual spin pumping tends to change the gap in between the two modes. As shown in Fig. S3(e,f), the values of two mutual -spin-pumping induced damping terms are chosen as 𝛼sp,12 = 0.013 and 𝛼sp,21 = 0.004, such that the gap of the initial phase of TR -MOKE signals is closed at high fields (region 3). Therefore, the initial phase of TR -MOKE signals provides certain measurement sensitivities to 𝛼sp,12 and 𝛼sp,21, which enables us to extract the values of 𝛼sp,𝑖𝑗 from measurement fitting. Here, we acknowledge that the measurement sensitivity to 𝛼sp,𝑖𝑗 from TR - MOKE is limited, which subsequently leads to relatively large error bars for 𝛼sp,𝑖𝑗 (see Table SI). 10 FIG. S3 Impact of mutual spin pumping on the phase with fixed damping values of 1 = 0.02 and 2 = 0.06. (a,c,e) The phase difference between the polar angles in two layers for HF and LF modes. (b,d,f) The calculated initial phase of TR -MOKE signals ( ) for each mode with 𝛼sp,12= 0 and 𝛼sp,21 = 0.01 (a,b), 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0 (c,d), and 𝛼sp,12 = 0.013 and 𝛼sp,21 = 0.004 (e,f). For the third case (e,f), the values of mutual spin pumping are chosen to close the gap in panel (f) for Hext > 15 kOe. The rest of the parameters used in this calculation can be found in TABLE SI. Dashed lines represent the reference case without mutual spin pumping ( 1 = 0.02, 2 = 0.06, and 𝛼sp,12= 𝛼sp,21 = 0). Supplemental Note 5: Region diagram s for p -SAFs with different degrees of asymmetries Figure S4 shows the region diagrams for p -SAFs with different degrees of asymmetries , represented by the difference of Hk,eff in two FM layers. Hk,eff,1 = Hk,eff,2 corresponds to the symmetric case (lowest asymmetry), as shown by Fig. S4(c). While the SAF in Fig. S4(a) has the highest asymmetry: Hk,eff,1 = 2 kOe, Hk,eff,2 = 6 kOe. Figure S4 clearly shows that |𝜃0,1−𝜃0,2|> 90° is a necessary but not sufficient condition for region 1 (CW precession). Because regions 2 or 3 also appear to the left of the red cu rve (where |𝜃0,1−𝜃0,2|>90°), especially when θH is close to 90° and Hk,eff,1 is close to Hk,eff,2. 11 FIG. S4 Region diagrams of p -SAFs with different degrees of asymmetries: Hk,eff,1 = 2 kOe, Hk,eff,2 = 6 kOe (a), Hk,eff,1 = 4 kOe, Hk,eff,2 = 6 kOe (b), Hk,eff,1 = 6 kOe, Hk,eff,2 = 6 kOe (c). The blue background represents region 1. The green background covers regions 2 and 3. The red curve shows the conditions where |𝜃0,1−𝜃0,2|=90°. |𝜃0,1−𝜃0,2|>90° to the left of the red curve. 𝛼1, 𝛼2, 𝛼sp,12, and 𝛼sp,21 are set as zero. 𝛾1=𝛾2=17.8 rad ns−1 kOe−1. Values of the rest parameters are the same as those in Table SI. References [1] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Angular dependence of ferromagnetic resonance in exchange -coupled Co/Ru/Co trilayer structures, Phys. Rev. B 50, 6094 (1994). [2] W. Wang, P. Li, C. Cao, F. Liu, R. Tang, G. Chai, and C. Jiang, Temperature dependence of interlayer exchange coupling and Gilbert damping in synthetic antiferromagnetic trilayers investigated using broadband ferromagnetic resonance, Appl. Phys. Lett. 113, 042401 (2018). [3] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M. Swagten, and B. Koopmans, Control of speed and efficiency of ultrafast demagnetization by direct transfer of spin angular momentum, Nat. Phys. 4, 855 (2008). [4] D. H. Zanette, Energy exchange between coupled mechanical oscil lators: linear regimes, J. Phys. Commun. 2, 095015 (2018).

2022-11-14

Understanding the rich physics of magnetization dynamics in perpendicular synthetic antiferromagnets (p-SAFs) is crucial for developing next-generation spintronic devices. In this work, we systematically investigate the magnetization dynamics in p-SAFs combining time-resolved magneto-optical Kerr effect (TR-MOKE) measurements with theoretical modeling. These model analyses, based on a Landau-Lifshitz-Gilbert approach incorporating exchange coupling, provide details about the magnetization dynamic characteristics including the amplitudes, directions, and phases of the precession of p-SAFs under varying magnetic fields. These model-predicted characteristics are in excellent quantitative agreement with TR-MOKE measurements on an asymmetric p-SAF. We further reveal the damping mechanisms of two procession modes co-existing in the p-SAF and successfully identify individual contributions from different sources, including Gilbert damping of each ferromagnetic layer, spin pumping, and inhomogeneous broadening. Such a comprehensive understanding of magnetization dynamics in p-SAFs, obtained by integrating high-fidelity TR-MOKE measurements and theoretical modeling, can guide the design of p-SAF-based architectures for spintronic applications.

Magnetization Dynamics in Synthetic Antiferromagnets with Perpendicular Magnetic Anisotropy

2211.07744v2

arXiv:2105.09879v2 [math.AP] 19 Mar 2024On the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity Alessandro Palmieri Department of Mathematics, University of Bari, Via E. Orabo na 4, 70125 Bari, Italy March 20, 2024 Abstract In this note, we derive a blow-up result for a semilinear gene ralized Tricomi equation with damping and mass terms having time-dependent coefficients. W e consider these coefficients with critical decay rates. Due to this threshold nature of the tim e-dependent coefficients (both for the damping and for the mass), the multiplicative constants app earing in these lower order terms strongly influence the value of the critical exponent, deter mining a competition between a Fujita- type exponent and a Strauss-type exponent. Keywords Critical exponent, Fujita exponent, Strauss exponent, Blo w-up, Power nonlinearity AMS Classification (2020) 35B33, 35B44, 35L15, 35L71. 1 Introduction In the present note, we prove a blow-up result for local in tim e weak solutions to the following semilinear Cauchy problem with power nonlinearity |u|p ∂2 tu−t2ℓ∆u+µt−1∂tu+ν2t−2u=|u|p, x ∈Rn, t> 1, u(1,x) =εu0(x), x ∈Rn, ∂tu(1,x) =εu1(x), x ∈Rn,(1) whereℓ>−1,µ,ν2are nonnegative real constants, p>1 andεis a positive constant describing the size of Cauchy data. We consider the case with initial data ta ken at the time t= 1, nevertheless, the results that we are going to prove are valid for data taken at a ny initial time t=t0>0. Forℓ= 0 andν2= 0, the linearized equation associated with the equation in (1) is known as Euler-Poisson-Darboux equation (see the introduction of [6] for a detailed overview on the li terature regarding this model), while for µ=ν2= 0 the second-order operator ∂2 t−t2ℓ∆ on the left-hand side of the equation in (1) is called generalized Tricomi operator . Motivated by these special cases and for the sake of brevity, we will call the equation in (1) semilinear Euler-Poisson-Darboux-Tricomi equation (semilinear EPDT equation). Over the last two decades, several papers have been devoted t o the study of semilinear models with power nonlinearity, that are special cases of (1) for gi ven values of the parameters ℓ,µ,ν2. Concerning the Cauchy problem associated with the semiline ar generalized Tricomi equation ∂2 tu−t2ℓ∆u=f(u,∂ tu), x ∈Rn, t> 0, u(0,x) =εu0(x), x ∈Rn, ∂tu(0,x) =εu1(x), x ∈Rn,(2) we recall [10, 42, 43] for the first results in the case with pow er nonlinearity f(u,∂ tu) =|u|p. More specifically, in [10] some blow-up results are proved for wea k solutions to the Cauchy problem associ- ated with the generalized Tricomi operator (and, more gener ally, for Grushin-type operators) by mean 1of the test function method. The so-called quasi-homogeneous dimension of the generalized Tricomi operator Q= (ℓ+ 1)n+ 1 made its appearance in the upper bound for pin these results (see [7, 25] for the definition of the quasi-homogeneous dimension for a m ore general partial differential opera- tor). On the other hand, in [42, 43] the fundamental solution for the generalized Tricomi operator (sometimes called also Gellerstedt operator in the literat ure) is employed to derive an integral repre- sentation formula, that is used in turn to prove (under suita ble assumptions on p) the local existence of solutions and the global existence of small data solution s, respectively. Afterwards, in the series of papers [16, 17, 18, 41, 19, 35] it was established that the critical exponent for (2) with f(u,∂ tu) = |u|pandn/greaterorequalslant2 is the Strauss-type exponent (cf. [34] and the literature citing this renowned paper), that we denote pStr(n,ℓ) in the present paper, given by the biggest root of the quadratic equation /parenleftbiggn−1 2+ℓ 2(ℓ+ 1)/parenrightbigg p2−/parenleftbiggn+ 1 2−3ℓ 2(ℓ+ 1)/parenrightbigg p−1 = 0. (3) We recall that for us the fact that pStr(n,ℓ) is the critical exponent for (2) with f(u,∂ tu) =|u|pmeans the following: for any 1 <p<p Str(n,ℓ) local in time solutions blow up in finite time under suitable sign assumptions for the Cauchy data and regardless of their size, while for p > p Str(n,ℓ) (technical upper bounds for pmay appear, depending on the space for the solutions) a globa l in time existence result for small data solutions holds. Very recently, even the cases with derivative type nonlinea rityf(u,∂ tu) =|∂tu|pand with mixed nonlinearity f(u,∂ tu) =|u|q+|∂tu|phave been studied from the point of view of blow-up dynamics i n [26, 3, 23, 13]. In particular, in [26] a blow-up result when f(u,∂ tu) =|∂tu|pis proved for 1 <p/lessorequalslantQ Q−2. Regarding the semilinear Euler-Poisson-Darboux equation and, more in general, the semilinear wave equation with scale-invariant damping and mass (i.e. ( 1) forℓ= 0), combining the contributions from many different authors (see [5, 8, 27, 20, 31, 33, 9, 24, 6, 4] and references therein) we may reasonably conjecture that for nonnegative µ,ν2such that the quantity δ.= (µ−1)2−4ν2(4) satisfiesδ/greaterorequalslant0 the critical exponent is given by max/braceleftBig pStr(n+µ,0), pFuj/parenleftBig n+µ−1 2−√ δ 2/parenrightBig/bracerightBig . (5) HerepFuj(n).= 1 +2 ndenotes the so-called Fujita exponent (cf. [11]) and its presence can be justified as a consequence of diffusion phenomena between the solution s to the corresponding linearized model and those to some suitable parabolic equation. We point out t hat the condition δ/greaterorequalslant0 implies somehow that the damping term µt−1∂tuhas a dominant influence over the mass term ν2t−2u. Although the proof of the necessity part of this conjecture is fully demon strated (see [31, 33, 24]), for the sufficiency part only the one-dimensional case was recently completely clarified [6], while in the higher dimensional case only a few special cases have been studied (often in the r adially symmetric case). We emphasize that for ℓ=−2 3andµ= 2,ν2= 0 the model in (1) is the semilinear wave equation in the Einstein-de Sitter spacetime with power nonlinearity (see [12]). Hence, for the semiline ar wave equation in the generalized Einstein-de Sitter spacetime, i.e., whenℓ∈(−1,0) andµ/greaterorequalslant0,ν2= 0 in (1), in the series of papers [29, 30, 37, 38] several blow-up r esults are proved provided that p >1 is below max/braceleftBig pStr/parenleftBig n+µ ℓ+1,ℓ/parenrightBig , pFuj((ℓ+ 1)n)/bracerightBig . (6) Moreover, under the same conditions for the parameters as ab ove, also the case with derivative type nonlinearity |∂tu|pis studied in [14, 15, 40]. Furthermore, we point out that in [ 39, 40] even the case ℓ/lessorequalslant−1 withν2= 0 is studied for different semilinear terms. The purpose of the present paper is twofold: on the one hand, w e want to prove the necessity part for the one-dimensional case in (1) that together with t he sufficiency part from [6] (cf. Corollary 5.1) will show the optimality of our result for n= 1; on the other hand, we want to generalize the condition that determines the critical exponent for (1) in t he generaln-dimensional case, obtaining as a candidate to be critical an exponent that is consistent wit h all previously mentioned special cases. Finally, we mention that in [2] the semilinear Cauchy proble m with derivative type nonlinearity and the same linear partial differential operator as in (1) is con sidered, and a Glassey-type exponent is found as the upper bound for the exponent in the blow-up range . 2Notations Throughout the paper we denote by φℓ(t).=tℓ+1 ℓ+ 1(7) the primitive of the speed of propagation tℓthat vanishes for t= 0. In particular, the amplitude of the light-cone for the Cauchy problem with data prescribed a t the initial time t0= 1 is given by the functionφℓ(t)−φℓ(1). The ball in Rnwith radius Raround the origin is denoted BR. The notation f/lessorsimilargmeans that there exists a positive constant Csuch thatf/lessorequalslantCgand, analogously, f/greaterorsimilarg. Finally, as in the introduction, we will denote by pFuj(n) the Fujita exponent and by pStr(n,ℓ) the Strauss-type exponent. 1.1 Main results Let us begin this section by introducing the notion of weak so lutions to (1) that we will employ throughout the entire paper. Definition 1.1. Letu0,u1∈L1 loc(Rn) such that supp u0,suppu1⊂BRfor someR>0. We say that u∈C/parenleftbig [1,T),W1,1 loc(Rn)/parenrightbig ∩C1/parenleftbig [1,T),L1 loc(Rn)/parenrightbig ∩Lp loc/parenleftbig (1,T)×Rn/parenrightbig is aweak solution to (1) on [1 ,T) ifu(1,·) =εu0inL1 loc(Rn),ufulfills the support condition suppu(t,·)⊂BR+φℓ(t)−φℓ(1) for anyt∈(1,T), (8) and the integral identity ˆ Rn∂tu(t,x)φ(t,x) dx+ˆt 1ˆ Rn/parenleftbig −∂tu(s,x)φs(s,x) +s2ℓ∇u(s,x)· ∇φ(s,x)/parenrightbig dxds +ˆt 1ˆ Rn/parenleftbig µs−1∂tu(s,x)φ(s,x) +ν2s−2u(s,x)φ(s,x)/parenrightbig dxds =εˆ Rnu1(x)φ(1,x) dx+ˆt 1ˆ Rn|u(s,x)|pφ(s,x) dxds (9) holds for any t∈(1,T) and any test function φ∈C∞ 0/parenleftbig [1,T)×Rn/parenrightbig . We notice that, performing further steps of integration by p arts in (9), we get the integral relation ˆ Rn/parenleftbig ∂tu(t,x)φ(t,x)−u(t,x)φt(t,x) +µt−1u(t,x)φ(t,x)/parenrightbig dx +ˆt 1ˆ Rnu(s,x)/parenleftbig φss(s,x)−s2ℓ∆φ(s,x)−µs−1φs(s,x) + (µ+ν2)s−2φ(s,x)/parenrightbig dxds =εˆ Rn/parenleftbig (u1(x) +µu0(x))φ(1,x)−u0(x)φt(1,x)/parenrightbig dx+ˆt 1ˆ Rn|u(s,x)|pφ(s,x) dxds (10) for anyt∈(1,T) and any test function φ∈C∞ 0/parenleftbig [1,T)×Rn/parenrightbig . Let us state our result in the sub-critical case. Theorem 1.2. Letℓ>−1andµ,ν2/greaterorequalslant0such thatδ/greaterorequalslant0. Let us assume that the exponent pof the nonlinear term satisfies 1<p< max/braceleftBig pStr/parenleftBig n+µ ℓ+1,ℓ/parenrightBig ,pFuj/parenleftBig (ℓ+ 1)n+µ−1 2−√ δ 2/parenrightBig/bracerightBig . Letu0,u1∈L1 loc(Rn)be nonnegative, nontrivial and compactly supported functi ons with supports contained in BRfor someR>0such that u1+µ−1−√ δ 2u0/greaterorequalslant0. (11) Letu∈C/parenleftbig [1,T),W1,1 loc(Rn)/parenrightbig ∩C1/parenleftbig [1,T),L1 loc(Rn)/parenrightbig ∩Lp loc/parenleftbig (1,T)×Rn/parenrightbig be a weak solution to (1) according to Definition 1.1 with lifespan T=T(ε). 3Then, there exists a positive constant ε0=ε0(u0,u1,n,p,ℓ,µ,ν2,R)such that for any ε∈(0,ε0] the weak solution ublows up in finite time. Furthermore, the upper bound estimat es for the lifespan T(ε)/lessorequalslant Cε−p(p−1) θ(n,ℓ,µ,p ) ifp<p Str/parenleftbig n+µ ℓ+1,ℓ/parenrightbig , Cε−/parenleftbig 2 p−1−/parenleftbig (ℓ+1)n+µ−1 2−√ δ 2/parenrightbig/parenrightbig−1 ifp<p Fuj/parenleftbig (ℓ+ 1)n+µ−1 2−√ δ 2/parenrightbig ,(12) holds, where the positive constant Cis independent of εand θ(n,ℓ,µ,p ).=ℓ+ 1 +/parenleftBig n+1 2(ℓ+ 1) +µ−3ℓ 2/parenrightBig p−/parenleftBig n−1 2(ℓ+ 1) +ℓ+µ 2/parenrightBig p2. (13) Remark 1.In the previous statement it might happen that the argument o f the Fujita exponent is a nonpositive number (however, only for ℓ <0 andµ∈[0,1)). Whenever this happens, we do not require any upper bound for p>1, meaning formally that pFuj(k) =∞fork/lessorequalslant0. Remark 2.The exponent pStr/parenleftbig n+µ ℓ+1,ℓ/parenrightbig is obtained from the Strauss-type exponent pStr(n,ℓ) defined through (3) by a shift of magnitudeµ ℓ+1in the space dimension. Equivalently, pStr/parenleftbig n+µ ℓ+1,ℓ/parenrightbig is the positive root to the quadratic equation /parenleftbiggn−1 2(ℓ+ 1) +ℓ+µ 2/parenrightbigg p2−/parenleftbiggn+ 1 2(ℓ+ 1) +µ−3ℓ 2/parenrightbigg p−(ℓ+ 1) = 0. Finally, we provide a blow-up result when we consider the cri tical Fujita-type exponent. Theorem 1.3. Letℓ>−1andµ,ν2/greaterorequalslant0such thatδ/greaterorequalslant0. Let us assume that the exponent pof the nonlinear term satisfies p=pFuj/parenleftBig (ℓ+ 1)n+µ−1 2−√ δ 2/parenrightBig . Letu0,u1∈L1 loc(Rn)be nonnegative, nontrivial and compactly supported functi ons with supports contained in BRfor someR>0. Letu∈C/parenleftbig [1,T),W1,1 loc(Rn)/parenrightbig ∩C1/parenleftbig [1,T),L1 loc(Rn)/parenrightbig ∩Lp loc/parenleftbig (1,T)×Rn/parenrightbig be a weak solution to (1)according to Definition 1.1 with lifespan T=T(ε). Then, there exists a positive constant ε0=ε0(u0,u1,n,p,ℓ,µ,ν2,R)such that for any ε∈(0,ε0] the weak solution ublows up in finite time. Moreover, the upper bound estimates f or the lifespan T(ε)/lessorequalslant/braceleftBigg exp/parenleftbig Eε−(p−1)/parenrightbig ifδ >0, exp/parenleftbig Eε−(p−1)/p/parenrightbig ifδ= 0,(14) holds, where the positive constant Eis independent of ε. 2 Proof of Theorem 1.2 In this section, we prove Theorem 1.2 by deriving a sequence o f lower bound estimates for the space average of a weak solution uto (1). More precisely, introducing the functional U(t).=ˆ Rnu(t,x) dx fort∈[1,T), our aim is to determine estimates from below for U. Letting to ∞the index of this sequence of lower bounds, we establish that the space average of ucannot be globally in time defined and we determine as a byproduct an upper bound estimate for the lifespan. In pa rticular, the first two steps that we need to carry out in order to apply the iteration argument are determining the iteration frame for U(namely, an integral inequality, where Uappears both on the left and right-hand side) and a first lower bound estimate for U. In order to establish such a first lower bound estimate for Uwe consider a suitable positive solution to the adjoint equation to the l inear EPDT equation. Finally, we employ this first lower bound estimate for Uto begin the iteration procedure, plugging it in the iterati on frame. Then, repeating iteratively the procedure we determ ine the desired sequence of lower bounds. 42.1 Derivation of the iteration frame In this subsection we derive the iteration frame for U. To this purpose we employ the double multiplier technique from [33] (see also [21, 32]). Given t∈(1,T), let us begin by choosing as test function in (9) a cut-off function that localizes the forward light-cone , namely, we take φ∈C∞ 0/parenleftbig [1,T)×Rn/parenrightbig such thatφ= 1 in {(s,x)∈[1,t]×Rn:|x|/lessorequalslantR+φℓ(s)−φℓ(1)}. Therefore, from (9) we get ˆ Rn∂tu(t,x) dx+ˆt 1ˆ Rn/parenleftbig µs−1∂tu(s,x) +ν2s−2u(s,x)/parenrightbig dxds =εˆ Rnu1(x) dx+ˆt 1ˆ Rn|u(s,x)|pdxds. Differentiating with respect to tthe previous equality, we obtain ˆ Rn|u(t,x)|pdx=ˆ Rn∂2 tu(t,x) dx+µt−1ˆ Rn∂tu(t,x) dx+ν2t−2ˆ Rnu(t,x) dx =U′′(t) +µt−1U′(t) +ν2t−2U(t). (15) If we denote by r1,r2the roots of the quadratic equation r2−(µ−1)r+ν2= 0, then, we can rewrite the right-hand side of the last relation as follows U′′(t) +µt−1U′(t) +ν2t−2U(t) =t−(r2+1)d dt/parenleftBig tr2+1−r1d dt/parenleftBig tr1U(t)/parenrightBig/parenrightBig . (16) We emphasize that the role of r1andr2is fully interchangeable in the previous identity. Combini ng (15) and (16), after some straightforward steps we arrive at U(t) =Ulin(t) +ˆt 1/parenleftBigs t/parenrightBigr1ˆs 1/parenleftBigτ s/parenrightBigr2+1ˆ Rn|u(τ,x)|pdxdτds, (17) where Ulin(t).=/braceleftBigg r1t−r2−r2t−r1 r1−r2U(1) +t−r2−t−r1 r1−r2U′(1) ifδ>0, t−r1(1 +r1lnt)U(1) +t−r1lntU′(1) ifδ= 0.(18) Let us set r1.=µ−1−√ δ 2, r 2.=µ−1+√ δ 2, where the definition of δis given in (4). From here on, these will be the fixed values of r1,r2. Consequently, from (17) we have a twofold result. On the one h and, we get the lower bound estimate forU U(t)/greaterorequalslantIεt−r1fort∈[1,T), (19) where the multiplicative constant Idepends on the positive quantities´ Rnu0(x) dxand´ Rnu1(x) dx. On the other hand, by using again the nonnegativity and nontr iviality ofu0,u1, we find U(t) =ˆt 1/parenleftBigs t/parenrightBigr1ˆs 1/parenleftBigτ s/parenrightBigr2+1ˆ Rn|u(τ,x)|pdxdτds (20) /greaterorsimilarˆt 1/parenleftBigs t/parenrightBigr1ˆs 1/parenleftBigτ s/parenrightBigr2+1 (R+φℓ(τ)−φℓ(1))−n(p−1)(U(τ))pdτds /greaterorsimilart−r1ˆt 1sr1−r2−1ˆs 1τr2+1(1 +τ)−n(ℓ+1)( p−1)(U(τ))pdτds. Note that in the last chain of inequalities we used the suppor t condition for uand Jensen’s inequality. Hence, we obtained the following the iteration frame for U U(t)/greaterorequalslantCt−r1ˆt 1sr1−r2−1ˆs 1τr2+1τ−n(ℓ+1)( p−1)(U(τ))pdτds (21) for a suitable positive constant Cthat depends on n,p,ℓ . 52.2 Solution of the adjoint equation In the previous subsection we established a first lower bound estimate for Uin (19). This estimate will be the starting point for the proof of the blow-up result for p<p Fuj/parenleftbig (ℓ+1)n+µ−1 2−√ δ 2/parenrightbig . However, to prove the blow-up result for p<p Str/parenleftbig n+µ ℓ+1,ℓ/parenrightbig we need to determine a further lower bound estimate forU. According to this purpose, we introduce a second auxiliary time-dependent functional, which is a certain weighted spatial average of u. In particular, we are going to choose the weight function to be a positive solution of the adjoint equation to the linea r EPDT equation, namely, ∂2 sψ−s2ℓ∆ψ−∂ ∂s(µs−1ψ) +ν2s−2ψ= 0. (22) In order to determine a suitable solution to (22), we follow t he approach from [30, Subsection 2.1]. We look for a solution to (22) with separate variables. As x-dependent function we consider the function ϕ(x).=/braceleftBigg ex+ e−xifn= 1,´ Sn−1ex·ωdσωifn/greaterorequalslant2, that has been introduce for the study of blow-up phenomena fo r semilinear hyperbolic models in [44]. The positive function ϕ∈C∞(Rn) satisfies ∆ ϕ=ϕand ϕ(x)∼cn|x|−n−1 2e|x|as|x| → ∞, (23) wherecnis a positive constant depending on n. On the other hand, as s-dependent function we look for a positive solution to the se cond-order linear ODE ̺′′(s)−s2ℓ̺(s)−µs−1̺′(s) + (µ+ν2)s−2̺(s) = 0. (24) Let us perform the change of variables σ=φℓ(s). Then,̺solves the previous equation if and only if σ2d2̺ dσ2+ℓ−µ ℓ+ 1σd̺ dσ+/parenleftbiggµ+ν2 (ℓ+ 1)2−σ2/parenrightbigg ̺= 0. (25) Next, we consider the transformation ̺(σ) =σαη(σ) withα.=µ+1 2(ℓ+1). Hence,̺solves (25) if and only ifηis a solution to σ2d2η dσ2+σdη dσ−/parenleftBig σ2+δ 4(ℓ+ 1)2/parenrightBig η= 0. (26) As solution to (26) we choose the modified Bessel function of t he second kind K√ δ 2(ℓ+1)(σ). Consequently, we set as positive solutions to (24) ̺(s).=sµ+1 2K√ δ 2(ℓ+1)(φℓ(s)). (27) Note that̺=̺(s;ℓ,µ,ν2), but for the sake of brevity we will skip the dependence on ℓ,µ,ν2in the notations hereafter. Therefore, we may define now the follow ing function as positive solution to (22) ψ(s,x).=̺(s)ϕ(x). (28) By using the weight function ψ, we introduce the auxiliary functional U0(t).=ˆ Rnu(t,x)ψ(t,x) dx. Notice that thanks to the support condition for the solution ufrom Definition 1.1, it is possible to employψas test function in (10). Consequently, using (22), we get ˆ Rn/parenleftbig ∂tu(t,x)ψ(t,x)−u(t,x)ψt(t,x) +µt−1u(t,x)ψ(t,x)/parenrightbig dx =εˆ Rn/parenleftbig ̺(1)(u1(x) +µu0(x))−̺′(1)u0(x)/parenrightbig ϕ(x)dx+ˆt 1ˆ Rn|u(s,x)|pψ(s,x) dxds. (29) 6Applying the recursive relation for the derivative of the mo dified Bessel function of the second kind ∂zKγ(z) =−Kγ+1(z) +γ zKγ(z) (cf. [28, Equations (10.29.1)]), we have ̺′(s) =−sµ+1 2+ℓK√ δ 2(ℓ+1)+1(φℓ(s)) +µ+1+√ δ 2sµ−1 2K√ δ 2(ℓ+1)(φℓ(s)). Thus, Iℓ,µ,ν2[u0,u1].=ˆ Rn/parenleftbig ̺(1)u1(x) + (̺(1)µ−̺′(1))u0(x)/parenrightbig ϕ(x)dx =ˆ Rn/parenleftBig K√ δ 2(ℓ+1)+1(φℓ(1))u0(x) + K√ δ 2(ℓ+1)(φℓ(1))/parenleftbig u1(x) +µ−1−√ δ 2u0(x)/parenrightbig/parenrightBig ϕ(x)dx>0, where we employed the nonnegativity and the nontriviality o fu0and (11). Hence, we can rewrite (29) as εIℓ,µ,ν2[u0,u1] +ˆt 1ˆ Rn|u(s,x)|pψ(s,x) dxds=U′ 0(t) +µt−1U0(t)−2̺′(t) ̺(t)U0(t) =̺2(t) tµd dt/parenleftbiggtµ ̺2(t)U0(t)/parenrightbigg . Since both ψand the nonlinear term are nonnegative, from the previous re lation we obtain U0(t)/greaterorequalslantεIℓ,µ,ν2[u0,u1]̺2(t) tµˆt 1sµ ̺2(s)ds>0 for any t∈[1,T). Thanks to the assumptions on the Cauchy data we have shown tha t the functional U0is nonnegative. Next, we shall determine a lower bound estimate for U0for large times. For this reason we recall the asymptotic behavior of K γfor large arguments, namely, K γ(z) =/radicalbig π/(2z)e−z(1 +O(z−1)) asz→ ∞ forz >0 (cf. [28, Equation (10.25.3)]). So, there exists T0=T0(ℓ,µ,ν2)>1 such that for s/greaterorequalslantT0it holds 1 4π(ℓ+ 1) e−2φℓ(s)sµ−ℓ/lessorequalslant̺2(s)/lessorequalslantπ(ℓ+ 1) e−2φℓ(s)sµ−ℓ. (30) Then, fort/greaterorequalslantT0we have U0(t)/greaterorequalslantεIℓ,µ,ν2[u0,u1]̺2(t) tµˆt T0sµ ̺2(s)ds /greaterorequalslantε 4Iℓ,µ,ν2[u0,u1]t−ℓe−2φℓ(t)ˆt T0sℓe2φℓ(s)ds. Fort/greaterorequalslant2T0, it results U0(t)/greaterorsimilarεIℓ,µ,ν2[u0,u1]t−ℓe−2φℓ(t)ˆt t/2sℓe2φℓ(s)ds /greaterorsimilarεIℓ,µ,ν2[u0,u1]t−ℓ/parenleftbig 1−e2φℓ(t/2)−2φℓ(t)/parenrightbig =εIℓ,µ,ν2[u0,u1]t−ℓ/parenleftBig 1−e−2 ℓ+1(2ℓ+1−1)tℓ+1/parenrightBig /greaterorsimilarεIℓ,µ,ν2[u0,u1]t−ℓ. We emphasize that in the last step we use the condition ℓ >−1 to estimate from below the factor containing the exponential term with a positive constant. Summarizing, we proved U0(t)/greaterorsimilarεt−ℓfort/greaterorequalslant2T0, (31) where the unexpressed multiplicative constant depends on t he Cauchy data and on the parameters ℓ,µ,ν2. Finally, we show how from (31) we derive a second lower bound estimate for U. By Hölder’s inequality, we find for t/greaterorequalslant2T0 εt−ℓ/lessorsimilarU0(t)/lessorequalslant/parenleftbiggˆ Rn|u(t,x)|pdx/parenrightbigg1/p/parenleftBiggˆ BR+φℓ(t)−φℓ(1)(ψ(t,x))p′dx/parenrightBigg1/p′ , 7wherep′denotes the conjugate exponent of p. Following [31, Section 3] and employing (30), we can estimate for t/greaterorequalslant2T0 ˆ Rn|u(t,x)|pdx/greaterorsimilarεpt−ℓp/parenleftBiggˆ BR+φℓ(t)−φℓ(1)(ψ(t,x))p′dx/parenrightBigg−(p−1) /greaterorsimilarεpt−ℓp(̺(t))−pe−p(R+φℓ(t)−φℓ(1))(R+φℓ(t)−φℓ(1))−(n−1)(p−1)+n−1 2p /greaterorsimilarεpt(−n−1 2(ℓ+1)−ℓ+µ 2)p+(n−1)(ℓ+1). Plugging this last estimate from below for the space integra l of the nonlinear term in (20), for any t/greaterorequalslantT1.= 2T0+ 1 we get U(t)/greaterorequalslantKεpt(−n−1 2(ℓ+1)−ℓ+µ 2)p+(n−1)(ℓ+1)+2, (32) where the multiplicative constant Kdepends on the Cauchy data and on n,p,ℓ,µ,ν2,R, which is the desired lower bound estimate for U. 2.3 Iteration argument In this section we establish the following sequence of lower bound estimates for U U(t)/greaterorequalslantCjt−αj(t−T1)βjfort/greaterorequalslantT1, (33) where {αj}j∈N,{βj}j∈Nand{Cj}j∈Nare sequences of positive real numbers that we will determin e iteratively during the proof. Let us begin by considering the case 1 <p<p Str/parenleftbig n+µ ℓ+1,ℓ/parenrightbig . As we have previously mentioned, we consider (32) as first lower bound estimate for Uforpbelow the Strauss-type exponent. Therefore, (33) forj= 0 is satisfied provided that we set C0.=Kεp,α0.= [n−1 2(ℓ+1)+ℓ+µ 2]pandβ0.= (n−1)(ℓ+1)+2. Next, we assume that (33) holds true for some j/greaterorequalslant0 and we want to prove it for j+ 1. If we plug the lower bound estimate for Uin (33) in the iteration frame (21), we get U(t)/greaterorequalslantCt−r1ˆt T1sr1−r2−1ˆs T1τr2+1−n(ℓ+1)( p−1)(U(τ))pdτds /greaterorequalslantCCp jt−r1ˆt T1sr1−r2−1ˆs T1(τ−T1)r2+1+ pβjτ−n(ℓ+1)( p−1)−pαjdτds /greaterorequalslantCCp jt−r2−1−n(ℓ+1)( p−1)−pαjˆt T1ˆs T1(τ−T1)r2+1+ pβjdτds /greaterorequalslantCCp j (r2+ 3 +pβj)2t−r2−1−n(ℓ+1)( p−1)−pαj(t−T1)r2+3+ pβj, where we used the relations r1−r2−1 =−(√ δ+ 1)<0 andr2+ 1>0. Hence, if we define Cj+1.=CCp j(r2+ 3 +pβj)−2, (34) αj+1.=r2+ 1 +n(ℓ+ 1)(p−1) +pαj, β j+1.=r2+ 3 +pβj, (35) then, we proved (33) for j+ 1 as well. Let us determine explicitly αjandβj. By using recursively the definition in (35), we find αj=r2+ 1 +n(ℓ+ 1)(p−1) +pαj−1=...= (r2+ 1 +n(ℓ+ 1)(p−1))j−1/summationdisplay k=0pk+pjα0 =/parenleftbiggr2+ 1 p−1+n(ℓ+ 1) +α0/parenrightbigg pj−r2+ 1 p−1−n(ℓ+ 1), (36) and, analogously, βj=/parenleftbiggr2+ 3 p−1+β0/parenrightbigg pj−r2+ 3 p−1. (37) 8In particular, combining (34) and (37), we get Cj=CCp j−1(r2+ 3 +pβj−1)−2=CCp j−1β−2 j/greaterorequalslantC/parenleftbiggr2+ 3 p−1+β0/parenrightbigg−2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright.=DCp j−1p−2j=DCp j−1p−2j for anyj/greaterorequalslant1. Applying the logarithmic function to both sides of the ine qualityCj/greaterorequalslantDCp j−1p−2jand employing iteratively the resulting relation, we have lnCj/greaterorequalslantplnCj−1−2jlnp+ lnD /greaterorequalslantp2lnCj−2−2(j+ (j−1)p) lnp+ (1 +p) lnD /greaterorequalslant.../greaterorequalslantpjlnC0−2 lnp/parenleftBiggj−1/summationdisplay k=0(j−k)pk/parenrightBigg + lnDj−1/summationdisplay k=0pk =pj/parenleftbigg lnC0−2plnp (p−1)2+lnD p−1/parenrightbigg +2 lnp p−1j+2plnp (p−1)2−lnD p−1, where we used the identity j−1/summationdisplay k=0(j−k)pk=1 p−1/parenleftbiggpj+1−p p−1−j/parenrightbigg . Letj0=j0(n,ℓ,µ,ν2,p)∈Nbe the smallest integer such that j0/greaterorequalslantlnD 2 lnp−p p−1. Then, for any j/greaterorequalslantj0it results lnCj/greaterorequalslantpj/parenleftbigg lnC0−2plnp (p−1)2+lnD p−1/parenrightbigg =pjln/parenleftbig/tildewideDεp/parenrightbig , (38) where/tildewideD.=KD1/(p−1)p−(2p)/(p−1)2. Finally, from (33) we can prove that Ublows up in finite time since the lower bound diverges asj→ ∞ and, besides, we can derive an upper bound estimate for the li fespan of the solution. Combining (36), (37) and (38), for j/greaterorequalslantj0andt/greaterorequalslantT1we obtain U(t)/greaterorequalslantexp/parenleftBig pjln/parenleftbig/tildewideDεp/parenrightbig/parenrightBig t−αj(t−T1)βj /greaterorequalslantexp/parenleftBig pj/parenleftBig ln/parenleftbig/tildewideDεp/parenrightbig −/parenleftBig r2+1 p−1+n(ℓ+ 1) +α0/parenrightBig lnt+/parenleftBig r2+3 p−1+β0/parenrightBig ln(t−T1)/parenrightBig/parenrightBig tr2+1 p−1+n(ℓ+1)(t−T1)−r3+1 p−1. Fort>2T1it holds the relation ln( t−T1)/greaterorequalslantlnt−ln 2, consequently, for j/greaterorequalslantj0we get U(t)/greaterorequalslantexp/parenleftBig pj/parenleftBig ln/parenleftbig/hatwideDεp/parenrightbig +/parenleftBig 2 p−1+β0−α0−n(ℓ+ 1)/parenrightBig lnt/parenrightBig/parenrightBig tr2+1 p−1+n(ℓ+1)(t−T1)−r3+1 p−1, where/hatwideD.= 2−r2+3 p−1−β0/tildewideD. Let us write explicitly the factor that multiplies ln tin the previous estimate 2 p−1+β0−α0−n(ℓ+ 1) =2 p−1+ (n−1)(ℓ+ 1) + 2 −/bracketleftBig n−1 2(ℓ+ 1) +ℓ+µ 2/bracketrightBig p−n(ℓ+ 1) =1 p−1/braceleftBig −/bracketleftBig n−1 2(ℓ+ 1) +ℓ+µ 2/bracketrightBig p(p−1)−(ℓ+ 1)(p−1) + 2p/bracerightBig =1 p−1/braceleftBig −/bracketleftBig n−1 2(ℓ+ 1) +ℓ+µ 2/bracketrightBig p2+/bracketleftBig n+1 2(ℓ+ 1) +µ−3ℓ 2/bracketrightBig p+ℓ+ 1/bracerightBig =θ(n,ℓ,µ,p ) p−1, whereθ(n,ℓ,µ,p ) is defined in (13) and is a positive quantity thanks to the con ditionp<p Str(n+µ ℓ+1,ℓ) on the exponent of the nonlinear term. Then, for j/greaterorequalslantj0andt/greaterorequalslant2T1we arrived at U(t)/greaterorequalslantexp/parenleftBig pj/parenleftBig ln/parenleftBig /hatwideDεptθ(n,ℓ,µ,p ) p−1/parenrightBig/parenrightBig/parenrightBig tr2+1 p−1+n(ℓ+1)(t−T1)−r3+1 p−1. (39) 9Let us fixε0=ε0(u0,u1,n,p,ℓ,µ,ν2,R) such that 0 <ε 0<(2T1)−θ(n,ℓ,µ,p )/(p(p−1))/hatwideD−1/p. Then, for anyε∈(0,ε0] and anyt>/hatwideD−(p−1)/θ(n,ℓ,µ,p )ε−p(p−1)/θ(n,ℓ,µ,p )we have t>2T1and ln/parenleftBig /hatwideDεptθ(n,ℓ,µ,p ) p−1/parenrightBig >0, so, lettingj→ ∞ in (39), the lower bound for Ublows up. Thus, we proved that Uis not finite for t/greaterorsimilarε−p(p−1)/θ(n,ℓ,µ,p ), that is, we showed the lifespan estimate in (12) for p<p Str(n+µ ℓ+1,ℓ). The case 1<p<p Fuj/parenleftbig (ℓ+ 1)n+µ−1 2−√ δ 2/parenrightbig can be treated in a completely analogous way. Indeed, employing (19) in place of (32), so that, C0=Iεandβ0−α0=1−µ 2+√ δ 2, we determine the following lower bound estimate for Uinstead of (39) U(t)/greaterorequalslantexp/parenleftBig pj/parenleftBig ln/parenleftBig /tildewideCεt2 p−1−(ℓ+1)n+β0−α0/parenrightBig/parenrightBig/parenrightBig tr2+1 p−1+n(ℓ+1)(t−T1)−r3+1 p−1 for anyt/greaterorequalslant2T1and forjgreater than a suitable j1(n,ℓ,µ,ν2,p)∈N, where/tildewideCis a certain positive constant. Thanks to the assumption on p, from this estimate we conclude the validity of the second upper bound estimate in (12) by repeating the same argument a s in the first case. Remark 3.In the case δ= 0 and for p < p Fuj((ℓ+ 1)n+r1), from (18) we see that we can actually improve the lower bound estimate (19) by a logarithmic facto r. By using a slicing procedure as in the next section, it is possible to prove the following upper bou nd estimate for the lifespan T(ε)2 p−1−((ℓ+1)n+r1)lnT(ε)/lessorsimilarε−1. 3 Proof of Theorem 1.3 In order to prove Theorem 1.3 we derive a sequence of lower bou nd estimates for U(t) with additional logarithmic factors. According to this purpose, we introdu ce{ℓj}j∈Nsuch thatℓj.= 2−2−(j+1). We are going to use this sequence to apply a slicing procedure in the iteration argument, following the ideas introduced in the paper [1]. More precisely, we establ ish the following estimates U(t)/greaterorequalslantKjt−r1/parenleftbigg ln/parenleftbiggt ℓj/parenrightbigg/parenrightbiggγj (40) for anyj∈Nand anyt/greaterorequalslantℓj, where {Kj}j∈N,{γj}j∈Nare sequences of nonnegative numbers to be determined iteratively. From (18) we obtain that U(t)/greaterorequalslant/braceleftBigg Iεt−r1 ifδ>0, Iεt−r1lntifδ= 0, which implies the validity of (40) for j= 0 provided that K0.=Iεand γ0.=/braceleftBigg 0 ifδ>0, 1 ifδ= 0. Let us proceed with the inductive step. We plug (40) for some j∈Nin (21) and we prove the validity of (40) forj+ 1, prescribing suitable recursive relations for the terms Kj+1andγj+1. Therefore, for t/greaterorequalslantℓjwe have U(t)/greaterorequalslantCKp jt−r1ˆt ℓjsr1−r2−1ˆs ℓjτr2+1−n(ℓ+1)( p−1)−r1p/parenleftBig ln/parenleftBig τ ℓj/parenrightBig/parenrightBigpγj dτds /greaterorequalslantCKp jt−r1ˆt ℓjsr1−r2−1−[(ℓ+1)n+r1](p−1)ˆs ℓjτr2−r1+1/parenleftBig ln/parenleftBig τ ℓj/parenrightBig/parenrightBigpγj dτds. Fort/greaterorequalslantℓj+1, we can shrink the interval of integration in the τ-integral as follows U(t)/greaterorequalslantCKp jt−r1ˆt ℓj+1sr1−r2−1−[(ℓ+1)n+r1](p−1)ˆs ℓjs ℓj+1τr2−r1+1/parenleftBig ln/parenleftBig τ ℓj/parenrightBig/parenrightBigpγj dτds /greaterorequalslantCKp jt−r1ˆt ℓj+1sr1−r2−1−[(ℓ+1)n+r1](p−1)/parenleftBig ln/parenleftBig s ℓj+1/parenrightBig/parenrightBigpγjˆs ℓjs ℓj+1/parenleftBig τ−ℓjs ℓj+1/parenrightBigr2−r1+1 dτds =C(r2−r1+ 2)−1/parenleftBig 1−ℓj ℓj+1/parenrightBigr2−r1+2 Kp jt−r1ˆt ℓj+1s1−[(ℓ+1)n+r1](p−1)/parenleftBig ln/parenleftBig s ℓj+1/parenrightBig/parenrightBigpγj ds. 10Due top=pFuj((ℓ+ 1)n+r1), the power of sis actually −1 in the last integral, so, for t/greaterorequalslantℓj+1we obtain U(t)/greaterorequalslantC(r2−r1+ 2)−1/parenleftBig 1−ℓj ℓj+1/parenrightBigr2−r1+2 Kp jt−r1ˆt ℓj+1s−1/parenleftBig ln/parenleftBig s ℓj+1/parenrightBig/parenrightBigpγj ds =C(r2−r1+ 2)−1/parenleftBig 1−ℓj ℓj+1/parenrightBigr2−r1+2 Kp j(pγj+ 1)−1t−r1/parenleftBig ln/parenleftBig t ℓj+1/parenrightBig/parenrightBigpγj+1 , which is exactly (40) for j+ 1, provided that γj+1=pγj+ 1, Kj+1=C(r2−r1+ 2)−1/parenleftBig 1−ℓj ℓj+1/parenrightBigr2−r1+2 (pγj+ 1)−1Kp j. Applying iteratively the recursive relation γj=pγj−1+ 1, we obtain γj=pjγ0+j−1/summationdisplay k=0pk=/parenleftBig γ0+1 p−1/parenrightBig pj−1 p−1(41) /lessorequalslant/parenleftBig γ0+1 p−1/parenrightBig pj. Moreover, 1 −ℓj−1 ℓj>2−(j+2). Consequently, combining the previous considerations, fo r anyj/greaterorequalslant1 we have Kj/greaterorequalslantMQ−jKp j−1, whereM.=C(r2−r1+ 2)−12−2(r2−r1+2)/parenleftBig γ0+1 p−1/parenrightBig−1 andQ.= 2r2−r1+2p. Applying the logarithmic function to both sides of the previous inequality and employ ing recursively the obtained estimate, we get lnKj/greaterorequalslantplnKj−1−jlnQ+ lnM /greaterorequalslantp2lnKj−2−(j+ (j−1)p) lnQ+ (1 +p) lnM /greaterorequalslant.../greaterorequalslantpjlnK0−lnQ/parenleftBiggj−1/summationdisplay k=0(j−k)pk/parenrightBigg + lnMj−1/summationdisplay k=0pk =pj/parenleftbigg lnK0−plnQ (p−1)2+lnM p−1/parenrightbigg +lnQ p−1j+plnQ (p−1)2−lnM p−1. Letj2=j2(n,µ,ν2,ℓ)∈Nbe the smallest integer such that j1/greaterorequalslantlnM lnQ−p p−1. Therefore, for any j∈N,j/greaterorequalslantj2it holds lnKj/greaterorequalslantpjln(/tildewiderMε), (42) where/tildewiderM.=IM1/(p−1)Q−p/(p−1)2. Finally, we combine (40), (41) and (42), so for any j/greaterorequalslantj2and anyt/greaterorequalslant2/greaterorequalslantℓjwe find U(t)/greaterorequalslantKjt−r1/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbig(γ0+1 p−1)pj−1 p−1 = exp/bracketleftBig lnKj+/parenleftBig γ0+1 p−1/parenrightBig pjln/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbig/bracketrightBig t−r1/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbig−1 p−1 /greaterorequalslantexp/bracketleftBig pj/parenleftBig ln(/tildewiderMε) +/parenleftBig γ0+1 p−1/parenrightBig ln/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbig/parenrightBig/bracketrightBig t−r1/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbig−1 p−1 = exp/bracketleftBig pjln/parenleftBig /tildewiderMε/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbigγ0+1/(p−1)/parenrightBig/bracketrightBig t−r1/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbig−1 p−1. Since ln(t 2)/greaterorequalslant1 2lntfort/greaterorequalslant4, forj/greaterorequalslantj2andt/greaterorequalslant4 we arrive at U(t)/greaterorequalslantexp/bracketleftBig pjln/parenleftBig /hatwiderMε(lnt)γ0+1/(p−1)/parenrightBig/bracketrightBig t−r1/parenleftbig ln/parenleftbigt 2/parenrightbig/parenrightbig−1 p−1, (43) where/hatwiderM.= 2−(γ0+1/(p−1))/tildewiderM. 11Let us fixε0=ε(u0,u1,n,ℓ,µ,ν2,R)>0 such that ε0/lessorequalslant(2 ln 2)−(γ0+1/(p−1))/hatwiderM−1. Then, for any ε∈(0,ε0] and anyt>exp/parenleftBig (/hatwiderMε)−(γ0+1/(p−1))−1/parenrightBig we have t/greaterorequalslant4 and ln/parenleftBig /hatwiderMε(lnt)γ0+1/(p−1)/parenrightBig >0, thus, letting j→ ∞ in (43) we obtain that the lower bound for U(t) blows up. Hence, we proved that for lnt/greaterorsimilarε−(γ0+1/(p−1))−1the average U(t) may not be finite. This completes the proof and shows the upper bound estimate (14) for the lifespan. 4 Concluding remarks In this final section, we analyze the result obtained in Theor em 1.2. Setting pc(n,ℓ,µ,ν2).= max/braceleftbigg pStr/parenleftBig n+µ ℓ+1,ℓ/parenrightBig ,pFuj/parenleftbigg (ℓ+ 1)n+µ−1 2−√ (µ−1)2−4ν2 2/parenrightbigg/bracerightbigg , from Theorem 1.2 we know that a blow-up result holds for 1 <p<p c(n,ℓ,µ,ν2) provided that δ/greaterorequalslant0 and that the compactly supported Cauchy data fulfill suitabl e sign assumptions. As in the case of the semilinear wave equation with scale-invariant damping and mass, the condition δ/greaterorequalslant0 implies that the damping term is dominant over the mass one. From a technical v iewpoint, this assumption guarantees the possibility to use the double multiplier technique (cf. [33, 21]) while deriving the iteration frame (21). Combining the blow-up result from this paper with the global existence result for small data solutions in [6, Corollary 5.1], it follows that pc(n,ℓ,µ,ν2) is the critical exponent for (1) when n= 1. Moreover, it is reasonable to conjecture that the exponent pc(n,ℓ,µ,ν2) is critical even for higher dimensions. Indeed, for µ=ν2= 0 we have that pc(n,ℓ,0,0) =pStr(n,ℓ) forn/greaterorequalslant2 andpc(n,ℓ,0,0) = pFuj(ℓ) forn= 1 (see [19, Remark 1.6] for the one-dimensional case) accor ding to the results for (2) with power nonlinearity that we recalled in the introductio n. In particular, when ℓ= 0 too we find thatpc(n,0,0,0) is the solution to the quadratic equation ( n−1)p2−(n+ 1)p−2 = 0, namely, the celebrated exponent named after the author of [34] which is t he critical exponent for the semilinear wave equation. Furthermore, we point out that for ℓ= 0 and for ν2= 0,ℓ∈(−1,0) the exponent pc(n,ℓ,µ,ν2) coincides with (5) and (6), respectively. 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[37] Tsutaya K., Wakasugi Y., Blow up of solutions of semilin ear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime. J. Math. Phys. 61, 091503 (2020). https://doi.org/10.1063/1.5139301 [38] Tsutaya K., Wakasugi Y., On heatlike lifespan of soluti ons of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime. J. Math. Anal. Appl. 500(2), 125133 (2021). https://doi.org/10.1016/j.jmaa.2021.125133 [39] Tsutaya K., Wakasugi Y., Blow up of solutions of semilin ear wave equations in accelerated expanding Friedmann-Lemaître-Robertson-Walker spaceti me.Reviews in Mathematical Physics 34(2): 2250003 (2022). https://doi.org/10.1142/S0129055X 22500039 [40] Tsutaya K., Wakasugi Y., On Glassey’s conjecture for se milinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime. Bound. Value. Probl. 2021 : 94 (2021) https://doi.org/10.1186/s13661-021-01571-0 [41] Tu Z., Lin J., Lifespan of semilinear generalized Trico mi equation with Strauss type exponent. 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2021-05-20

In this note, we derive a blow-up result for a semilinear generalized Tricomi equation with damping and mass terms having time-dependent coefficients. We consider these coefficients with critical decay rates. Due to this threshold nature of the time-dependent coefficients (both for the damping and for the mass), the multiplicative constants appearing in these lower-order terms strongly influence the value of the critical exponent, determining a competition between a Fujita-type exponent and a Strauss-type exponent.

On the the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity

2105.09879v2

A SECOND-ORDER NUMERICAL METHOD FOR LANDAU-LIFSHITZ-GILBERT EQUATION WITH LARGE DAMPING PARAMETERS YONGYONG CAI, JINGRUN CHEN, CHENG WANG, AND CHANGJIAN XIE Abstract. A second order accurate numerical scheme is proposed and imple- mented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of equations with constant coecients where fast solvers are available, so that the numerical eciency has been greatly im- proved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great eciency improvement will be veried by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and nds physically reasonable structures while many existing methods have failed. For the do- main wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic eld will be obtained through the reported simulations. 1.Introduction Ferromagnetic materials are widely used for data storage due to the bi-stable states of the intrinsic magnetic order or magnetization. The dynamics of magneti- zation has been modeled by the Landau-Lifshitz-Gilbert (LLG) equation [9,13]. In particular, two terms are involved in the dynamics of the LLG equation: the gyro- magnetic term, which is energetically conservative, and the damping term, which is energetically dissipative. The damping term is important since it strongly aects the energy required and the speed at which a magnetic device operates. A recent experiment on a magnetic- semiconductor heterostructure [25] has indicated that the Gilbert damping constant can be adjusted. At the microscopic level, the electron scattering, the itinerant electron relaxation [11], and the phonon-magnon coupling [16, 17] are responsible to the damping, which can be obtained from electronic structure calculations [19]. For the application purpose, tuning the damping parameter allows one to optimize the magneto-dynamic properties in the material, such as lowering the switching current and increasing the writing speed of magnetic memory devices [23]. While most experiments have been devoted to small damping parameters [4,14, 22], large damping eects are observed in [10,18]. The magnetization switching time Date : May 11, 2021. 2010 Mathematics Subject Classication. 35K61, 65N06, 65N12. Key words and phrases. Micromagnetics simulations, Landau-Lifshitz-Gilbert equation, second-order method, large damping parameter. 1arXiv:2105.03576v1 [physics.comp-ph] 8 May 20212 Y. CAI, J. CHEN, C. WANG, AND C. XIE tends to be shorter in the presence of the large damping constant [18]. Extremely large damping parameters ( 9) are presented in [10]. The LLG equation is a vectorial and nonlinear system with the xed length of magnetization in a point-wise sense. Signicant eorts have been devoted to design ecient and stable numerical methods for micromagnetics simulations; see [6, 12] for reviews and references therein. Among the existing numerical works, semi- implicit schemes have been very popular since they avoid a complicated nonlinear solver while preserving the numerical stability; see [2, 7, 24], etc. In particular, the second-order accurate backward dierentiation formula (BDF) scheme is con- structed in [24], with a one-sided interpolation. In turn, a three-dimensional lin- ear system needs to be solved at each time step, with non-constant coecients. Moreover, a theoretical analysis of the second order convergence estimate has been established in [5] for such a BDF2 method. As another approach, a linearly implicit method in [2] introduces the tangent space to deal with the length constraint of magnetization, with the rst-order temporal accuracy. As a further extension, high- order BDF schemes have been constructed and analyzed in a more recent work [1]. An unconditionally unique solvability of the semi-implicit schemes has been proved in [1,5], while the convergence analysis has required a condition that the temporal step-size is proportional to the spatial grid-size. However, an obvious disadvantage has been observed for these semi-implicit schemes: the vectorial structure of the LLG equation leads to a non-symmetric linear system at each time step, which cannot be implemented by an FFT-based fast solver. In fact, the GMRES is often used, while its eciency depends heavily on the temporal step-size and the spatial grid-size, and extensive numerical experiments have indicated much more expensive computational costs than standard Poisson solvers [24]. The Gauss-Seidel projection method (GSPM) is another popular set of numerical algorithms since only linear systems with constant coecients need to be solved at each time step [8,15,21]. This method is based on a combination of a Gauss-Seidel update of an implicit solver for the gyromagnetic term, the heat ow of the harmonic map, and a projection step to overcome the stiness and the nonlinearity associated to the LLG equation. In this numerical approach, the implicit discretization is only applied to the scalar heat equation implicitly several times; therefore, the FFT- based fast solvers become available, due to the symmetric, positive denite (SPD) structures of the linear system. The original GSPM method [20] turns out to be unstable for small damping parameters, while this issue has been resolved in [8] with more updates of the stray eld. Its numerical eciency has been further improved by reducing the number of linear systems per time step [15]. One little deciency of GSPM is its rst-order accuracy in time. Meanwhile, in spite of these improvements, the GSPM method is computation- ally more expensive than the standard Poisson solver, because of the Gauss-Seidel iteration involved in the algorithm. An additional deciency of the GSPM is its rst-order accuracy in time. Moreover, most of the above-mentioned methods have been mainly focused on small damping parameters with the only exception in a theoretical work [1]. In other words, there has been no numerical method designed specically for real micromagnetics simulations with large damping parameters. In this paper, we propose a second-order accurate numerical method to solve the LLG equation with large damping parameters, whose complexity is also comparable toA SECOND-ORDER METHOD FOR LLG EQUATION 3 solving the scalar heat equation. To achieve this goal, the LLG system is refor- mulated, in which the damping term is rewritten as a harmonic mapping ow. In turn, the constant-coecient Laplacian part is treated by a standard BDF2 tem- poral discretization, and the associated dissipation will form the foundation of the numerical stability. Meanwhile, all the nonlinear parts, including both the gyro- magnetic term and the remaining nonlinear expansions in the damping term, are computed by a fully explicit approximation, which is accomplished by a second order extrapolation formula. Because of this fully explicit treatment for the nonlin- ear parts, the resulting numerical scheme only requires a standard Poisson solver at each time step. This fact will greatly facilitate the computational eorts, since the FFT-based fast solver could be eciently applied, due to the SPD structure of the linear system involved at each time step. In addition, the numerical stability has been demonstrated by extensive computational experiments, and these experiments has veried the idea that the dissipation property of the heat equation part would be able to ensure the numerical stability of the nonlinear parts, with large damping parameters. The rest of this paper is organized as follows. In section 2, the micromagnetics model is reviewed, and the numerical method is proposed, as well as its comparison with the GSPM and the semi-implicit projection method (SIPM). Subsequently, the numerical results are presented in section 3, including the temporal and spa- tial accuracy check in both the 1D and 3D computations, the numerical eciency investigation (in comparison with the GSPM and SIPM algorithms), the stability study with respect to the damping parameter, and the dependence of domain wall velocity on the damping parameter and the external magnetic eld. Finally, some concluding remarks are made in section 4. 2.The physical model and the numerical method 2.1.Landau-Lifshitz-Gilbert equation. The LLG equation describes the dy- namics of magnetization which consists of the gyromagnetic term and the damping term [3,13]. In the nondimensionalized form, this equation reads as mt=mhem(mhe) (2.1) with the homogeneous Neumann boundary condition (2.2)@m @ @ = 0; where is a bounded domain occupied by the ferromagnetic material and is unit outward normal vector along @ . In more details, the magnetization m: Rd!R3;d= 1;2;3 is a three- dimensional vector eld with a pointwise constraint jmj= 1. The rst term on the right-hand side in (2.1) is the gyromagnetic term and the second term stands for the damping term, with >0 being the dimensionless damping coecient. The eective eld heis obtained by taking the variation of the Gibbs free energy of the magnetic body with respect to m. The free energy includes the exchange energy, the anisotropy energy, the magnetostatic energy, and the Zeeman energy: (2.3)F[m] =0M2 s 2Z jrmj2+q m2 2+m2 3
2he
mhs
m
dx :4 Y. CAI, J. CHEN, C. WANG, AND C. XIE Therefore, the eective eld includes the exchange eld, the anisotropy eld, the stray eldhs, and the external eld he. For a uniaxial material, it is clear that he=
mq(m2e2+m3e3) +hs+he; (2.4) where the dimensionless parameters become =Cex=(0M2 sL2) andq=Ku=(0M2 s) withLthe diameter of the ferromagnetic body and 0the permeability of vacuum. The unit vectors are given by e2= (0;1;0),e3= (0;0;1), and
denotes the standard Laplacian operator. For the Permalloy, an alloy of Nickel (80%) and Iron (20%), typical values of the physical parameters are given by: the exchange constantCex= 1:31011J/m, the anisotropy constant Ku= 100 J/m3, the sat- uration magnetization constant Ms= 8:0105A/m. The stray eld takes the form hs=1 4rZ r1 jxyj
m(y)dy: (2.5) If is a rectangular domain, the evaluation of (2.5) can be eciently done by the Fast Fourier Transform (FFT) [20]. For brevity, the following source term is dened f=Q(m2e2+m3e3) +hs+he: (2.6) and the original PDE system (2.1) could be rewritten as mt=m(
m+f)mm(
m+f): (2.7) Thanks to point-wise identity jmj= 1, we obtain an equivalent form: (2.8)mt=(
m+f) + jrmj2m
f
mm(
m+f): In particular, it is noticed that the damping term is rewritten as a harmonic map- ping ow, which contains a constant-coecient Laplacian diusion term. This fact will greatly improve the numerical stability of the proposed scheme. For the numerical description, we rst introduce some notations for discretization and numerical approximation. Denote the temporal step-size by k, andtn=nk, nT k withTthe nal time. The spatial mesh-size is given by hx=hy=hz= h= 1=N, andmn i;j;`stands for the magnetization at time step tn, evaluated at the spatial location ( xi1 2;yj1 2;z`1 2) withxi1 2= i1 2
hx,yj1 2= j1 2
hyand z`1 2= `1 2
hz(0i;j;`N+ 1). In addition, a third order extrapolation formula is used to approximate the homogeneous Neumann boundary condition. For example, such a formula near the boundary along the zdirection is given by mi;j;1=mi;j;0;mi;j;N +1=mi;j;N: The boundary extrapolation along other boundary sections can be similarly made. The standard second-order centered dierence applied to
mresults in
hmi;j;k=mi+1;j;k2mi;j;k+mi1;j;k h2x +mi;j+1;k2mi;j;k+mi;j1;k h2y +mi;j;k+12mi;j;k+mi;j;k1 h2z;A SECOND-ORDER METHOD FOR LLG EQUATION 5 and the discrete gradient operator rhmwithm= (u;v;w )Treads as rhmi;j;k=2 64ui+1;j;kui1;j;k hxvi+1;j;kvi1;j;k hxwi+1;j;kwi1;j;k hxui;j+1;kui;j1;k hyvi;j+1;kvi;j1;k hywi;j+1;kwi;j1;k hyui;j;k +1ui;j;k1 hzvi;j;k +1vi;j;k1 hzwi;j;k +1wi;j;k1 hz3 75: Subsequently, the GSPM and the SIPM numerical methods need to be reviewed, which could be used for the later comparison. 2.2.The Gauss-Seidel projection method. The GSPM is based on a combi- nation of a Gauss-Seidel update of an implicit solver for the gyromagnetic term, the heat ow of the harmonic map, and a projection step. It only requires a series of heat equation solvers with constant coecients; as a result, the FFT-based fast solvers could be easily applied. This method is rst-order in time and second-order in space. Below is the detailed outline of the GSPM method in [8]. Step 1. Implicit Gauss-Seidel: gn i= (I
t
h)1(mn i+
tfn i); i= 2;3; g i= (I
t
h)1(m i+
tf i); i= 1;2; (2.9) (2.10)0 @m 1 m 2 m 31 A=0 @mn 1+ (gn 2mn 3gn 3mn 2) mn 2+ (gn 3m 1g 1mn 3) mn 3+ (g 1m 2g 2m 1)1 A: Step 2. Heat ow without constraints: (2.11) f=Q(m 2e2+m 3e3) +h s+he; (2.12)0 @m 1 m 2 m 31 A=0 @m 1+
t(
hm 1+f 1) m 2+
t(
hm 2+f 2) m 3+
t(
hm 3+f 3)1 A: Step 3. Projection onto S2: (2.13)0 @mn+1 1 mn+1 2 mn+1 31 A=1 jmj0 @m 1 m 2 m 31 A: Heremdenotes the intermediate values of m, and stray elds hn sandh sare evaluated at mnandm, respectively. Remark 2.1. Two improved versions of the GSPM have been studied in [15], which turn out to be more ecient than the original GSPM. Meanwhile, it is found that both improved versions become unstable when  > 1, while the original GSPM (outlined above) is stable even when 10. Therefore, we shall use the original GSPM in [8]for the numerical comparison in this work.6 Y. CAI, J. CHEN, C. WANG, AND C. XIE 2.3.Semi-implicit projection method. The SIPM has been outlined in [5,24]. This method is based on the second-order BDF temporal discretization, combined with an explicit extrapolation. It is found that SIPM is unconditionally stable and is second-order accurate in both space and time. The algorithmic details are given as follows. (2.14)8 >>>>>>< >>>>>>:3 2~mn+2 h2mn+1 h+1 2mn h k=^mn+2 h 
h~mn+2 h+^fn+2 h
^mn+2 h ^mn+2 h(
h~mn+2 h+^fn+2 h) ; mn+2 h=~mn+2 h j~mn+2 hj; where ~mn+2 his an intermediate magnetization, and ^mn+2 h,^fn+2 hare given by the following extrapolation formula: ^mn+2 h= 2mn+1 hmn h; ^fn+2 h= 2fn+1 hfn h; withfn h=Q(mn 2e2+mn 3e3) +hn s+hn e. The presence of cross product in the SIPM yields a linear system of equations with non-symmetric structure and vari- able coecients. In turn, the GMRES solver has to be applied to implement this numerical system. The numerical evidence has revealed that, the convergence of GMRES solver becomes slower for larger temporal step-size kor smaller spatial grid-sizeh, which makes the computation more challenging. 2.4.The proposed numerical method. The SIPM in (2.14) treats both the gyromagentic and the damping terms in a semi-implicit way, i.e.,
mis computed implicitly, while the coecient functions are updated by a second order accurate, explicit extrapolation formula. The strength of the gyromagnetic term is controlled by
m+fsince the length of mis always 1. Meanwhile, the strength of the damping term is controlled by the product of
m+fand the damping parameter . For small , say1, it is reasonable to treat both the gyromagentic and the damping terms semi-implicitly. However, for large , an alternate approach would be more reasonable, in which the whole gyromagentic term is computed by an explicit extrapolation, while the nonlinear parts in the damping term is also updated by an explicit formula, and only the constant-coecient
mpart in the damping term is implicitly updated. This idea leads to the proposed numerical method. To further simplify the presentation, we start with (2.8), and the numerical algorithm is proposed as follows. (2.15)8 >>>>>>>>>>< >>>>>>>>>>:3 2~mn+2 h2mn+1 h+1 2mn h k=^mn+2 h 
h^mn+2 h+^fn+2 h + 
h~mn+2 h+^fn+2 h + jrh^mn+2 hj2^mn+2 h
^fn+2 h ^mn+2 h; mn+2 h=~mn+2 h j~mn+2 hj;A SECOND-ORDER METHOD FOR LLG EQUATION 7 where ^mn+2 h= 2mn+1 hmn h; ^fn+2 h= 2fn+1 hfn h: Table 1 compares the proposed method, the GSPM and the SIPM in terms of number of unknowns, dimensional size, symmetry pattern, and availability of FFT-based fast solver of linear systems of equations, and the number of stray eld updates. At the formal level, the proposed method is clearly superior to both the GSPM and the SIPM algorithms. In more details, this scheme will greatly improve the computational eciency, since only three Poisson solvers are needed at each time step. Moreover, this numerical method preserves a second-order accuracy in both space and time. The numerical results in section 3 will demonstrate that the proposed scheme provides a reliable and robust approach for micromagnetics simu- lations with high accuracy and eciency in the regime of large damping parameters. Table 1. Comparison of the proposed method, the Gauss-Seidel projection method, and the semi-implicit projection method. Property or number Proposed method GSPM SIPM Linear systems 3 7 1 Size N3N33N3 Symmetry Yes Yes No Fast Solver Yes Yes No Accuracy O(k2+h2)O(k+h2)O(k2+h2) Stray eld updates 1 4 1 Remark 2.2. To kick start the proposed method, one can apply a rst-order al- gorithm, such as the rst-order BDF method, in the rst time step. An overall second-order accuracy is preserved in this approach. 3.Numerical experiments In this section, we present a few numerical experiments with a sequence of damp- ing parameters for the proposed method, the GSPM [8] and the SIPM [24], with the accuracy, eciency, and stability examined in details. Domain wall dynamics is studied and its velocity is recorded in terms of the damping parameter and the external magnetic eld. 3.1.Accuracy and eciency tests. We set= 1 andf= 0 in (2.8) for conve- nience. The 1D exact solution is given by me= (cos(X) sint;sin(X) sint;cost)T; and the corresponding exact solution in 3D becomes me= (cos(XYZ ) sint;sin(XYZ ) sint;cost)T; whereX=x2(1x)2,Y=y2(1y)2,Z=z2(1z)2. In fact, the above exact solutions satisfy (2.8) with the forcing term g=@tme
mejrmej2+me
me, as well as the homogeneous Neumann boundary condition.8 Y. CAI, J. CHEN, C. WANG, AND C. XIE For the temporal accuracy test in the 1D case, we x the spatial resolution ash= 5D4, so that the spatial approximation error becomes negligible. The damping parameter is taken as = 10, and the nal time is set as T= 1. In the 3D test for the temporal accuracy, due to the limitation of spatial resolution, we take a sequence of spatial and temporal mesh sizes: k=h2 x=h2 y=h2 z=h2= 1=N0 for the rst-order method and k=hx=hy=hz=h= 1=N0for the second- order method, with the variation of N0indicated below. Similarly, the damping parameter is given by = 10, while the nal time Tis indicated below. In turn, the numerical errors are recorded in term of the temporal step-size kin Table 2. It is clear that the temporal accuracy orders of the proposed numerical method, the GSPM, and the SIPM are given by 2, 1, and 2, respectively, in both the 1D and 3D computations. The spatial accuracy order is tested by xing k= 1D5,= 10,T= 1 in 1D andk= 1D3,= 10,T= 1 in 3D. The numerical error is recorded in term of the spatial grid-size hin Table 3. Similarly, the presented results have indicated the second order spatial accuracy of all the numerical algorithms, including the proposed method, the GSPM, and the SIPM, respectively, in both the 1D and 3D computations. To make a comparison in terms of the numerical eciency, we plot the CPU time (in seconds) vs. the error norm kmhmek1. In details, the CPU time is recorded as a function of the approximation error in Figure 1a in 1D and in Figure 1b in 3D, with a variation of kand a xed value of h. Similar plots are also displayed in Figure 1c in 1D and Figure 1d in 3D, with a variation of hand a xed value of k. In the case of a xed spatial resolution h, the proposed method is signicantly more ecient than the GSPM and the SIPM in both the 1D and 3D computations. The SIPM is slightly more ecient than the GSPM, while such an advantage depends on the performance of GMRES, which may vary for dierent values of kandh. In the case of a xed time step size k, the proposed method is slightly more ecient than the GSPM, in both the 1D and 3D computations, and the GSPM is more ecient than the SIPM. 3.2.Stability test with large damping parameters. To check the numerical stability of these three methods in the practical simulations of micromagnetics with large damping parameters, we consider a thin lm of size 480 48020 nm3with grid points 1001004. The temporal step-size is taken as k= 1 ps. A uniform state along the xdirection is set to be the initial magnetization and the external magnetic eld is set to be 0. Three dierent damping parameters, = 0:01;10;40, are tested with stable magnetization proles shown in Figure 2. In particular, the following observations are made. The proposed method is the only one that is stable for very large damping parameters; All three methods are stable for moderately large ; The proposed method is the only one that is unstable for small . In fact, a preliminary theoretical analysis reveals that, an optimal rate convergence estimate of the proposed method could be theoretically justied for >3. Mean- while, extensive numerical experiments have implied that  > 1 is sucient to ensure the numerical stability in the practical computations.A SECOND-ORDER METHOD FOR LLG EQUATION 9 Table 2. The numerical errors for the proposed method, the GSPM and the SIPM with = 10 andT= 1. Left: 1D with h= 5D4; Right: 3D with k=h2 x=h2 y=h2 z=h2= 1=N0 for GSPM and k=hx=hy=hz=h= 1=N0for the proposed method and SIPM, with N0specied in the table. 1D 3D kk
k1k
k 2k
kH1k=hk
k1k
k 2k
kH1 4.0D-2 4.459D-4 5.226D-4 5.588D-4 1/20 6.171D-4 4.240D-4 4.246D-4 2.0D-2 1.147D-4 1.345D-4 1.436D-4 1/24 4.381D-4 3.010D-4 3.014D-4 1.0D-2 2.899D-5 3.402D-5 3.631D-5 1/28 3.268D-4 2.245D-4 2.248D-4 5.0D-3 7.192D-6 8.529D-6 9.119D-6 1/32 2.531D-4 1.739D-4 1.741D-4 2.5D-3 1.699D-6 2.321D-6 2.518D-6 1/36 2.017D-4 1.386D-4 1.387D-4 order 2.007 1.961 1.957 { 1.902 1.903 1.903 (a)Proposed method 1D 3D kk
k1k
k 2k
kH1k=h2k
k1k
k 2k
kH1 2.5D-3 2.796D-4 2.264D-4 1.445D-3 1/36 4.194D-4 2.683D-4 2.815D-4 1.25D-3 1.425D-4 1.174D-4 7.720D-4 1/64 2.388D-4 1.399D-4 1.500D-4 6.25D-4 7.170D-5 5.940D-5 4.026D-4 1/144 1.069D-4 6.106D-5 6.736D-5 3.125D-4 3.591D-5 2.971D-5 2.069D-4 1/256 6.021D-5 3.442D-5 3.860D-5 1.5625D-4 1.799D-5 1.488D-5 1.054D-4 1/400 3.855D-5 2.208D-5 2.501D-5 order 0.991 0.984 0.945 { 0.992 1.032 1.000 (b)GSPM 1D 3D kk
k1k
k 2k
kH1k=hk
k1k
k 2k
kH1 4.0D-2 4.315D-4 5.111D-4 8.774D-4 1/20 6.170D-4 4.240D-4 4.249D-4 2.0D-2 1.128D-4 1.334D-4 2.255D-4 1/24 4.380D-4 3.010D-4 3.016D-4 1.0D-2 2.872D-5 3.399D-5 5.706D-5 1/28 3.268D-4 2.245D-4 2.251D-4 5.0D-3 7.174D-6 8.552D-6 1.433D-5 1/32 2.531D-4 1.739D-4 1.743D-4 2.5D-3 1.721D-6 2.333D-6 3.784D-6 1/36 2.017D-4 1.386D-4 1.389D-4 order 1.991 1.951 1.969 { 1.902 1.903 1.902 (c)SIPM Under the same setup outlined above, we investigate the energy dissipation of the proposed method, the GSPM, and the SIPM. The stable state is attainable at t= 2 ns, while the total energy is computed by (2.3). The energy evolution curves of dierent numerical methods with dierent damping parameters, = 2;5;8;10, are displayed in Figure 3. One common feature is that the energy dissipation rate turns out to be faster for larger , in all three schemes. Meanwhile, a theoretical derivation also reveals that the energy dissipation rate in the LLG equation (2.1) depends on , and a larger leads to a faster energy dissipation rate. Therefore, the numerical results generated by all these three numerical methods have made a nice agreement with the theoretical derivation.10 Y. CAI, J. CHEN, C. WANG, AND C. XIE Table 3. The numerical errors of the proposed method, the GSPM and the SIPM with = 10 andT= 1. Left: 1D with k= 1D5; Right: 3D with k= 1D3. 1D 3D hk
k1k
k 2k
kH1hk
k1k
k 2k
kH1 4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3 2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4 1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4 5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5 order 2.000 2.000 2.000 { 2.035 2.047 2.037 (a)Proposed method 1D 3D hk
k1k
k 2k
kH1hk
k1k
k 2k
kH1 4.0D-2 7.388D-3 7.392D-3 8.244D-3 1/2 4.256D-3 2.470D-3 2.470D-3 2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.810D-4 5.589D-4 5.744D-4 1.0D-2 4.619D-4 4.622D-4 5.158D-4 1/8 2.447D-4 1.388D-4 1.423D-4 5.0D-3 1.153D-4 1.156D-4 1.302D-4 1/16 6.103D-5 3.468D-5 3.613D-5 order 2.000 2.000 1.995 { 2.037 2.047 2.030 (b)GSPM 1D 3D hk
k1k
k 2k
kH1hk
k1k
k 2k
kH1 4.0D-2 7.388D-3 7.392D-3 8.243D-3 1/2 4.261D-3 2.472D-3 2.472D-3 2.0D-2 1.848D-3 1.848D-3 2.061D-3 1/4 9.822D-4 5.595D-4 5.753D-4 1.0D-2 4.621D-4 4.621D-4 5.153D-4 1/8 2.453D-4 1.390D-4 1.424D-4 5.0D-3 1.155D-4 1.155D-4 1.288D-4 1/16 6.137D-5 3.471D-5 3.554D-5 order 2.000 2.000 2.000 { 2.035 2.047 2.037 (c)SIPM Meanwhile, we choose the same sequence of values for , and display the energy evolution curves in terms of time up to T= 2 ns in Figure 4. It is found that the proposed method have almost the same energy dissipation pattern with the other two methods for moderately large damping parameters = 2;5;8. In the case of = 10, the SIPM has a slightly dierent energy dissipation pattern from the other two numerical methods. 3.3.Domain wall motion. A Ne el wall is initialized in a nanostrip of size 800  1004 nm3with grid points 128 644. An external magnetic eld of he= 5 mT is then applied along the positive xdirection and the domain wall dynamics is simulated up to 2 ns with = 2;5;8. The corresponding magnetization proles are visualized in Figure 5. Qualitatively, the domain wall moves faster as the value of increases. Quantitatively, the corresponding dependence is found to be linear; see Figure 6. The slopes tted by the least-squares method in terms of andhe are recorded in Table 4.A SECOND-ORDER METHOD FOR LLG EQUATION 11 10-610-510-410-3100101102 Proposed method GSPM SIPM (a)Varyingkin 1D up to T= 1 1.8 2 2.2 2.4 2.6 2.8 3 3.2 10-7101102103 Proposed method GSPM SIPM(b)Varyingkin 3D up to T= 0:1 10-510-410-310-2101102103 Proposed method GSPM SIPM (c)Varyinghin 1D up to T= 1 10-410-310-210-1100101102103 Proposed method GSPM SIPM(d)Varyinghin 3D up to T= 1 Figure 1. CPU time needed to achieve the desired numerical ac- curacy, for the proposed method, the GSPM and the SIPM, in both the 1D and 3D computations. The CPU time is recorded as a function of the approximation error by varying korhindepen- dently. CPU time with varying k: proposed method <SIPM< GSPM; CPU time with varying h: proposed method /GSPM< SIPM. 4.Conclusions In this paper, we have proposed a second-order accurate numerical method to solve the Landau-Lifshitz-Gilbert equation with large damping parameters. For the numerical convenience, the LLG system is reformulated so that in which the damp- ing term is rewritten as a harmonic mapping ow .This numerical scheme is based on the second-order backward-dierentiation formula approximation for the temporal derivative, combined with an implicit treatment of the constant-coecient diusion term, and the fully explicit extrapolation approximation of the nonlinear terms, in- cluding the gyromagnetic term and the nonlinear part of the harmonic mapping ow. Thanks to the large damping parameter, the proposed method is veried12 Y. CAI, J. CHEN, C. WANG, AND C. XIE Figure 2. Stable structures in the absence of magnetic eld at 2 ns when= 0:01;10;40. The color denotes the angle between the rst two components of the magnetization vector. Top: Proposed method; Middle: GSPM; Bottom: SIPM. Left: = 40; Middle: = 10; Right: = 0:01. (a)Proposed (b)GSPM (c)SIPM Figure 3. Energy evolution curves of three numerical methods, with dierent damping constants, = 2;5;8;10, up tot= 2 ns in the absence of external magnetic eld. Left: Proposed numerical method; Middle: GSPM; Right: SIPM. One common feature is that the energy dissipation rate is faster for larger , which is physically reasonable.A SECOND-ORDER METHOD FOR LLG EQUATION 13 (a)= 2 (b)= 5 (c)= 8 (d)= 10 Figure 4. Energy evolution curves in terms of time, for the nu- merical results created by three numerical methods up to t= 2 ns in the absence of external magnetic eld for (a) = 2, (b)= 5, (c)= 8, and (d) = 10. The energy dissipation pattern of the proposed method is consistent with the other two methods for (a), (b), and (c), and the SIPM has a slightly dierent energy dissipa- tion pattern from the other two methods for (d). to be unconditionally stable. The proposed method is much more ecient than other semi-implicit schemes since only symmetric, positive denite linear systems of equations with constant coecients need to be solved. Meanwhile, the proposed method is more accurate than the standard Gauss-Seidel projection method, due to its second-order accuracy in time. Numerical results in 1D and 3D are pro- vided to demonstrate the accuracy and the eciency of the proposed numerical method. In addition, micromagnetics simulations using the proposed method have provided physically reasonable structures and captured the linear dependence of the domain wall velocity with respect to the damping parameter. Therefore, the proposed method could be eciently used for challenging practical simulations of micromagnetics with large damping parameters.14 Y. CAI, J. CHEN, C. WANG, AND C. XIE (a)Magnetization for initial state (b)Magnetization with = 2 at 2 ns (c)Magnetization with = 5 at 2 ns (d)Magnetization with = 8 at 2 ns Figure 5. Magnetization proles of Ne el wall motion in the pres- ence of a magnetic eld he= 5 mT, with = 2;5;8 at 2 ns for the proposed numerical method. The in-plane arrow denotes the rst two components of the magnetization vector. The wall moves faster for larger values of and its velocity depends linearly on . Figure 6. Linear dependence of the wall velocity with respect to the damping parameter (left) and the external magnetic eld he (right). Acknowledgments This work is supported in part by the grants NSFC 11971021 (J. Chen), NSF DMS-2012669 (C. Wang), NSFC 11771036 (Y. Cai).A SECOND-ORDER METHOD FOR LLG EQUATION 15 Table 4. Linear dependence of the domain wall velocity Vin terms of the external magnetic eld heand the damping parameter . V(m/s) he(mT)5 6 7 8 9 10 Slope 3 76 91 109 123 139 154 1.024 4 105 118 139 157 179 196 0.928 5 129 145 169 192 217 244 0.932 6 153 169 200 227 256 286 0.927 7 177 196 232 263 294 333 0.927 8 200 222 263 303 333 385 0.954 9 230 250 294 345 385 435 0.954 10 253 270 323 370 417 476 0.943 Slope 0.984 0.910 0.910 0.933 0.917 0.950 { References [1] G. Akrivis, M. Feischl, B. Kov acs, and C. Lubich, Higher-order linearly implicit full dis- cretization of the Landau-Lifshitz-Gilbert equation , Math. Comp. 90(2021), 995{1038. [2] F. Alouges and P. Jaisson, Convergence of a nite element discretization for the Landau- Lifshitz equations in micromagnetism , Math. Models Methods Appl. Sci. 16(2006), no. 02, 299{316. [3] W.F. Brown, Micromagnetics , Interscience Tracts on Physics and Astronomy. Interscience Publishers (John Wiley and Sons), New York-London, 1963. [4] S. Budhathoki, A. Sapkota, K.M. Law, B. Nepal, S. Ranjit, S. Kc, T. Mewes, and A. Hauser, Low Gilbert damping and linewidth in magnetostrictive FeGa thin lms , J. Magn. Magn. Mater. 496(2020), 165906. [5] J. Chen, C. Wang, and C. Xie, Convergence analysis of a second-order semi-implicit pro- jection method for Landau-Lifshiz equation , Appl. Numer. Math. (2021). Submitted and in review. [6] I. Cimr ak, A survey on the numerics and computations for the Landau-Lifshitz equation of micromagnetism , Arch. Comput. Methods Eng. 15(2008), no. 3, 277{309. [7] H. Gao, Optimal error estimates of a linearized Backward Euler FEM for the Landau-Lifshitz equation , SIAM J. Numer. Anal. 52(2014), no. 5, 2574{2593. [8] C.J. Garca-Cervera and W. E, Improved Gauss-Seidel projection method for micromagnetics simulations , J. Comput. Phys. 171(2001), no. 1, 357{372. [9] T.L. Gilbert, Phys. Rev. 100(1955), 1243. [Abstract only; full report, Armor Research Foun- dation Project No. A059, Supplementary Report, May 1, 1956 (unpublished)]. [10] T.L. Gilbert and J.M. Kelly, Anomalous rotational damping in ferromagnetic sheets , Armour Research Foundation of Illinois Institute of Technology (1955). (unpublished). [11] B. Heinrich, D. Fra tov a, and V. Kambersk y, The in uence of s-d exchange on relaxation of magnons in metals , Phys. Stat. Solidi B-basic Solid Stat. Phys. 23(1967), 501{507. [12] M. Kruz k and A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism , SIAM Rev. 48(2006), no. 3, 439{483. [13] L.D. Landau and E.M. Lifshits, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies , Phys. Z. Sowjet. 63(1935), no. 9, 153{169. [14] D.M. Lattery, D. Zhang, J. Zhu, X. Hang, J. Wang, and X. Wang, Low Gilbert damping constant in perpendicularly magnetized W/CoFeB/MgO lms with high thermal stability , Sci. Rep. 8(2018), 13395. [15] P. Li, C. Xie, R. Du, J. Chen, and X. Wang, Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation , J. Comput. Phys. 401(2020), 109046. [16] T. Nan, Y. Lee, S. Zhuang, Z. Hu, J. Clarkson, X. Wang, C. Ko, H. Choe, Z. Chen, D. Budil, J. Wu, S. Salahuddin, J. Hu, R. Ramesh, and N. Sun, Electric-eld control of spin dynamics during magnetic phase transitions , Sci. Adv. 6(2020), no. 40, eabd2613.16 Y. CAI, J. CHEN, C. WANG, AND C. XIE [17] H. Suhl, Theory of the magnetic damping constant , IEEE Trans. Magn. 34(1998), 1834{ 1838. [18] T. Tanaka, S. Kashiwagi, Y. Otsuka, Y. Nozaki, Y. Hong, and K. Matsuyama, Microwave- assisted magnetization reversal of exchange-coupled composite nanopillar with large Gilbert damping constant , IEEE Tran. Magn. 50(2014), 1{3. [19] H. Tang and K. Xia, Gilbert damping parameter in MgO-based magnetic tunnel junctions from rst principles , Phys. Rev. Applied 7(2017), 034004. [20] C. Wang and J.-G. Liu, Convergence of gauge method for incompressible ow , Math. Comp. 69(2000), 1385{1407. [21] X. Wang, C.J. Garc a-Cervera, and W. E, A Gauss-Seidel projection method for micromag- netics simulations , J. Comput. Phys. 171(2001), no. 1, 357{372. [22] R. Weber, D. Han, I. Boventer, S. Jaiswal, R. Lebrun, G. Jakob, and M. Kl aui, Gilbert damping of CoFe-alloys , J. Phys. D 52(2019), 325001. [23] D. Wei, Micromagnetics and recording materials , Springer Briefs in Apllied Sciences and Technology, Springer Berlin Heidelberg, 2012. [24] C. Xie, C.J. Garc a-Cervera, C. Wang, Z. Zhou, and J. Chen, Second-order semi-implicit methods for micromagnetics simulations , J. Comput. Phys. 404(2020), 109104. [25] D. Zhang, M. Li, L. Jin, C. Li, Y. Rao, X. Tang, and H. Zhang, Extremely large magnetiza- tion and Gilbert damping modulation in NiFe/GeBi bilayers , ACS Appl. Electron. Mater. 2 (2020), no. 1, 254{259. School of Mathematical Sciences, Beijing Normal University, Beijing, China. Email address :yongyong.cai@bnu.edu.cn School of Mathematical Sciences, Soochow University, Suzhou, China. Email address :jingrunchen@suda.edu.cn Mathematics Department, University of Massachusetts, North Dartmouth, MA 02747, USA. Email address :cwang1@umassd.edu School of Mathematical Sciences, Soochow University, Suzhou, China. Email address :20184007005@stu.suda.edu.cn

2021-05-08

A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of equations with constant coefficients where fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.

A second-order numerical method for Landau-Lifshitz-Gilbert equation with large damping parameters

2105.03576v1

A single-ion nonlinear mechanical oscillator N. Akerman, S. Kotler, Y. Glickman, Y. Dallal, A. Keselman, and R. Ozeri Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel We study the steady state motion of a single trapped ion oscillator driven to the nonlinear regime. Damping is achieved via Doppler laser-cooling. The ion motion is found to be well described by the Dung oscillator model with an additional nonlinear damping term. We demonstrate a unique ability of tuning both the linear as well as the nonlinear damping coecients by controlling the cooling laser parameters. Our observations open a way for the investigation of nonlinear dynamics on the quantum-to-classical interface as well as mechanical noise squeezing in laser-cooling dynamics. PACS numbers: 37.10.Ty 37.10.Vz Nonlinear dynamics prevails in many dynamical sys- tems in nature, introducing a rich behavior such as criti- cality, bifurcations and chaos. Nonlinear dynamics on the microscopic scale is especially interesting as it can shed light on the quantum-to-classical transition as well as provide a mean to suppress thermal and quantum noise. All mechanical oscillators will show nonlinearity when driven far enough from equilibrium. The simplest such nonlinear oscillator is the Dung oscillator which in- cludes a cubic term in the restoring force [1]. Recently, such Dung nonlinear dynamics has been extensively studied with nano-electromechanical beam resonators. The basins of attraction of a nano beam oscillator were mapped [2]. Noise squeezing and stochastic resonances were observed close to the Dung instability [3, 4]. Noise squeezing was predicted to enable mass and force detec- tion with precision below the standard thermal limit [5{ 7] and possibly below the standard quantum limit when operating close to the oscillator ground state [8]. The mechanical motion of trapped ions is highly con- trollable and can be eciently laser-cooled to the quan- tum ground state [9]. High delity production of Fock, squeezed, and Schr odinger-cat states was demonstrated with a single trapped-ion [10, 11]. At the temperature range obtained with laser-cooling techniques, quadruple RF Paul traps are excellently approximated as harmonic. Nonlinearity in ion motion was observed when several ions are trapped due to their mutual Coulomb repulsion. Here nonlinearity couples between the ion-crystal normal modes, even at the single quantum level [12, 13]. Trap nonlinearities are important in the context of resonance ejection in high resolution mass spectrometry [14]. How- ever, in these experiments ions are typically not laser- cooled and furthermore the eect of Coulomb nonlinear- ities in the large ion cloud is intertwined with that of the trap. Recently, amplication saturation of a single- ion \phonon laser", resulting from optical forces that are nonlinear in the ions velocity, was demonstrated [15]. Here, we study the nonlinear mechanical response of a single laser-cooled88Sr+ion, in a linear RF-Paul trap. The nonlinearity originates from the higher than quadrupolar order terms in the trapping potential. We EMCCD Lasers S1/2 422nm P1/2 D3/2 1092nm PMT 88 Sr+ Drive force Trap RF FIG. 1: Schematic diagram of the experimental set-up and the relevant energy levels of88Sr+ion. The positively biased trap end-caps produce a static trapping potential in the axial direction with a small anharmonicity. The ion oscillator is driven to the nonlinear regime by a small oscillating voltage on one of the trap end-caps. Violet and infra-red laser beams provide laser cooling and optical pumping. Scattered violet photons are collected by an imaging system and directed ei- ther to an EMCCD camera or to a PMT. nd that the ion steady state response is well described by the Dung model with an additional nonlinear damping term [16]. Unlike other realizations of nonlinear mechani- cal oscillators, both the linear and the nonlinear damping components can be precisely controlled. Our trap has the canonical linear four rods and two end-caps conguration shown in Fig. 1. The distance of the ion to the end-caps and rod-electrodes is 0 :65 mm and 0:27 mm respectively. Here we examine only the motion along the axial direction of the trap. In this direction, trapping is dominated by the static electric potential due to a positive constant voltage on the trap end-caps. This potential is well approximated to be harmonic with !0=2= 438 KHz. However, as the trap end-caps do not satisfy the pure electric quadruple boundary condition, a small octupolar contribution to the electric eld results a positive cubic term in the restoring force and an energy level dierence of ~!0+~
nlnwherenis the harmonic oscillator quantum number and
nl=2= 0:8 mHz is thearXiv:1003.1577v1 [quant-ph] 8 Mar 20102 nonlinear dispersion. This nonlinearity becomes increas- ingly important with growing oscillation amplitude. The ion is driven to the nonlinear regime by adding a small oscillating voltage to one of the trap end-caps. The ion is Doppler-cooled by scattering photons from a single laser beam, slightly red-detuned from the S1=2!P1=2tran- sition at 422 nm. To prevent population accumulation in theD3=2meta-stable level we repump the ion on the D3=2!P1=2transition at 1092 nm. We measure the steady-state oscillation amplitude of the ion as we slowly scan the drive frequency, !, across the harmonic resonance, !0. The scan is from lower to higher frequencies (positive sweep) or vice versa (negative sweep). Photons scattered during the cooling process are collected by an imaging system (N.A. = 0.31), and are either directed towards an Electron-Multiplying CCD (EMCCD) camera or a Photo Multiplier Tube (PMT). We measure the amplitude of motion by taking time- averaged images of the ion as shown in Fig.2(a). The image is then integrated along the direction perpendicu- lar to motion to produce a single curve. Every column in Fig. 2(b) corresponds to a curve produced this way, for a positive frequency sweep. As seen, the oscillation am- plitude increases as the drive frequency approaches !0, continues to increase passed !0, until at a given critical drive frequency, !m, abruptly collapses to a signicantly lower value. We extract the ion oscillation amplitude by tting the curve to the expected time-averaged po- sition distribution. The blue and red lines in Fig. 3 are the measured amplitudes for positive and negative sweeps respectively. Dierent curves correspond to dif- ferent drive amplitudes. The asymmetry and hysteresis as well as the abrupt amplitude changes at specic criti- cal drive frequencies are a clear deviation from the driven harmonic oscillator response. As expected from a posi- tive nonlinearity, the oscillator self-frequency is \pulled" to higher values at higher oscillation amplitudes. In order to measure the phase dierence between the ion-oscillator and the driving force, we time-stamp each photon measured by the PMT within a single drive pe- riod to allocate it with a corresponding drive phase. The instantaneous photon scattering rate from the cool- ing laser beam is determined by the ions' instantaneous velocity through its associated Doppler shift [17]. A histogram of the measured photon phases is shown in Fig.2(c). A clear sinusoidal oscillation of the photon scat- tering rate yields the ion-oscillator phase. The columns in Fig.2(d) are photon phase histograms for a positive drive frequency sweep. As seen, at the critical frequency, !m, a phase jump of 1 :2 radians in the oscillator motion accompanies the sudden change in oscillation amplitude. Our observations are well accounted for by the Dung oscillator model. The Dung equation of motion is, x+ 2_x+!2 0x+x3=kcos(!t): (1) 10001200140016001800 0 π 2π Drive phase[rad]Counts(c) σ/2π[KHz]Drive phase[rad](d) 0 π2π −1−0.500.5 σ/2π[KHz]X [µm](b) −1−0.500.5−20−1001020 (a) Figure 1: 1FIG. 2: Driven ion-oscillator amplitude and phase. (a) Time- averaged ion images taken at various drive frequencies. (b) Columns are time-averaged images, integrated along the di- rection perpendicular to ion-motion, during a positive fre- quency scan. (c) A histogram of the number of photons de- tected at dierent driving force phases. (d) Columns are pho- ton phase histograms taken during a positive frequency scan. The solid line is the theoretical phase given by Eq.3 shifted by a constant to match the peak in the histograms. Herexis the displacement of the ion from its equilib- rium position, is the an-harmonic coecient, is the linear damping coecient and kis the drive amplitude. The recoil noise inherent to the spontaneous photon scat- tering process, which would appear as a Langevin force term, is neglected. An approximate solution to Eq.1 can be obtained by the multiple scale method [1]. Here the solution has the from x(t) =a(t) cos(!t) , wherea(t) is a slowly-varying oscillation amplitude and is the os- cillator phase. The steady-state solution for asolves, =3 8!0a2s k2 4!2 0a22; (2) where=!!0is the drive detuning. The steady-state solution for a, at a xed k, vs. drive frequency is shown by the black line in the inset of Fig.3. Above a criti- cal amplitude ac,atrifurcates into three solutions. One solution with small and one with large amplitude, are sta- ble, while the third, intermediate amplitude solution, is unstable and is positioned on the state-space separatrix. This bistablity persists until the high amplitude solution reaches a maximal value, am, at which the drive force is overwhelmed by damping and the oscillator is forced into a single stable solution. Positive and negative frequency scans carry the oscillator into the bistability region along dierent stable attractors leading to the observed hys- teresis as illustrated by the arrows in the inset. To com- pare with our data we independently measure all the pa-3 rameters in Eq.1. The driving force amplitude, k, is mea- sured by observing ion displacement vs. end-cap voltage, !0is measured via ion response in the linear regime. A value of=42= 1:240:03
1018Hz2=m2is measured using the observed dependence of amonm=!m!0, the instability detuning, am=p 8!0m=3. A value of=2= 39:20:3 Hz, which result in a quality factor Q= 5590, is evaluated using the variation of amwith the drive amplitude, am=k=(2!0). The blue and red cir- cles in the inset are the measured amplitudes, for positive and negative scans respectively, showing good agreement with the theoretical curve. −1−0.500.51012345 02040 k[Hz2m] σ/2π[KHz]a[µm] −1−0.5 00.511.5010203040 σ/2π[KHz]a[µm] x104σm am FIG. 3: Measured oscillator amplitude vs. drive frequency for various driving force amplitudes and both positive (blue) and negative (red) scans. The inset shows the Dung model calculation (black solid line) and our measured amplitudes (blue circles - positive scan; red circles - negative scan) The Dung oscillator steady state phase is given by, tan() =8!0 3a2!0: (3) The white solid line in Fig.2(d) shows the theoretical phase curve vs. drive frequency for our experimental pa- rameters, showing good agreement with our data. Linear damping is a very good approximation for most mechanical oscillators, as typically dissipation originates from coupling of the oscillator to an ohmic bath. Re- cently, the contribution of non-linear damping to the mo- tion of a nano beam resonator was studied [19]. In our experiment damping results from the change in radia- tion pressure vs. ion velocity. When the laser frequency is tuned below the cooling transition, the leading con- tribution is indeed linear in the ions' velocity. However, as the oscillation amplitude increases or the cooling-laser detuning reduced, the eect of damping force terms that are nonlinear in the ions' velocity increases [17]. To account for nonlinear damping, Eq.1 is modied to include a term which is cubic in the oscillator velocity, x+ 2_x+ _x3+!2 0x+x3=kcos(!t): (4)Here is the cubic damping coecient. The steady-state amplitude is now a solution of [18, 19], 9 16(2+ 2!6 0)a6+3!0( !3 0)a4 +4!2 0(2+2)a2k2= 0:(5) When > 0, nonlinear damping acts to eectively in- crease dissipation for larger oscillation amplitudes. Un- like the linear damping case, amdoes not increase linearly withkbut is rather limited by the growing dissipation. We ndand by a maximum likelihood t of the mea- suredamvs.kcurve to the solution of Eq. 5. It is instructive to look at the responsivity, = 2!0a=k, in order to distinguish linear from nonlinear damping [18]. In Fig.4 we plot the measured for positive scans and various drive amplitudes, k, for two dierent cooling-laser detuning values, . In Fig.4(a) =2=420 MHz, = 0 and the maximal responsivity is seen to be independent of k. In Fig.4(b) =2=160 MHz,!2 0 =2= 0:090:002 m2Hz and the maximal responsivity decreases as kin- creases. The linear dissipation term, , is similar in both cases. The solid lines are the solutions of Eq.5 showing good agreement with the data. −0.500.511.500.20.40.60.81 σ/2π[KHz]χ(a) −0.5 00.5 100.20.40.60.81 σ/2π[KHz]χ(b) FIG. 4: Calculated and measured responsivity, = 2!0a=k, for positive drive scans. Dierent curves correspond to dier- ent drive amplitudes. Due to small drifts in !0(<100Hz) each curve was separately shifted on the frequency axis to t the theoretical curve. (a) Linear damping, the cooling laser detuning=2=420 MHz and = 0. The maximal re- sponsivity is independent of drive amplitude k. (b) Nonlinear damping,=2=160 MHz and !2 0 =2= 0:09m2Hz. The maximal responsivity decreases as kincreases. We next repeat the measurement of and for various cooling-laser detunings at a xed repump-laser frequency and lasers intensities. The measured and vs.are shown in gures 5(a) and 5(b) respectively. To com- pare with the theoretically predicted values we write the cooling-laser scattering force, Fs( _x) =~kcp(c+kc_x;r+kr_x); (6) wherekc=randc=rare the wave-vectors and detunings of the cooling and repump lasers respectively, = 2  21 MHz is the spectral linewidth of the P1=2level and4 pis theP1=2population. The damping coecients are therefore given by the appropriate derivatives, =1 2mdFs d_x; =1 6md3Fs d_x3: (7) Heremis the ion mass. We calculate Pby numerically solving the eight coupled Bloch equations, corresponding to the population in all states in the S1=2,P1=2andD3=2 levels coupled by the cooling and repump lasers. The cubic damping coecient is highly sensitive to dierent laser parameters due to the presence of dark resonances. The solid lines in gures 5(a) and 5(b) are the calculated and showing good agreement with our measured val- ues. The two lasers intensities and the repump-laser de- tuning were used as t parameters, yielding values that agree within 20% with their measured value. −200−150−100020406080100120 δc/2π[MHz]µ/2π [Hz](a) −200−150−10000.050.10.150.20.25 δc/2π[MHz]ω02γ/2π [µm−2Hz](b) FIG. 5: (a) Linear and (b) cubic damping coecients for various cooling-beam detunings. Filled circles are measured values and solid lines are calculated using equations 6 and 7. An additional nonlinear damping term, proportional tox2_x, results from the laser beam nite size (100 m FWHM) and has an identical eect to that of on the steady-state motion [16, 18]. This term was calculated to be small relative to [20] and was taken into account in Fig.5. In conclusion, we have driven a single-ion oscillator to the nonlinear regime. The ion steady-state motion, show- ing a bifurcation into two stable attractors and hysteresis, is well described by the Dung oscillator model with an additional nonlinear damping term. Unlike previously studied nonlinear mechanical oscillators, here both the linear and nonlinear parts of dissipation can be tuned with the cooling laser parameters. The study of the nonlinear motion of trapped laser- cooled ions opens several exciting research avenues. Since trapped atomic-ions can be cooled to the quantum ground state, they are an excellent platform to study nonlinear behavior in the quantum regime. As shown in [21], unlike the simple harmonic oscillator, a Dung oscillator will demonstrate a clear quantum-to-classical transition even when classically driven. Moreover, as theion-spin can be entangled with its motion, it will be pos- sible to form a coherent superposition of the two attrac- tors states of motion. Laser-cooling of a nonlinear driven ion-oscillator has several interesting aspects that can be further explored. Since the Doppler shifts associated with the oscillation amplitudes in the nonlinear regime are signicant compared with the cooling transition line- width, the laser-cooling force is largely nonlinear in the oscillator velocity. Furthermore, the thermal state gen- erated by laser-cooling is the result of balance between the damping force and the inherent heating due to the recoil noise from spontaneous photon scattering. Close to the Dung instability, the ion-oscillator response to noise is quadrature dependent. One noise quadrature is largely enhanced whereas the other quadrature is sup- pressed [3]. Laser-cooling in this case is likely to produce squeezed states of motion. This work was partially supported by the ISF Morasha program and the Minerva foundation. [1] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, ser.Wiley Classics Library. New York: Wiley, 1995. [2] I. Kozinsky et. al. , Phys. Rev. Lett. 99, 207201 (2007). [3] R. Almog, S. Zaitsev, O. Shtempluck, and E. Buks, Phys. Rev. Lett. 98, 078103 (2007). [4] R. L. Badzey and P. Mohanty, Nature 437, 995 (2005). [5] B. Yurke, D. S. Greywall, A. N. Pargellis, and P. A. Busch, Phys. Rev. A 51, 4211 (1995). [6] E. Buks and B. Yurke, Phys. Rev. E 74, 046619 (2006). [7] J. S. Aldridge and A. N. Cleland, Phys. Rev. Lett. 94, 156403 (2005). [8] E. Babourina-Brooks, A. Doherty, and G. J. Milburn, New. J. Phys. 10, 105020 (2008). [9] D. Leibfried, R. Blatt, C. Monroe, and D. J. Wineland, Rev. Mod. Phys. 75, 281 (2003). [10] D. M. Meekhof et. al. , Phys. Rev. Lett. 76, 1796 (1996). [11] C. Monroe, D. M. Meekhof, B. E. King, D. J. Wineland, Science 272, 1131 (1996). [12] C. Marquet, F. Schmidt-Kaler, D. F. V. James, Appl. Phys. B 76, 199 (2003). [13] C. F. Roos et. al. , Phys. Rev. A 77, 040302(R) (2008). [14] A. A. Makarov, Anal. Chem. 68, 4257 (1996) [15] K. Vahala et. al. , Nature Physics 5, 682 (2009). [16] B. Ravindra and A. K. Mallik Phys. Rev. E 49, 4950 (1994) [17] As the life-time of the P 1=2level (8 ns) is much shorter than the oscillation period (228 ns), the photon scattering rate instantaneously adjusts as the ions' velocity varies. [18] R. Lifshitz and M. C. Cross, Review of Nonlinear Dy- namics and Complexity 1, 1 (2008) [19] S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb arXiv:cond-mat/053130v1 (2005). [20] The ratio between the two terms depends on the cooling laser detuning and is always below 0 :2. [21] I. Katz, A. Retzker, R. Straub, and R. Lifshitz, Phys. Rev. Lett. 99, 040404 (2007).

2010-03-08

We study the steady state motion of a single trapped ion oscillator driven to the nonlinear regime. Damping is achieved via Doppler laser-cooling. The ion motion is found to be well described by the Duffing oscillator model with an additional nonlinear damping term. We demonstrate a unique ability of tuning both the linear as well as the nonlinear damping coefficients by controlling the cooling laser parameters. Our observations open a way for the investigation of nonlinear dynamics on the quantum-to-classical interface as well as mechanical noise squeezing in laser-cooling dynamics.

A single-ion nonlinear mechanical oscillator

1003.1577v1

Thermal fluctuation field for current-induced domain wall motion Kyoung-Whan Kim and Hyun-Woo Lee PCTP and Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea /H20849Received 18 May 2010; revised manuscript received 23 August 2010; published 20 October 2010 /H20850 Current-induced domain wall motion in magnetic nanowires is affected by thermal fluctuation. In order to account for this effect, the Landau-Lifshitz-Gilbert equation includes a thermal fluctuation field and literatureoften utilizes the fluctuation-dissipation theorem to characterize statistical properties of the thermal fluctuationfield. However, the theorem is not applicable to the system under finite current since it is not in equilibrium. Toexamine the effect of finite current on the thermal fluctuation, we adopt the influence functional formalismdeveloped by Feynman and Vernon, which is known to be a useful tool to analyze effects of dissipation andthermal fluctuation. For this purpose, we construct a quantum-mechanical effective Hamiltonian describingcurrent-induced domain wall motion by generalizing the Caldeira-Leggett description of quantum dissipation.We find that even for the current-induced domain wall motion, the statistical properties of the thermal noise isstill described by the fluctuation-dissipation theorem if the current density is sufficiently lower than theintrinsic critical current density and thus the domain wall tilting angle is sufficiently lower than /H9266/4. The relation between our result and a recent result /H20851R. A. Duine, A. S. Núñez, J. Sinova, and A. H. MacDonald, Phys. Rev. B 75, 214420 /H208492007/H20850/H20852, which also addresses the thermal fluctuation, is discussed. We also find interesting physical meanings of the Gilbert damping /H9251and the nonadiabaticy parameter /H9252; while /H9251charac- terizes the coupling strength between the magnetization dynamics /H20849the domain wall motion in this paper /H20850and the thermal reservoir /H20849or environment /H20850,/H9252characterizes the coupling strength between the spin current and the thermal reservoir. DOI: 10.1103/PhysRevB.82.134431 PACS number /H20849s/H20850: 75.78.Fg, 75.60.Ch, 05.40.Ca I. INTRODUCTION Current-induced domain wall /H20849DW/H20850motion in a ferro- magnetic nanowire is one of representative examples tostudy the effect of spin-transfer torque /H20849STT/H20850. The motion of DW is generated by the angular momentum transfer betweenspace-time-dependent magnetization m /H6023/H20849x,t/H20850and conduction electrons, of which spins interact with m/H6023by the exchange coupling. This system is usually described by the Landau-Lifshitz-Gilbert /H20849LLG/H20850equation, 1–3 /H11509m/H6023 /H11509t=/H92530H/H6023eff/H11003m/H6023+/H9251 msm/H6023/H11003/H11509m/H6023 /H11509t+jp/H9262B ems/H20875/H11509m/H6023 /H11509x−/H9252 msm/H6023/H11003/H11509m/H6023 /H11509x/H20876, /H208491/H20850 where /H92530is the gyromagnetic ratio, jpis the spin-current density, ms=/H20841m/H6023/H20841is the saturation magnetization, and /H9262Bis the Bohr magneton. /H9251is the Gilbert damping coefficient, and /H9252 is the nonadiabatic coefficient representing the magnitude ofthe nonadiabatic STT. 4In Eq. /H208491/H20850, the effective magnetic field Heffis given by H/H6023eff=A/H116122m/H6023+H/H6023ani+H/H6023th, /H208492/H20850 where Ais stiffness constant, H/H6023anidescribes the effect of the magnetic anisotropy, and H/H6023this the thermal fluctuation field describing the thermal noise. In equilibrium situations, the magnitude and spatiotemporal correlation of H/H6023thare gov- erned by the fluctuation-dissipation theorem,5–7 /H20855Hth,i/H20849x/H6023,t/H20850Hth,j/H20849x/H6023/H11032,t/H11032/H20850/H20856=4/H9251kBT /H6036/H9267/H9254/H20849x/H6023−x/H6023/H11032/H20850/H9254/H20849t−t/H11032/H20850/H9254ij,/H208493/H20850 where /H20855¯/H20856represents the statistical average, i,jdenote x,y, orzcomponent, kBis the Boltzmann constant, Tis the tem-perature, and /H9267=ms//H9262Bis the spin density. Equation /H208493/H20850 plays an important role for the study of the magnetizationdynamics at finite temperature, 8 Equation /H208493/H20850has been also used in literature9–13to exam- ine effects of thermal fluctuations on the current-inducedDW motion. In nonequilibrium situations, however, the fluctuation-dissipation theorem does not hold generally.Since the system is not in equilibrium any more when thecurrent is applied, it is not clear whether Eq. /H208493/H20850may be still used. Recalling that H /H6023this estimated to affect the magnetiza- tion dynamics considerably in many experimentalsituations 14–17of the current-driven DW motion, it is highly desired to properly characterize H/H6023thin situations with non- zero jp. Recently, Duine18attempted this characterization and showed that Eq. /H208493/H20850is not altered by the spin current up to first order in the spin-current magnitude. This analysis how-ever is limited to situations where the spin-flip scattering isthe main mechanism responsible for /H9252. In this paper, we generalize this analysis by using a completely different ap-proach which does not assume any specific physical origin of /H9252. Htharises from extra degrees of freedom /H20849other than mag- netization /H20850, which are not included in the LLG equation. The extra degrees of freedom /H20849phonons for instance /H20850usually have much larger number of degrees of freedom than magnetiza-tion and thus form a heat reservoir. Thus properties of H thare determined by the heat reservoir. The heat reservoir playsanother role. In the absence of the extra degrees of freedom,the Gilbert damping coefficient /H9251should be zero since the total energy should be conserved when all degrees of free-dom are taken into account. Thus the heat reservoir is re-sponsible also for finite /H9251. These dual roles of the heat res- ervoir are the main idea behind the Einstein’s theory of thePHYSICAL REVIEW B 82, 134431 /H208492010/H20850 1098-0121/2010/82 /H2084913/H20850/134431 /H2084916/H20850 ©2010 The American Physical Society 134431-1Brownian motion.19There are also claims that /H9251is correlated with/H9252/H20849Refs. 18and20–22/H20850in the sense that mechanisms, which generate /H9252, also contribute to /H9251. Thus the issue of H/H6023th and the issue of /H9251and/H9252are mutually connected. Recalling that the main mechanism responsible for /H9251varies from ma- terial to material, it is reasonable to expect that the main mechanism for H/H6023thand/H9252may also vary from material to material. Recently, various mechanisms of /H9252were examined such as momentum transfer,23–25spin mistracking,26,27spin- flip scattering,18,21,22,25,28and the influence of a transport current.29This diversity of mechanisms will probably apply toH/H6023thas well. Instead of examining each mechanism of H/H6023thone by one, we take an alternative approach to address this issue. In1963, Feynman and Vernon 30proposed the so-called influ- ence functional formalism, which allows one to take accountof damping effects without detailed accounts of dampingmechanisms. This formalism was later generalized by Smith and Caldeira. 31This formalism has been demonstrated to be a useful tool to address dissipation effects /H20849without specific accounts of detailed damping mechanisms /H20850on, for instance, quantum tunneling,32nonequilibrium dynamic Coulomb blockade,33and quantum noise.34To take account of damp- ing effects which are energy nonconserving processes in gen-eral, the basic idea of the influence functional formalism is tointroduce infinite number of degrees of freedom /H20849called en- vironment /H20850behaves like harmonic oscillators which couple with the damped system. /H20851See Eq. /H2084913/H20850./H20852Caldeira and Leggett 32suggested the structure of the spectrum of environ- ment Eq. /H2084913/H20850and integrated out the degrees of freedom of environment to find the effective Hamiltonian describing theclassical damping Eq. /H2084912/H20850. For readers who are not familiar with the Caldeira-Leggett’s theory of quantum dissipation,we present the summary of details of the theory in Sec. II B. In order to address the issue of H /H6023th, we follow the idea of the influence functional formalism and construct an effectiveHamiltonian describing the magnetization dynamics. The ef-fective Hamiltonian describes not only energy-conservingprocesses but also energy-nonconserving processes such asdamping and STT. From this approach, we find that Eq. /H208493/H20850 holds even in nonequilibrium situations with finite j p, pro- vided that jpis sufficiently smaller than the so-called intrin- sic critical current density23so that the DW tilting angle /H9278 /H20849to be defined below /H20850is sufficiently smaller than /H9266/4. We remark that in the special case where the spin-flip scattering mechanism of /H9252is the main mechanism of H/H6023th, our finding is consistent with Ref. 18, which reports that the spin flip scat- tering mechanism does not alter Eq. /H208493/H20850at least up to the first order in jp. But our calculation indicates that Eq. /H208493/H20850holds not only in situations where the spin flip scattering is the dominant mechanism of H/H6023thand/H9252but also in more diverse situations as long as the heat reservoir can be described bybosonic excitations /H20849such as electron-hole pair excitations or phonon /H20850, i.e., the excitations effectively behave like har- monic oscillators to be described by Caldeira-Leggett’stheory. We also remark that in addition to the derivation ofEq./H208493/H20850in nonequilibrium situations, our calculation also re- veals an interesting physical meaning of /H9252, which will be detailed in Sec. III.This paper is organized as follows. In Sec. II, we first introduce the Caldeira-Leggett’s version of the influencefunctional formalism and later generalize this formalism sothat it is applicable to our problem. This way, we construct aHamiltonian describing the DW motion. In Sec. III, some implications of this model is discussed. First, a distinct in-sight on /H9252is emphasized. Second, as an application, statisti- cal properties of the thermal fluctuation field are calculatedin the presence of nonzero j p, which verifies the validity of Eq./H208493/H20850when jpis sufficiently smaller than the intrinsic criti- cal density. It is believed that many experiments16,17are in- deed in this regime. Finally, in Sec. IV, we present some concluding remarks. Technical details about the quantumtheory of the DW motion and methods to obtain solutions areincluded in Appendices. II. GENERALIZED CALDEIRA-LEGGETT DESCRIPTION A. Background Instead of full magnetization profile m/H6023/H20849x,t/H20850, the DW dy- namics is often described2,23,35–37by two collective coordi- nates, DW position x/H20849t/H20850and DW tilting angle /H9278/H20849t/H20850. When expressed in terms of these collective coordinates, the LLGEq./H208491/H20850reduces to the so-called Thiele equations, dx dt=jp/H9262B ems+/H9251/H9261d/H9278 dt+/H92530K/H9261 mssin 2/H9278+/H9257x/H20849t/H20850,/H208494a/H20850 /H9261d/H9278 dt=−/H9251dx dt+/H9252jp/H9262B ems+/H9257p/H20849t/H20850. /H208494b/H20850 Here Kis the hard-axis anisotropy, /H9261is the DW thickness. /H9257x/H20849t/H20850and/H9257p/H20849t/H20850are functions describing thermal noise field Hth,i/H20849x,t/H20850. By definition, the statistical average of the thermal noise field Hth,i/H20849x,t/H20850is zero and similarly the statistical aver- ages of /H9257x/H20849t/H20850and/H9257p/H20849t/H20850should also vanish regardless of whether the system is in equilibrium. The question of theircorrelation function is not trivial however. If the thermalnoise field H th,i/H20849x,t/H20850satisfies the correlation in Eq. /H208493/H20850,i tc a n be derived from Eq. /H208493/H20850that/H9257x/H20849t/H20850and/H9257p/H20849t/H20850satisfy the cor- relation relation12 /H20855/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856/H11008/H9251kBT/H9254ij/H9254/H20849t−t/H11032/H20850, /H208495/H20850 for/H20853i,j/H20854=/H20853x,p/H20854. But as mentioned in Sec. I, Eq./H208493/H20850is not guaranteed generally in the presence of the nonzero current.Then Eq. /H208495/H20850is not guaranteed either. The question of what should be the correlation function /H20855 /H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in such a situ- ation will be discussed in Sec. III. When the spin-current density jpis sufficiently smaller than the so-called intrinsic critical density /H20841e/H92530K/H9261//H9262B/H20841,23/H9278 stays sufficiently smaller than /H9266/4. In many experimental situations,38–40this is indeed the case,41so we will confine ourselves to the small /H9278regime in this paper. Then, one can approximate sin 2 /H9278/H110152/H9278to convert the equations into the following form:42 dx dt=vs+/H9251S 2KMdp dt+p M+/H92571/H20849t/H20850, /H208496a/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-2dp dt=−2/H9251KM Sdx dt+2/H9252KM Svs+/H92572/H20849t/H20850, /H208496b/H20850 where p=2KM/H9261/H9278/S,Sis the spin angular momentum at each individual magnetic site, and vs=jp/H9262B/emsis the adia- batic velocity,43which is a constant of velocity dimension and proportional to jp. The yet undetermined constant Mis the effective DW mass42–44which will be fixed so that the new variable pbecomes the canonical conjugate to x./H92571/H20849t/H20850 and/H92572/H20849t/H20850are the same as /H9257x/H20849t/H20850and/H9257p/H20849t/H20850except for propor- tionality constants. When the thermal noises /H92571/H20849t/H20850and/H92572/H20849t/H20850are ignored, one obtains from Eq. /H208496/H20850the time dependence of the DW posi- tion, x/H20849t/H20850=x/H208490/H20850+/H9252 /H9251vst+S 2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850 /H11003/H20851/H9251p/H208490/H20850−Mvs/H20849/H9251−/H9252/H20850/H20852. /H208497/H20850 Note that after a short transient time, the DW speed ap- proaches the terminal velocity /H9252vs//H9251. Thus the ratio /H9252//H9251is an important parameter for the DW motion. When the ther-mal noises are considered, they generate a correction to Eq./H208497/H20850. However, from Eq. /H208496/H20850, it is evident that the statistical average of x/H20849t/H20850should still follow Eq. /H208497/H20850. Thus as far as the temporal evolution of the statistical average is concerned, wemay ignore the thermal noises. In the rest of Sec. II,w ea i m to derive a quantum mechanical Hamiltonian, which repro-duces the same temporal evolution as Eq. /H208497/H20850in the statistical average level. In Sec. III, we use the Hamiltonian to derive the correlation function /H20855 /H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in the presence of the nonzero current. Now, we begin our attempt to construct an effective Hamiltonian that reproduces the DW dynamics Eq. /H208496/H20850/H20851or equivalently Eq. /H208497/H20850/H20852. We first begin with the microscopic quantum-mechanical Hamiltonian Hs-d, Hs-d=−J/H20858 iS/H6023i·S/H6023i+1−A/H20858 i/H20849S/H6023i·zˆ/H208502+K/H20858 i/H20849S/H6023i·yˆ/H208502+HcS, /H208498/H20850 which has been used in previous studies20of the DW dynam- ics. Here Jrepresents the ferromagnetic exchange interac- tion, AandKrepresent longitudinal /H20849easy-axis /H20850and trans- verse/H20849hard-axis /H20850anisotropy, respectively. The last term HcS represents the coupling of the spin system with the spin- polarized current, HcS=−/H20858 i,/H9251=↑,↓/H20851t/H20849ci/H9251†ci+1/H9251+ci+1/H9251†ci/H9251/H20850−/H9262ci/H9251†ci/H9251/H20852−JH/H20858 iS/H6023ci·S/H6023i, /H208499/H20850 where JHis the exchange interaction between conduction electron and the localized spins, ci/H9251is the annihilation opera- tor of the conduction electron at the site i,S/H6023ciis the electron- spin operator, tis the hopping integral, and /H9262is the chemical potential of the system.Recently Kim et al.43analyzed Hs-din detail in the small tilting angle regime and found that Hs-dcontains gapless low-lying excitations and also high-energy excitations with afinite energy gap. The gapless excitations of H s-dare de- scribed by a simple Hamiltonian H0, H0=vsP+P2 2M/H2084910/H20850 while the high-energy excitations have a finite energy gap 2S/H20881A/H20849A+K/H20850.I nE q . /H2084910/H20850,Pis the canonical momentum of the DW position operator Q, and M=/H60362 K/H208812A Ja4is the effective DW mass called Döring mass.44Here, ais the lattice spacing between two neighboring spins. /H20849See, for details, Appendix A./H20850Below we will neglect the high energy excitations and focus on the low-lying excitations described by Eq. /H2084910/H20850. For the analysis of the high-energy excitation effects on the DW,See Ref. 42. From Eq. /H2084910/H20850, one obtains the following Heisenberg’s equation of motion: dQ dt=vs+P M, /H2084911a/H20850 dP dt=0 . /H2084911b/H20850 Note that the current /H20849proportional to vs/H20850appears in the equa- tion fordQ dt. Thus the current affects the DW dynamics by introducing a difference between the canonical momentum P and the kinematic momentum P+Mvs. In this sense, the ef- fect of the current is similar to a vector potential /H20851canonical momentum P/H6023and kinematic momentum P/H6023+/H20849e/c/H20850A/H6023/H20852. The vector potential /H20849difference between the canonical momen- tum and the kinetic momentum /H20850allows the system in the initially zero momentum state to move without breaking thetranslational symmetry of the system. In other words, thecurrent-induced DW motion is generated without any forceterm in Eq. /H2084911b/H20850violating the translational symmetry of the system. This should be contrasted with the effect of the mag-netic field or magnetic defects, which generates a force termin Eq. /H2084911b/H20850. The solution of Eq. /H2084911/H20850is trivial, /H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856 +/H20849/H20855P/H208490/H20850/H20856/M+ vs/H20850t. Here, the statistical average /H20855¯/H20856is de- fined as /H20855¯/H20856=Tr/H20849/H9267¯/H20850/Tr/H20849/H9267/H20850, where /H9267denotes the density matrix at t=0. Associating /H20855Q/H20849t/H20850/H20856=x/H20849t/H20850,/H20855P/H20849t/H20850/H20856=p/H20849t/H20850, one finds that Eq. /H2084911/H20850is identical to Eq. /H208496/H20850if/H9251=/H9252=0. This implies that the effective Hamiltonian H0/H20851Eq./H2084910/H20850/H20852fails to capture effects of nonzero /H9251and/H9252. In the next three sections, we attempt to resolve this problem. B. Caldeira-Leggett description of damping To solve the problem, one should first find a way to de- scribe damping. A convenient way to describe finite dampingwithin the effective Hamiltonian approach is to adopt theCaldeira-Leggett description 32of the damping. Its main idea is to introduce a collection of additional degrees of freedom/H20849called environment /H20850and couple them to the original dy- namic variables so that energy of the dynamic variables canTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-3be transferred to the environment. For instance, for a one- dimensional /H208491D/H20850particle subject to damped dynamics, dx dt=p M, /H2084912a/H20850 dp dt=−dV/H20849x/H20850 dx−/H9253dx dt. /H2084912b/H20850 Caldeira and Leggett32demonstrated that its quantum- mechanical Hamiltonian can be constructed by adding damp-ing Hamiltonian H 1to the undamped Hamiltonian H0 =P2/2M+V/H20849Q/H20850. The damping Hamiltonian H1contains a collection of environmental degrees of freedom /H20853xi,pi/H20854be- having like harmonic oscillators /H20851see Eq. /H2084914/H20850/H20852, which couple to the particle through the linear coupling term /H20858iCixiQbe- tween Qand the environmental variables xi. Here, Ciis the coupling constant between xiandQ. The implication of the coupling is twofold: /H20849i/H20850the coupling to the environment gen- erates damping, whose precise form depends on Ci,mi, and /H9275i. It is demonstrated in Ref. 32that the coupling generates the simple damping of the form in Eq. /H2084912b/H20850ifCi,mi, and/H9275i satisfy the following relation of the spectral function J/H20849/H9275/H20850: J/H20849/H9275/H20850/H11013/H9266 2/H20858 iCi2 mi/H9275i/H9254/H20849/H9275−/H9275i/H20850=/H9253/H9275. /H2084913/H20850 /H20849ii/H20850The coupling also modified the potential Vby generating an additional contribution − /H20858iCi2Q2/2mi/H9275i2. This implies that V/H20849x/H20850in Eq. /H2084912b/H20850should not be identified with V/H20849Q/H20850inH0 /H20849even though the same symbol Vis used /H20850but should be iden- tified instead with the total potential that includes the contri-bution from the environmental coupling. If we express thetotal Hamiltonian Hin terms of the effective V/H20849x/H20850that ap- pears in Eq. /H2084912b/H20850, it reads H=H 0+H1, /H2084914a/H20850 H0=P2 2M+V/H20849Q/H20850, /H2084914b/H20850 H1=/H20858 i/H20875pi2 2mi+1 2mi/H9275i2/H20873xi+Ci mi/H9275i2Q/H208742/H20876./H2084914c/H20850 By identifying x/H20849t/H20850=/H20855Q/H20849t/H20850/H20856,p/H20849t/H20850=/H20855P/H20849t/H20850/H20856, the equations of motion obtained from Eqs. /H2084913/H20850and/H2084914/H20850reproduce Eq. /H2084912/H20850. C. Generalization to the DW motion: /H9251term Here we aim to apply the Caldeira-Leggett approach to construct an effective Hamiltonian of the DW dynamics sub-ject to finite damping /H20849 /H9251/HS110050/H20850. To simplify the problem, we first focus on a situation, where only /H9251is relevant and /H9252is irrelevant. This situation occurs if there is no current /H20849vs =0/H20850. Then Eq. /H208496/H20850reduces to dx dt=/H9251S 2KMdp dt+p M, /H2084915a/H20850dp dt=−2/H9251KM Sdx dt. /H2084915b/H20850 Note that /H9252does not appear. Note also that these equations are slightly different from Eq. /H2084912/H20850, where a damping term is contained only in the equation ofdp dt. However, in the equa- tions of the DW /H20851Eq./H2084915/H20850/H20852, damping terms appear not only in the equation ofdp dt/H20851Eq./H2084915b/H20850/H20852but also in the equation ofdx dt /H20851Eq./H2084915a/H20850/H20852. Thus the Caldeira-Leggett description in the preceding section is not directly applicable and should be generalized.To get a hint, it is useful to recall the conjugate relation between QandP. The equations ofdQ dtanddP dtare obtained by differentiating Hwith respect to Pand − Q, respectively. Of course, it holds for /H20849xi,pi/H20850, also. Thus, one can obtain another set of Heisenberg’s equation of motion by exchang-ing/H20849Q,x i/H20850↔/H20849−P,−pi/H20850. By this canonical transformation, the position coupling /H20858iCixiQchanges to a momentum coupling term, and the damping term in the equation ofdP dtis now in that ofdQ dt. This mathematical relation that the momentum coupling generates a damping term in the equation ofdQ dt makes it reasonable to expect that the momentum coupling /H20858iDipiPis needed45to generate the damping in the equation fordQ dt. Here Diis the coupling constant between Pandpi. The reason why, in the standard Caldeira-Leggett approach,the damping term appears only in Eq. /H2084912b/H20850is that Eq. /H2084914/H20850 contains only position coupling terms /H20858 iCixiQ. It can be eas- ily verified that the implications of the momentum couplingare again twofold: /H20849i/H20850the coupling indeed introduces the damping term in the equation ofdQ dt./H20849ii/H20850it modifies the DW mass. The mass renormalization arises from the fact that inthe presence of the momentum coupling /H20858 iDipiP, the kine- matic momentum midxi dtof an environmental degree of free- dom xiis now given by /H20849pi+DimiP/H20850instead of pi. Then the term/H20858i/H20851pi2 2mi+DipiP/H20852can be decomposed into two pieces, /H20858i/H20849pi+DimiP/H208502 2mi, which is the kinetic energy associated with xi, and/H20851−/H20858iDi2mi 2/H20852P2. Note that the second piece has the same form as the DW kinetic termP2 2M. Thus this second piece generates the renormalization of the DW mass. Due to thismass renormalization effect, Min Eq. /H2084915/H20850should be inter- preted as the renormalized mass that contains the contribu-tion from the environmental coupling. If MinH 0in Eq. /H2084910/H20850 is interpreted as the renormalized mass, the environmentHamiltonian H 2for the DW dynamics becomes H2=/H20858 i/H208751 2mi/H20849pi+DimiP/H208502+1 2mi/H9275i2/H20873xi+Ci mi/H9275i2Q/H208742/H20876. /H2084916/H20850 Here,/H20858i/H20849pi+DimiP/H208502/2micoupling is equivalent to the origi- nal form /H20858i/H20849pi2/2mi+DipiP/H20850under the mass renormalization 1/M→1/M−/H20858iDi2mi/2. Note that in H2, the collective co- ordinates QandPof the DW couple to the environmental degrees of freedom /H20853xi,pi/H20854through two types of coupling, /H20858iCixiQand/H20858iDipiP. Finally, one obtains the total Hamiltonian describing the DW motion in the absence of the current,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-4H=H0/H20841vs=0+H2=P2 2M+/H20858 i/H208751 2mi/H20849pi+DimiP/H208502 +1 2mi/H9275i2/H20873xi+Ci mi/H9275i2Q/H208742/H20876. /H2084917/H20850 Now, the renormalized mass Min the above equation is iden- tical to the mass in Eq. /H2084915/H20850. To make the physical meaning ofxiclearer, we perform the canonical transformation, xi→−Ci mi/H9275i2xi,pi→−mi/H9275i2 Cipi. /H2084918/H20850 Defining /H9253i=CiDi /H9275i2, and redefining a new miasmi/H20849new/H20850=Ci2 mi/H9275i4, the Hamiltonian becomes simpler as H=P2 2M+/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876. /H2084919/H20850 Now, the translational symmetry of the system and the physi- cal meaning of xibecome obvious. The next step is to impose proper constraints on /H9253iandmi, so that the damping terms arising from Eq. /H2084919/H20850agree exactly with those in Eq. /H2084915/H20850. For this purpose, it is convenient to introduce Laplace transformed variables Q˜/H20849/H9261/H20850,P˜/H20849/H9261/H20850,x˜i/H20849/H9261/H20850, p˜i/H20849/H9261/H20850, where Q˜/H20849/H9261/H20850=/H208480/H11009e−/H9261t/H20855Q/H20849t/H20850/H20856dt, and other transformed variables are defined in a similar way. Then the variables x˜i andp˜ican be integrated out easily /H20849see Appendix B /H20850. After some tedious but straightforward algebra, it is verified thatwhen the following three constraints on /H9253i,/H9275i,miare satis- fied for any positive /H9261, /H20858 i/H9253i/H9275i2 /H92612+/H9275i2=0 , /H2084920a/H20850 /H20858 i/H9253i2/H9261 mi/H20849/H92612+/H9275i2/H20850=/H9251S 2KM, /H2084920b/H20850 /H20858 imi/H9275i2/H9261 /H92612+/H9275i2=2/H9251KM S, /H2084920c/H20850 the DW dynamics satisfies the following equation: /H20898/H9261 −1 M−/H9251S/H9261 2KM 2/H9251KM S/H9261 /H9261/H20899/H20873Q˜ P˜/H20874 =/H20873/H20855Q/H208490/H20850/H20856 /H20855P/H208490/H20850/H20856/H20874+/H20898−/H9251S 2KM/H20855P/H208490/H20850/H20856 2/H9251KM S/H20855Q/H208490/H20850/H20856/H20899, /H2084921/H20850 which is nothing but the Laplace transformation of the DW equation /H20851Eq./H2084915/H20850/H20852if/H20855Q/H20856and/H20855P/H20856are identified with xandp. Thus we verify that the Hamiltonian Hin Eq. /H2084919/H20850indeed provides a generalized Caldeira-Leggett-type quantumHamiltonian for the DW motion. As a passing remark, we mention that in the derivation of Eq. /H2084921/H20850, the environmental degrees of freedom at the initial moment /H20849t=0/H20850are assumed to be in their thermal equilibrium so that /H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856, /H2084922a/H20850 /H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856. /H2084922b/H20850 Equation /H2084922/H20850can be understood as follows. First, one ob- tains Eq. /H2084922/H20850by following Appendix D which describes the statistical properties of Eq. /H2084919/H20850at high temperature. In Ap- pendix D, /H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856is reduced to an in- tegration of an odd function so it is shown to vanish. Thesecond way is probably easier to understand and does notrequire the classical limit or high-temperature limit. TheHamiltonian /H20851Eq./H2084919/H20850/H20852is symmetric under the canonical transformation Q/H208490/H20850→−Q/H208490/H20850,P/H208490/H20850→−P/H208490/H20850,x i/H208490/H20850→−xi/H208490/H20850, and pi/H208490/H20850→−pi/H208490/H20850. Due to this symmetry, one obtains /H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855Q/H208490/H20850−xi/H208490/H20850/H20856and/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856=/H20855/H9253iP/H208490/H20850 −pi/H208490/H20850/H20856, which lead to /H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856and/H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856, respectively. Here physical origin of the momentum coupling /H20849/H9253/H20850be- tween the DW and environment deserves some discussion.Equation /H2084919/H20850is reduced to the original Caldeira-Leggett Hamiltonian if /H9253i=0. However, Eq. /H2084920b/H20850implies that the momentum coupling as well as the position coupling is in-dispensable to describe the Gilbert damping. To understandthe origin of the momentum coupling /H9253i, it is useful to recall that since P/H11008/H9278/H11008/H20849tilting/H20850, one can interpret Pand Qas transverse and longitudinal spin fluctuation of the DW state,respectively. /H20849See, for explicit mathematical relation, Appen- dix A. /H20850Thus, if there is rotational symmetry on spin interac- tion with the heat bath /H20849or environment /H20850, the existence of the position coupling requires the existence of the momentumcoupling. Thus the appearance of the damping terms both inEqs./H2084915a/H20850and/H2084915b/H20850is natural in view of the rotational sym- metry of the spin exchange interaction and also in view ofthe physical meaning of PandQas transverse and longitu- dinal spin fluctuations. D. Coupling with the spin current: /H9252term In this section, we aim to construct a Caldeira-Leggett- type effective quantum Hamiltonian that takes account of notonly /H9251but also /H9252. Since /H9252becomes relevant only when there exists finite spin current, we have to deal with situations withfinite current /H20849 vs/HS110050/H20850. Then the system is notin thermal equi- librium. As demonstrated in Eq. /H2084910/H20850, the spin current couples with the DW linear momentum, i.e., vsP. Here, adiabatic velocity vsacts as the coupling constant proportional to spin current. The spin current may also couple directly to the environmen-tal degrees of freedom. Calling this coupling constant v, one introduces the corresponding coupling term /H20858ivpi. Later we find that this coupling is crucial to account for nonzero /H9252.A t this point we will not specify the value of v. Now, the total effective Hamiltonian in the presence of the spin current ob-tained by adding the coupling term /H20858 ivpito Eq. /H2084919/H20850. Then,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-5Htot=H+Hcurrent =P2 2M+vsP+/H20858 ivpi +/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876./H2084923/H20850 In order to illustrate the relation between Eqs. /H208496/H20850and /H2084923/H20850, we consider a situation, where the current is zero until t=0 and turned on at t=0 to a finite value. This situation is described by the following time-dependent Hamiltonian: Htot=P2 2M+vs/H20849t/H20850P+/H20858 iv/H20849t/H20850pi +/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876,/H2084924/H20850 where vs/H20849t/H20850=vs/H9008/H20849t/H20850andv/H20849t/H20850=v/H9008/H20849t/H20850. And /H9008/H20849t/H20850is /H9008/H20849t/H20850=/H208771fort/H110220, 0fort/H110210./H20878 /H2084925/H20850 To make a quantitative comparison between Eqs. /H208496/H20850and /H2084924/H20850, one needs to integrate out environmental degrees of freedom /H20853xi,pi/H20854, which requires one to specify their initial conditions. Since the system is in thermal equilibrium untilt=0, we may still impose the constraint in Eq. /H2084922/H20850to exam- ine the DW dynamics for t/H110220. By following a similar pro- cedure as in Sec. II C and by using the constraints in Eq. /H2084920/H20850, 46one finds that the effective Hamiltonian H/H20851Eq./H2084924/H20850/H20852 predicts/H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856+vt+S 2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850 /H11003/H20851/H9251/H20855P/H208490/H20850/H20856−M/H9251/H20849vs−v/H20850/H20852. /H2084926/H20850 This is exactly the same as Eq. /H208497/H20850if /H9252 /H9251=v vs. /H2084927/H20850 So by identifying vwith vs/H9252//H9251, we obtain a Caldeira- Leggett-type effective quantum Hamiltonian of the DW dy-namics. One needs to consider an external force on Eq. /H208496b/H20850/H20851or Eq./H208494b/H20850/H20852when the translational symmetry of the system is broken by some factors such as external magnetic field andmagnetic defects. To describe this force, one can add a posi-tion dependent potential V/H20849Q/H20850/H20849Ref. 47/H20850to Eq. /H2084924/H20850. Consid- ering the Heisenberg’s equation, the potential V/H20849Q/H20850generates the term − V /H11032/H20849Q/H20850in Eq. /H208496b/H20850. III. IMPLICATIONS A. Insights on the physical meaning of /H9252 Equation /H2084927/H20850provides insights on the physical meaning of/H9252./H9252depends largely on the coupling between the envi- ronment and current, not on the damping form. Recallingthat vsdescribes the coupling between the current and the DW, we find that /H9252//H9251, which describes the asymptotic be- havior of the DW motion, is the ratio between the current-magnetization /H20849DW in the present case /H20850coupling and current-environment coupling. That is, /H9252 /H9251=/H20849Coupling between the current and the environment /H20850 /H20849Coupling between the current and the DW /H20850. /H2084928/H20850 To make the physical meaning of Eq. /H2084928/H20850more transpar- ent, it is useful to examine consequences of the nonzero cou-pling vbetween the current and the environment. One of the immediate consequences of the nonzero vappears in the ve- locities of the environmental degrees of freedom. It can beverified easily that the initial velocities of environmental co- ordinates are given by exactly v,/H20855x˙i/H208490/H20850/H20856=v. Recalling that the terminal velocity of the DW, /H20855Q˙/H20849t/H20850/H20856approaches vs/H9252//H9251, one finds from Eq. /H2084927/H20850that the terminal velocity of the DW is nothing but the environment velocity. This result is verynatural since the total Hamiltonian H tot/H20851Eq./H2084923/H20850/H20852is Galilean invariant and the total mass of the environment /H20849or reservoir /H20850 is much larger than the DW mass.48A very similar conclu- sion is obtained by Garate et al.29By analyzing the Kamber- sky mechanism,49which is reported50to be the dominant damping mechanism in transition metals such as Fe, Co, Ni,they found that the ratio /H9252//H9251is approximately given by the ratio between the drift velocity of the Kohn-Sham quasipar- ticles and vs. Since the collection of Kohn-Sham quasiparti-cles play the role of the environment in case of the Kamber- sky mechanism, the result in Ref. 29is consistent with ours. It is interesting to note that our calculation, which is largelyindependent of details of damping mechanism, reproducesthe result for the specific case. 29This implies that the result in Ref. 29can be generalized if the drift velocity of the Kohn-Sham quasiparticles is replaced by the general cou-pling constant vbetween the current and the environment. Our claim that the origin of /H9252is the direct coupling be- tween the current and environment has an interesting con-ceptual consistency with the work by Zhang and Li. 4Zhang and Li derived the nonadiabatic term by introducing a spin-relaxation term in the equation of motion of the conduction electrons. A clear consistency arises from generalizing thespin relaxation in Ref. 4to the coupling with environment in our work. In Ref. 4, Gilbert damping /H20849 /H9251/H20850and the nonadia- batic STT /H20849/H9252/H20850are identified as the spin relaxation of magne- tization and conduction electrons, respectively. Generalizingthe spin relaxation to environmental coupling, /H9251and/H9252areKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-6now identified as the coupling of the environment with the magnetization /H20849i.e., the DW in our model /H20850and the coupling of the environment with current, respectively. It is exactlyhow we identified /H9251and/H9252, and this gives the conceptual consistency between our work and Ref. 4. As an additional comment, while some magnitudes and origins of /H9252claimed in different references, such as Refs. 4and29, seem to be based on completely independent phenomena, our work andinterpretation on /H9252provide a connection between them through the environmental degrees of freedom. B. Effect of environment on stochastic forces Until now, our considerations has been limited to the evo- lution of the expectation values /H20855Q/H20849t/H20850/H20856and/H20855P/H20849t/H20850/H20856and thus thermal fluctuation effects have been ignored. In this section,we address the issue of thermal fluctuations. For this pur-pose, we need to go beyond the expectation values and so wederive the following operator equations from the Hamil- tonian Eq. /H2084923/H20850: Q˙= vs+/H9251S 2KMP˙+P M+/H92571/H20849t/H20850, /H2084929a/H20850 P˙=−2/H9251KM SQ˙+2/H9251KM Sv+/H92572/H20849t/H20850, /H2084929b/H20850 where /H92571/H20849t/H20850=/H20858 i/H9253i/H9275i/H20873/H9004xisin/H9275it−/H9004pi mi/H9275icos/H9275it/H20874,/H2084930a/H20850 /H92572/H20849t/H20850=/H20858 i/H20849mi/H9275i2/H9004xicos/H9275it+/H9275i/H9004pisin/H9275it/H20850./H2084930b/H20850 Here/H9004xi/H11013xi/H208490/H20850−Q/H208490/H20850and/H9004pi/H11013pi/H208490/H20850−/H9253iP/H208490/H20850. The deriva- tion of Eqs. /H2084929/H20850and/H2084930/H20850utilizes constraints Eqs. /H2084920/H20850and /H2084927/H20850. We remark that the result in Sec. II D can be recovered from Eqs. /H2084929/H20850and/H2084930/H20850by taking the expectation values of the operators. When Eq. /H2084929/H20850is compared to Eq. /H208496/H20850,i ti s evident that /H92571/H20849t/H20850and/H92572/H20849t/H20850defined in Eq. /H2084930/H20850carry the information about the thermal noise. It is easy to verify thatthe expectation values of /H92571/H20849t/H20850and/H92572/H20849t/H20850vanish, thus repro- ducing the results in the earlier section. Here it should benoticed that Eq. /H2084930/H20850relates /H92571/H20849t/H20850and/H92572/H20849t/H20850in the nonequi- librium situations /H20849after the current is turned on or t/H110220/H20850to the operators /H9004xiand/H9004pi, which are defined in the equilib- rium situation /H20849right before the current is turned on or t=0/H20850. Thus by combining Eq. /H2084930/H20850with the equilibrium noise char- acteristics of /H9004xiand/H9004pi, we can determine the thermal noise characteristic in the nonequilibrium situation /H20849t/H110220/H20850. To extract information about the noise, one needs to evaluate the correlation functions /H20855/H20853/H9257i/H20849t/H20850,/H9257j/H20849t/H20850/H20854/H20856 /H20849i,j=1,2/H20850, where /H20853,/H20854denotes the anticommutator. Due to the relations in Eq./H2084930/H20850, the evaluation of the correlation function reduces to the expectation value evaluation of the operator products/H20853x i/H208490/H20850,pj/H208490/H20850/H20854,xi/H208490/H20850xj/H208490/H20850, and pi/H208490/H20850pj/H208490/H20850in the equilibrium situation governed by the equilibrium Hamiltonian /H20851Eq. /H2084919/H20850/H20852. In the classical limit /H20849/H6036→0, see the next paragraph to find out when the classical limit is applicable /H20850, Eq./H2084919/H20850is just acollection of independent harmonic oscillators of /H20853/H9004xi,/H9004pi/H20854. Hence, the equipartition theorem determines their correla-tions, /H20855/H9004x i/H20856=/H20855/H9004pi/H20856=/H20855/H9004xi/H9004pi/H20856=0 , /H2084931a/H20850 /H20855/H9004xi/H9004xj/H20856=kBT mi/H9275i2/H9254ij, /H2084931b/H20850 /H20855/H9004pi/H9004pj/H20856=mikBT/H9254ij. /H2084931c/H20850 Equation /H2084920/H20850and/H2084931/H20850give the correlations of /H92571/H20849t/H20850and /H92572/H20849t/H20850. After some algebra, one straightforwardly gets /H20855/H9257i/H20849t/H20850/H20856=0 , /H2084932a/H20850 /H20855/H92571/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=0 , /H2084932b/H20850 /H20855/H92571/H20849t/H20850/H92571/H20849t/H11032/H20850/H20856=/H9251S 2KMkBT/H9254/H20849t−t/H11032/H20850, /H2084932c/H20850 /H20855/H92572/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=2/H9251KM SkBT/H9254/H20849t−t/H11032/H20850. /H2084932d/H20850 These relations are consistent with Eq. /H208495/H20850when/H92571/H20849t/H20850and /H92572/H20849t/H20850in Eq. /H2084930/H20850are identified with those in Eq. /H208496/H20850. Thus they confirm that the relations /H20851Eq./H2084932/H20850/H20852assumed in many papers9–13indeed hold rather generally in the regime where the tilting angle remains sufficiently smaller than /H9266/4. Next we consider the regime where the condition of the classical limit is valid. Since statistical properties of the sys- tem at finite temperature is determined bykBT /H6036, the classical limit/H20849/H6036→0/H20850is equivalent to the high-temperature limit /H20849T →/H11009/H20850. Thus, in actual experimental situations, the above cor- relation relations, Eq. /H2084932/H20850, will be satisfied at high tempera- ture. In this respect, we find that most experimental situa-tions belong to the high-temperature regime. See AppendixD for the estimation of the “threshold” temperature, abovewhich Eq. /H2084932/H20850is applicable. In Appendix D, the correlations in the high temperatures are derived more rigorously. Finally we comment briefly on the low-temperature quan- tum regime. In this regime, one cannot use the equipartitiontheorem since the system is not composed of independentharmonic oscillators, that is, /H20851/H9004x i,/H9004pj/H20852=i/H6036/H20849/H9254ij+/H9253j/H20850. Note that the commutator contains an additional term i/H6036/H9253j. Here, the additional term i/H6036/H9253jcomes from the commutator /H20851−Q, −/H9253jP/H20852. Then, Eq. /H2084932/H20850, which is assumed in other papers,9–13 is not guaranteed any more. IV . CONCLUSION In this paper, we examine the effect of finite current on thermal fluctuation of current-induced DW motion by con-structing generalized Caldeira-Leggett-type Hamiltonian ofthe DW dynamics, which describes not only energy-conserving dynamics processes but also the Gilbert dampingand STT. Unlike the classical damping worked out by Cal-deira and Leggett, 32the momentum coupling is indispensable to describe the Gilbert damping. This is also related to theTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-7rotational symmetry of spin-interaction nature. It is demon- strated that the derived Caldeira-Leggett-type quantum-mechanical Hamiltonian reproduces the well-known DWequations of motion. Our Hamiltonian also illustrates that the nonadiabatic STT is closely related with the coupling of the spin current tothe environment. Thus, the environmental degrees of free-dom are responsible for both the Gilbert damping /H20849 /H9251/H20850and the nonadiabatic STT /H20849/H9252/H20850. By this process, the ratio of /H9252and/H9251 was derived to be the ratio of current-DW coupling and current-environment coupling. The nonadiabatic term isnothing but the result of the direct coupling between thecurrent and environment in our theory. By using the Calderia-Leggett-type Hamiltonian, which describes the time evolution of the system, we obtained theexpression of stochastic forces caused by thermal noise inthe presence of the finite current. By calculating the equilib-rium thermal fluctuation at high temperature, we verify thatwhen j pis sufficiently smaller than the intrinsic critical den- sity, jpdoes not modify the correlation relations of thermal noise unless the temperature is extremely low. The upperbound of the critical temperature, below which the aboveconclusion does not apply, is obtained by reexamining thesystem with Feynman path integral. The bound is muchlower than the temperature in most experimental situations. Lastly we remark that the Joule heating 51is an important factor that affects the thermal fluctuation field since it raisesthe temperature of the nanowire. The degree of the tempera-ture rise depends on the thermal conductivities and heat ca-pacities of not only the nanowire but also its surroundingmaterials such as substrate layer materials of the nanowire.Such factors are not taken into account in this paper. Simul-taneous account of the Joule heating dynamics and the ther-mal fluctuation field /H20849in the presence of current /H20850goes beyond the scope of the paper and may be a subject of future re-search. ACKNOWLEDGMENTS We acknowledge critical comment by M. Stiles, who pointed out the importance of the momentum coupling andinformed us of Ref. 29. This work was financially supported by the NRF /H20849Grants No. 2007-0055184, No. 2009-0084542, and No. 2010-0014109 /H20850and BK21. K.W.K. acknowledges the financial support by the TJ Park. APPENDIX A: EFFECTIVE HAMILTONIAN OF THE DW MOTION FROM 1D s-dMODEL (Ref. 52) The starting point is 1D s-dmodel, Hs-d=−J/H20858 iS/H6023i·S/H6023i+1−A/H20858 i/H20849S/H6023i·zˆ/H208502+K/H20858 i/H20849S/H6023i·yˆ/H208502+HcS, /H20849A1/H20850 as mentioned in Sec. II A. In order to consider the DW dynamics, one first introduce the classical DW profile initially given by /H20855S/H6023i·xˆ/H20856=Ssin/H9258/H20849zi/H20850, /H20849A2a/H20850/H20855S/H6023i·yˆ/H20856=0 , /H20849A2b/H20850 /H20855S/H6023i·zˆ/H20856=Scos/H9258/H20849zi/H20850, /H20849A2c/H20850 where ziis the position of the ith localized spin, and /H9258/H20849z/H20850 =2 cot−1e−/H208812A/Ja2/H20849z−q/H20850. Here qis the classical position of the DW. Small quantum fluctuations of spins on top of the clas-sical DW profile can be described by the Holstein-Primakoffboson operator b i, to describe magnon excitations. Kim et al.43found eigenmodes of these quantum fluctuations in the presence of the classical DW background, which amount toquantum mechanical version of the classical vibration eigen-modes in the presence of the DW background reported longtime ago by Winter. 53The corresponding eigenstates of this Hamiltonian are composed of spin-wave states with the finite eigenenergy Ek=/H20881/H20849JSa2k2+2AS/H20850/H20849JSa2k2+2AS+2KS/H20850 /H20849/H113502S/H20881A/H20849A+K/H20850/H20850and so-called bound magnon states with zero energy Ew=0. Here, kis the momentum of spin wave states and ais the lattice spacing between two neighboring spins. Let akandbwdenote proper linear combinations of bi andbi†, which represent the boson annihilation operators of finite-energy spin-wave states and zero-energy bound mag-non states, respectively. In terms of these operators, Eq. /H208498/H20850 reduces to H s-d=P2 2M+/H20858 kEkak†ak+HcS, /H20849A3/H20850 where higher-order processes describing magnon-magnon in- teractions are ignored. Here Mis the so-called Döring mass,44defined as M=/H60362 K/H208812A Ja4, and Pis defined as −i/H6036/H208492AS2 Ja4/H208501/4/H20849bw†−bw/H20850. According to Ref. 43,Pis a translation generator of the DW position, that is, exp /H20849iPq 0//H6036/H20850shifts the DW position by q0. Thus Pcan be interpreted as a canonical momentum of the DW translational motion. The first term inEq./H20849A3/H20850, which amounts to the kinetic energy of the DW translational motion, implies that Mis the DW mass. We identify this Mwith the undetermined constant Min Eq. /H208496/H20850. According to Ref. 43,Pis also proportional to the degree of the DW tilting, that is, /H20849b w†−bw/H20850/H11008Siy. In the adiabatic limit, that is, when the DW width /H9261is sufficiently large in view of the electron dynamics, the re-maining term H cScan be represented in a simple way in terms of the bound magnon operators and the adiabatic ve-locity of the DW, 20,43 HcS=vsP. /H20849A4/H20850 Then the effective s-dHamiltonian of the DW motion be- comes Hs-d=P2 2M+vsP+/H20858 kEkak†ak. /H20849A5/H20850 Note that the bound magnon part and the spin-wave part are completely decoupled in Eq. /H20849A5/H20850since Pcontains only the bound magnon operators, which commute with the spin-wave operators. The DW position operator should satisfy the following two properties: geometrical relation /H20855Q/H20856−q=a 2S/H20858i/H20855S/H6023i·zˆ/H20856andKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-8canonical relation /H20851Q,P/H20852=i/H6036. Then, one can show that Q =q−/H20849Ja4 32AS2/H208501/4/H20849bw†+bw/H20850satisfies these two properties. Note thatQis expressed in terms of the bound magnon operators. Then as far as the Heisenberg equations of motion for Qand Pare concerned, the last term in Eq. /H20849A5/H20850does not play any role. This term will be ignored from now on. Thus, the ef-fective Hamiltonian for the DW motion is reduced to H 0=P2 2M+vsP /H20849A6/H20850 so we got Eq. /H2084910/H20850, the effective Hamiltonian of the DW motion. APPENDIX B: SOLUTION FOR A GENERAL QUADRATIC DAMPING This section provides the solution of the equation of mo- tion for a general quadratic damping. This is applicable notonly for the generalized Caldeira-Leggett description in thispaper but also for any damping type which quadraticallyinteracts with the DW. In general, let us consider a general quadratic damping Hamiltonian, H=P 2 2M+vsP+/H20858 i/H9273iTAi/H9273i, /H20849B1/H20850 where /H9273i=/H20849QPx ipi/H20850T, and Aii sa4/H110034 Hermitian matrix. Now, one straightforwardly gets the corresponding coupledequations, dQ dt=P M+vs+/H20858 i/H20849B21iQ+B22iP+B23ixi+B24ipi/H20850, /H20849B2a/H20850 dP dt=−/H20858 i/H20849B11iQ+B12iP+B13ixi+B14ipi/H20850,/H20849B2b/H20850 dxi dt=B41iQ+B42iP+B43ixi+B44ipi, /H20849B2c/H20850 dpi dt=−/H20849B31iQ+B32iP+B33ixi+B34ipi/H20850./H20849B2d/H20850 Here, Bii sa4 /H110034 real symmetric matrix defined as Bi =2 Re /H20851Ai/H20852, and Bjkiis the element of Biinjth row and kth column. With the Laplace transform of the expectation values of each operator, for example, Q˜/H20849/H9261/H20850/H11013L/H20851Q/H20849t/H20850/H20852/H20849/H9261/H20850=/H20885 0/H11009 /H20855Q/H20849t/H20850/H20856e−/H9261tdt, /H20849B3/H20850 the set of coupled equations transforms as /H9261Q˜−/H20855Q/H208490/H20850/H20856=P˜ M+vs /H9261+/H20858 i/H20849B21iQ˜+B22iP˜+B23ix˜i+B24ip˜i/H20850, /H20849B4a/H20850/H9261P˜−/H20855P/H208490/H20850/H20856=−/H20858 i/H20849B11iQ˜+B12iP˜+B13ix˜i+B14ip˜i/H20850, /H20849B4b/H20850 /H9261x˜i−/H20855xi/H208490/H20850/H20856=B41iQ˜+B42iP˜+B43ix˜i+B44ip˜i,/H20849B4c/H20850 /H9261p˜i−/H20855pi/H208490/H20850/H20856=−/H20849B31iQ˜+B32iP˜+B33ix˜i+B34ip˜i/H20850. /H20849B4d/H20850 Rewriting these in matrix forms, the equations become sim- pler as /H9261/H20873Q˜ P˜/H20874−/H20898/H20855Q/H208490/H20850/H20856+vs /H9261 /H20855P/H208490/H20850/H20856/H20899=/H20902/H2089801 M 00/H20899+/H20858 i/H20873B21iB22i −B11i−B12i/H20874/H20903 /H11003/H20873Q˜ P˜/H20874+/H20858 i/H20873B23iB24i −B13i−B14i/H20874 /H11003/H20873x˜i p˜i/H20874, /H20849B5a/H20850 /H9261/H20873x˜i p˜i/H20874−/H20873/H20855xi/H208490/H20850/H20856 /H20855pi/H208490/H20850/H20856/H20874=/H20873B41iB42i −B31i−B32i/H20874/H20873Q˜ P˜/H20874 +/H20873B43iB44i −B33i−B34i/H20874/H20873x˜i p˜i/H20874. /H20849B5b/H20850 From Eq. /H20849B5b/H20850, one can calculate /H20849x˜ip˜i/H20850Tin terms of Q˜and P˜, /H20873x˜i p˜i/H20874=/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873B41iB42i −B31i−B32i/H20874/H20873Q˜ P˜/H20874 +/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856 /H20855pi/H208490/H20850/H20856/H20874. /H20849B6/H20850 From Eqs. /H20849B5a/H20850and/H20849B6/H20850, one finally gets the equation of /H20849Q˜P˜/H20850T,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-9/H20900/H20898/H9261−1 M 0/H9261/H20899−/H20858 i/H20877/H20873B21iB22i −B11i−B12i/H20874+/H20873B23iB24i −B13i−B14i/H20874/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873B41iB42i −B31i−B32i/H20874/H20878/H20901/H20873Q˜ P˜/H20874 =/H20898/H20855Q/H208490/H20850/H20856+vs /H9261 /H20855P/H208490/H20850/H20856/H20899+/H20858 i/H20873B23iB24i −B13i−B14i/H20874/H20873/H9261−B43i−B44i B33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856 /H20855pi/H208490/H20850/H20856/H20874. /H20849B7/H20850 Inverting the matrix in front of /H20849Q˜P˜/H20850T, one can get the solu- tion of /H20849Q˜P˜/H20850T. Then, finally, the solution /H20849/H20855Q/H20856/H20855P/H20856/H20850Tis ob- tained by the inverse Laplace transform of /H20849Q˜P˜/H20850T, /H20873/H20855Q/H20849t/H20850/H20856 /H20855P/H20849t/H20850/H20856/H20874=L−1/H20875/H20873Q˜/H20849/H9261/H20850 P˜/H20849/H9261/H20850/H20874/H20876. /H20849B8/H20850 APPENDIX C: SOLUTION OF EQ. ( 24) In the special case that current is applied at t=0,/H9008/H20849t/H20850in Eq./H2084925/H20850becomes Heaviside step function. This is the case we are interested in. In a real DW system, the DW velocityjumps from 0 to a finite value at the moment that the spincurrent starts to be applied. This jumping comes from thediscontinuity in Eq. /H2084925/H20850which makes the Hamiltonian dis- continuous. Right before the current is applied, the DW re-mains on the stable /H20849or equilibrium /H20850state described by Eq. /H2084922/H20850. Suppose that Eq. /H2084920a/H20850also holds for /H9261=0. Then, Eq. /H2084924/H20850 transforms as /H20849up to constant /H20850 H tot=P2 2M+vs/H20849t/H20850P+/H20858 i/H208751 2mi/H20849pi−/H9253iP+miv/H20849t/H20850/H208502/H20876 +/H20858 i1 2mi/H9275i2/H20849xi−Q/H208502. /H20849C1/H20850 Performing the canonical transform pi→pi−miv/H20849t/H20850, one can transform this Hamiltonian in the form of Eq. /H20849B1/H20850, Htot=P2 2M+vs/H20849t/H20850P+/H20858 i/H208751 2mi/H20849pi−/H9253iP/H208502+1 2mi/H9275i2/H20849xi−Q/H208502/H20876. Here, one of the constraints Eq. /H2084920a/H20850is generalized to hold even for /H9261=0, so that /H20858i/H9253i=0. Note that the discontinuity due tov/H20849t/H20850is absorbed in the new pi. Thus, Eq. /H2084922b/H20850should be written as /H20855pi/H208490+/H20850/H20856=/H20855pi/H208490−/H20850/H20856+miv=/H9253i/H20855P/H208490/H20850/H20856+miv./H20849C2/H20850 The initial condition of xiis the same as Eq. /H2084922a/H20850. Now, using these initial conditions and Eqs. /H20849B7/H20850and/H20849B8/H20850under the constraints in Eq. /H2084920/H20850, one gets the solution of this sys- tem as Eq. /H2084926/H20850.APPENDIX D: CORRELATIONS OF STOCHASTIC FORCES AT HIGH TEMPERATURE This section provides the quantum derivation of correla- tion relations of stochastic forces at high temperature. Theclassical correlation relations in Eq. /H2084932/H20850are valid quantum mechanically at high temperature. Since Eq. /H2084931/H20850implies Eq. /H2084932/H20850, it suffices to show Eq. /H2084931/H20850in this section. The basic strategy is studying statistical properties of the HamiltonianEq./H2084919/H20850/H20849under quadratic potential bQ 2/H2085054by the Feynman path integral along the imaginary-time axis. The Feynmanpath integral of a system described by a quadratic Lagrang-ian is proportional to the exponential of the action valueevaluated at the classical solution. Hence, the key point ofthe procedure is to get the classical solution with imaginarytime. 1. General relations a. Classical action under high-temperature limit Define a column vector /H9273=/H20849Qx1x2¯/H20850T. Let the Euclidean Lagrangian of the system be LE=1 2/H9273˙TA/H9273˙+1 2/H9273B/H9273, where A andBare symmetric matrices. /H20849The symbols “ A” and “ B” are not the same as those in Appendix B. /H20850Explicitly, L =1 2/H20858nmx˙nAnmx˙m+1 2/H20858nmxnBnmxm. Here x0/H11013Q./H11509LE /H11509x˙n=/H20858mAnmx˙m =A/H9273˙and/H11509LE /H11509xn=/H20858mBnmxm=B/H9273lead to the classical equation of motion, A/H9273¨=B/H9273. /H20849D1/H20850 The classical action value Sc/H20849evaluated at the classical path /H20850 is then, Sc=/H208480/H9270LEdt=1 2/H208480/H9270/H20849/H9273˙TA/H9273˙+/H9273B/H9273/H20850dt=1 2/H9273TA/H9273˙/H208410/H9270 +/H208480/H9270/H20849−/H9273TA/H9273¨+/H9273B/H9273/H20850dt=1 2/H9273TA/H9273˙/H208410/H9270. Here, /H9270=/H6036/kBT. Now, the only thing one needs is to find /H9273˙at boundary points. In the case of Eq. /H2084919/H20850,Ais invertible. Hence, the equa- tion becomes /H9273¨=A−1B/H9273. Suppose that A−1Bis diagonaliz- able, that is A−1B=C−1DC. Here Dnm=/H9261n/H9254nmis diagonal ma- trix and /H9261nisnth eigenvalue of A−1B. Define a new vector /H9264=C/H9273. Finally, we get the equation, /H9264¨=/H20898/H926100¯ 0/H92611¯ ]]/GS/H20899/H9264. /H20849D2/H20850 Imposing the boundary condition /H9273/H208490/H20850=/H9273i,/H9273/H20849/H9270/H20850=/H9273fand de- fining the corresponding /H9264i=C/H9273i,/H9264f=C/H9264f, then one gets the solution of /H9264and its derivative straightforwardly,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-10/H9264n=/H9264fn+/H9264in 2cosh/H20881/H9261n/H20873t−/H9270 2/H20874 cosh/H20881/H9261n/H9270 2+/H9264fn−/H9264in 2sinh/H20881/H9261n/H20873t−/H9270 2/H20874 sinh/H20881/H9261n/H9270 2, /H20849D3/H20850 /H9264˙n=/H20881/H9261n/H20900/H9264fn+/H9264in 2sinh/H20881/H9261n/H20873t−/H9270 2/H20874 cosh/H20881/H9261n/H9270 2 +/H9264fn−/H9264in 2cosh/H20881/H9261n/H20873t−/H9270 2/H20874 sinh/H20881/H9261n/H9270 2/H20901. /H20849D4/H20850 Now, /H9264˙at boundary points are obtained as /H9264˙n/H208490/H20850=/H20881/H9261n/H20873−/H9264fn+/H9264in 2tanh/H20881/H9261n/H9270 2+/H9264fn−/H9264in 2coth/H20881/H9261n/H9270 2/H20874, /H20849D5/H20850 /H9264˙n/H20849/H9270/H20850=/H20881/H9261n/H20873/H9264fn+/H9264in 2tanh/H20881/H9261n/H9270 2+/H9264fn−/H9264in 2coth/H20881/H9261n/H9270 2/H20874. /H20849D6/H20850 If/H20881/H20841/H9261n/H20841/H9270 2=/H20881/H20841/H9261n/H20841/H6036 2kBT/H112701, tanh/H20881/H9261n/H9270 2/H11015/H20881/H9261n/H9270 2. Then, /H9264˙n/H208490/H20850/H11015−/H9264fn+/H9264in 2/H9261n/H9270 2+/H9264fn−/H9264in /H9270, /H20849D7/H20850 /H9264˙n/H20849/H9270/H20850/H11015/H9264fn+/H9264in 2/H9261n/H9270 2+/H9264fn−/H9264in /H9270. /H20849D8/H20850 In matrix form, /H9264˙/H208490/H20850/H11015−D/H9264f+/H9264i 2/H9270 2+/H9264f−/H9264i /H9270=−DC/H9273f+/H9273i 2/H9270 2+C/H9273f−/H9273i /H9270, /H20849D9/H20850 /H9264˙/H20849/H9270/H20850/H11015D/H9264f+/H9264i 2/H9270 2+/H9264f−/H9264i /H9270=DC/H9273f+/H9273i 2/H9270 2+C/H9273f−/H9273i /H9270. /H20849D10/H20850 Using A−1B=C−1DC, it leads to /H9273˙/H208490/H20850/H11015−A−1B/H9273f+/H9273i 2/H9270 2+/H9273f−/H9273i /H9270, /H20849D11/H20850 /H9273˙/H20849/H9270/H20850/H11015A−1B/H9273f+/H9273i 2/H9270 2+/H9273f−/H9273i /H9270. /H20849D12/H20850 Finally one can obtain the classical action,Sc=1 2/H9273TA/H9273˙/H208410/H9270=/H20873/H9273f+/H9273i 2/H20874T B/H20873/H9273f+/H9273i 2/H20874/H9270 2 +/H20873/H9273f−/H9273i 2/H20874T A/H20873/H9273f−/H9273i 2/H208742 /H9270. /H20849D13/H20850 This is valid even if some eigenvalues are zero. /H20849By taking limit of /H9261i→0, cosh and sinh becomes constant and linear, respectively. /H20850 b. Propagator and its derivatives The propagator is given by the Feynman path integral, K/H20849/H9273f,/H9273i;/H9270/H20850=/H20855/H9273f/H20841e−H/kBT/H20841/H9273i/H20856=/H20848D/H9273e−/H20848LEdt//H6036, where D/H9273=/H20863iDxi. For quadratic Lagrangian, it is well known that/H20848D /H9273e−/H20848LEdt/H6036=F/H20849/H9270/H20850e−Sc//H6036. Here F/H20849/H9270/H20850is a smooth function de- pendent on /H9270only. Now we aim to calculate K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850. It is easy to obtain the corresponding classical action by replacing /H9273f =/H9273i+/H9254/H9273in Eq. /H20849D13/H20850, Sc/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=/H9270 2/H9273iTB/H9273i+/H9270 2/H9273iTB/H9254/H9273+/H9270 8/H9254/H9273TB/H9254/H9273 +1 2/H9270/H9254/H9273TA/H9254/H9273. /H20849D14/H20850 Then, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850is/H20849up to second order of /H9254/H9273/H20850, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−Sc//H6036=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i /H11003/H208751−1 /H6036/H20873/H9270 2/H9273iTB/H9254/H9273+/H9270 8/H9254/H9273TB/H9254/H9273 +1 2/H9270/H9254/H9273TA/H9254/H9273/H20874+1 2/H60362/H20873/H9270 2/H9273iTB/H9254/H9273/H208742/H20876. Zeroth order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i. First order:−/H9270 2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H9273iTB/H9254/H9273 =−/H9270 2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858nmxinBnm/H9254xm. Second order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 /H6036/H20873/H9270 8/H9254/H9273TB/H9254/H9273 +1 2/H9270/H9254/H9273TA/H9254/H9273/H20874+1 2/H60362/H20873/H9270 2/H9273iTB/H9254/H9273/H208742/H20878 =F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 2/H6036/H20858 nm/H9254xn/H20873/H9270 4Bnm+1 /H9270Anm/H20874/H9254xm +/H92702 8/H60362/H20873/H20858 klmnxikBkn/H9254xnxilBlm/H9254xm/H20874/H20878. /H20849D15/H20850 By the relation, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=K/H20849/H9273i,/H9273i;/H9270/H20850+/H20858m/H11509K /H11509xfm/H9254xm +/H20858nm1 2/H115092K /H11509xfn/H11509xfm/H9254xn/H9254xm+O/H20849/H9254/H92733/H20850, K/H20849/H9273i,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i, /H20849D16/H20850THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-11/H20879/H11509K /H11509xfm/H20879 /H9273i=/H9273f=−/H9270 2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858 nBnmxin,/H20849D17/H20850 /H20879/H115092K /H11509xfn/H11509xfm/H20879 /H9273i=/H9273f=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 /H6036/H20873/H9270 4Bnm+1 /H9270Anm/H20874 +/H92702 4/H60362/H20873/H20858 klBknxikBlmxil/H20874/H20878 =F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1 /H6036/H20873/H9270 4Bnm+1 /H9270Anm/H20874 +/H92702 4/H60362/H20873/H20858 kBknxik/H20874/H20873/H20858 kBkmxik/H20874/H20878./H20849D18/H20850 c. Correlations Statistical average of an operator Ais given byTr/H20849Ae−H/kBT/H20850 Tr/H20849e−H/kBT/H20850. What we want to find are the averages of /H9004xn/H9004xm,/H9004pn/H9004pm, and/H20853/H9004xn,/H9004pm/H20854for/H9004xn/H11013xn−Qand/H9004pn/H11013pn−/H9253nP, Tr/H20849/H9004xn/H9004xme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004xme−H/kBT/H20841/H9273i/H20856 =/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856 =/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850K/H20849/H9273i,/H9273i;/H9270/H20850, /H20849D19/H20850 Tr/H20849/H9004pn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004pn/H9004pme−H/kBT/H20841/H9273i/H20856 =−/H60362/H20885/H20879d/H9273i/H20873/H11509 /H11509xfn−/H9253n/H11509 /H11509Qf/H20874 /H11003/H20873/H11509 /H11509xfm−/H9253m/H11509 /H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879 /H9273i=/H9273f, /H20849D20/H20850 Tr/H20849/H9004xn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004pme−H/kBT/H20841/H9273i/H20856 =−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509 /H11509xfm −/H9253m/H11509 /H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879 /H9273i=/H9273f, /H20849D21/H20850 Tr/H20849e−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856=/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850, /H20849D22/H20850 where d/H9273i=/H20863ndxin.2. Correlations under quadratic potential Under potential bQ2, the matrices AandBcorresponding the Hamiltonian Eq. /H2084919/H20850are A=/H20898MM /H92531 M/H92532 ¯ M/H92531M/H925312+m1M/H92531/H92532¯ M/H92532M/H92532/H92531M/H925322+m2¯ ]] ] /GS /H20899,/H20849D23/H20850 B=/H20898b+/H20858 nmn/H9275n2 −m1/H927512−m2/H927522¯ −m1/H927512m1/H927512 0 ¯ −m2/H9275220 m2/H927522¯ ]] ] /GS/H20899./H20849D24/H20850 Then, e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273iis written as e−/H20849/H9270/2/H6036/H20850/H20851/H20858nmn/H9275n2/H20849Qi−xin/H208502+bQi2/H20852. a. x-x correlations Since K/H20849/H9273,/H9273;/H9270/H20850is an even function of /H20849xn−Qi/H20850,i ti s trivial that Tr /H20849/H9004xn/H9004xme−H/kBT/H20850=0 unless n=m. For n=m,T r /H20849/H9004xn2e−H/kBT/H20850=/H20848d/H9273i/H20849xin−Qi/H208502K/H20849/H9273i,/H9273i;/H9270/H20850. Thus, Tr/H20849/H9004xn2e−H/kBT/H20850 Tr/H20849e−H/kBT/H20850=/H20885dxin/H20849xin−Qi/H208502e−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502 /H20885dxine−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502 =/H6036 /H9270mnwn2=kBT mnwn2. /H20849D25/H20850 So, finally one gets /H20855/H9004xn/H9004xm/H20856=kBT mnwn2/H9254nm. b. x-p correlations Explicitly rewriting the derivative of K, /H20879/H11509K /H11509xfm/H20879 /H9273i=/H9273f=−/H9270 2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850mm/H9275m2/H20849xim−Qi/H20850for/H20849m /HS110050/H20850, /H20849D26/H20850 /H20879/H11509K /H11509Qf/H20879 /H9273i=/H9273f=−/H9270 2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850/H20877bQi2+/H20858 nmn/H9275n2/H20849Qi−xin/H20850/H20878. /H20849D27/H20850 Using the above relations,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-12Tr/H20849/H9004xn/H9004pme−H/kBT/H20850=−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509 /H11509xfm−/H9253m/H11509 /H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879 /H9273i=/H9273f =−i/H9270 2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253mbQi2+/H9253m/H20858 lml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850 =−i/H9270 2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253m/H20858 lml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850 =i/H9270 2/H9253m/H20858 lml/H9275l2Tr/H20849/H20849/H9004xin/H9004xile−H/kBT/H20850/H20850+mm/H9275m2Tr/H20849/H9004xin/H9004xime−H/kBT/H20850. /H20849D28/H20850 In the third line, it is used that /H20848dxin/H20849xin−Qi/H20850 /H11003/H20851even function of /H20849xin−Qi/H20850/H20852=0. One can now write the x-pcorrelations in terms of x-x correlations. /H20855/H9004xn/H9004pm/H20856=i/H9270 2/H20873/H9253m/H20858 lml/H9275l2/H20855/H9004xin/H9004xil/H20856+mm/H9275m2/H20855/H9004xin/H9004xim/H20856/H20874 =i/H9270kBT 2/H20873/H9253m/H20858 l/H9254nl+/H9254nm/H20874=i/H6036 2/H20849/H9253m+/H9254nm/H20850,/H20849D29/H20850 which is purely imaginary. Thus, /H20855/H20853/H9004xn,/H9004pm/H20854/H20856=/H20855/H9004xn/H9004pm/H20856 +/H20855/H9004xn/H9004pm/H20856/H11569=0. c. p-p correlations It is convenient to calculate /H20848d/H9273i/H115092K /H11509xfn/H11509xfm/H20841/H9273i=/H9273f. The trickiest part is /H20848d/H9273i/H20858kBknxik/H20858kBkmxikK/H20849/H9273i,/H9273i;/H9270/H20850, n/HS110050,m/HS110050:/H20858 kBknxik/H20858 kBkmxik=mn/H9275n2/H20849xin−Qi/H20850mm/H9275m2/H20849xim −Qi/H20850, n=0 , m/HS110050:/H20858 kBknxik/H20858 kBkmxik=/H20873/H20858 kmk/H9275k2/H20849Qi−xik/H20850 +bQi/H20874mm/H9275m2/H20849xim−Qi/H20850, n=0 , m=0 :/H20858 kBknxik/H20858 kBkmxik=/H20873/H20858 kmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874 /H11003/H20873/H20858 kmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874. After integrating over xik, odd terms with respect to /H20849xik −Q/H20850vanish. Taking only even terms, one obtains n/HS110050,m/HS110050→mn2/H9275n4/H20849xin−Qi/H208502/H9254nm=mm/H9275m2/H20849xin−Qi/H208502Bnm, n=0 , m/HS110050→−mm2/H9275m4/H20849Qi−xim/H208502=mm/H9275m2/H20849xim−Qi/H208502Bnm,n=0 , m=0→/H20858 kmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2. Integrating out and using the identity /H20848duu2e−u2/2/H9251 =/H9251/H20848due−u2/2/H9251for/H9251/H110220, one finds n/HS110050,m/HS110050:/H20885d/H9273imm/H9275m2/H20849xin−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850, n=0 , m/HS110050:/H20885d/H9273imm/H9275m2/H20849xim−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850, n=0 , m=0 :/H20885d/H9273i/H20875/H20858 kmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2/H20876K/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270/H20873/H20858 kmk/H9275k2+b/H20874/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850 =/H6036 /H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850. The result is/H6036 /H9270Bnm/H20848d/H9273iKindependent of the cases. Finally, one can obtain /H20885/H20879d/H9273i/H115092K /H11509xfn/H11509xfm/H20879 /H9273i=/H9273f=/H20877−1 /H6036/H20873/H9270 4Bnm+1 /H9270Anm/H20874 +/H9270 4/H6036Bnm/H20878/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850 =−Anm /H9270/H6036/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,/H20849D30/H20850 or equivalently,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-13/H20883/H115092 /H11509xn/H11509xm/H20884=−kBT /H60362Anm=−kBT /H60362/H20849M/H9253n/H9253m+mn/H9254nm/H20850, /H20849D31/H20850 where /H92530=1,m0=0. Finally, p-pcorrelation is obtained /H20855/H9004pn/H9004pm/H20856=−/H60362/H20883/H20873/H11509 /H11509xn−/H9253n/H11509 /H11509Q/H20874/H20873/H11509 /H11509xm−/H9253m/H11509 /H11509Q/H20874/H20884 =kBT/H20849Anm−/H9253mAn0−/H9253nAm0+/H9253n/H9253mA00/H20850 =kBT/H20849M/H9253n/H9253m+mn/H9254mn−M/H9253m/H9253n−M/H9253n/H9253m +M/H9253n/H9253m/H20850=mnkBT/H9254mn. /H20849D32/H20850 The above three results of x-x,x-p, and p-pcorrelations are the same as Eq. /H2084931/H20850.3. Sufficient condition for “high” temperature We assumed the high-temperature approximationkBT /H6036 /H11271/H20881/H20841/H9261n/H20841 2. Indeed, the temperature should satisfykBT /H6036/H11271/H20881/H9261M 2, where /H9261Mis the absolute value of maximum eigenvalue of A−1B. It is known that, for eigenvalue /H9261of a matrix A,/H20841/H9261/H20841is not greater than maximum column /H20849or row /H20850sum,55 /H20841/H9261/H20841/H11349max j/H20858 i/H20841aij/H20841/H11013/H20648A/H20648. /H20849D33/H20850 According to the above definition of /H20648·/H20648, It is not hard to see that/H20648AB/H20648/H11349/H20648A/H20648/H20648B/H20648. The above argument says /H9261M/H11349/H20648A−1B/H20648/H11349/H20648A−1/H20648/H20648B/H20648. /H20849D34/H20850 It is not hard to obtain A−1with the following LDU factorization. /H20898MM /H92531 M/H92532 ¯ M/H92531M/H925312+m1M/H92531/H92532¯ M/H92532M/H92532/H92531M/H925322+m2¯ ]] ] /GS /H20899=/H2089810 0 ¯ /H9253110 ¯ /H9253201 ¯ ]] ]/GS/H20899/H20898M 0 0¯ 0m10¯ 00 m2¯ ]]]/GS/H20899/H208981/H92531/H92532¯ 01 0 ¯ 00 1 ¯ ]] ]/GS/H20899. /H20849D35/H20850 Inverting the factorized matrices, A−1=/H208981/H92531/H92532¯ 01 0 ¯ 00 1 ¯ ]] ]/GS/H20899−1 /H20898M 0 0¯ 0m10¯ 00 m2¯ ]]]/GS/H20899−1 /H2089810 0 ¯ /H9253110 ¯ /H9253201 ¯ ]] ]/GS/H20899−1 =/H208981−/H92531−/H92532¯ 01 0 ¯ 00 1 ¯ ]] ]/GS/H20899/H208981 M0 0¯ 01 m10¯ 001 m2¯ ]]] /GS/H20899 /H11003/H2089810 0 ¯ −/H9253110 ¯ −/H9253201 ¯ ]] ] /GS/H20899=/H208981 M+/H20858 n/H9253n2 mn−/H92531 m1−/H92532 m2¯ −/H92531 m11 m10¯ −/H92532 m201 m2¯ ]] ] /GS/H20899. /H20849D36/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-14Thus, the maximum column sum of A−1is /H20648A−1/H20648= max n/H208731 M+/H20858 i/H9253i2 mi+/H20858 i/H20841/H9253i/H20841 mi,1+/H20841/H9253n/H20841 mn/H20874./H20849D37/H20850 If/H9253iare on the order of 1 or larger,1 M+/H20858i/H9253i2 mi+/H20858i/H20841/H9253i/H20841 miis the maximum value. And, in this limit, it is smaller than1 M +2/H20858i/H9253i2 mi. So one can get /H20648A/H20648/H113491 M+2/H20858 i/H9253i2 mi. /H20849D38/H20850 Since Bis given by B=/H20898b+/H20858 nmn/H9275n2 −m1/H927512−m2/H927522¯ −m1/H927512m1/H927512 0 ¯ −m2/H9275220 m2/H927522¯ ]] ] /GS/H20899,/H20849D39/H20850 the maximum column sum of Bis /H20648B/H206481= max n/H20873b+2/H20858 imiwi2,2mnwn2/H20874/H11349/H20841b/H20841+2/H20858 imiwi2. /H20849D40/H20850 Finally, one obtains the upper bound of /H9261M, /H9261M/H11349/H20648A−1/H20648/H20648B/H20648/H11349/H208731 M+2/H20858 i/H9253i2 mi/H20874/H20873/H20841b/H20841+2/H20858 imi/H9275i2/H20874. /H20849D41/H20850 In order to evaluate the expression on the right-hand side of the inequality Eq. /H20849D41/H20850, we use the constraints Eq. /H2084920/H20850.T o convert the summations to known quantities, we generalizethe constraint to the Caldeira-Legget-type continuous formwith the following definitions of spectral functions, J p/H20849/H9275/H20850/H11013/H9266 2/H20858 i/H9253i2/H9275i mi/H9254/H20849/H9275i−/H9275/H20850=/H9251S 2KM/H9275,/H20849D42/H20850 Jx/H20849/H9275/H20850/H11013/H9266 2/H20858 imi/H9275i3/H9254/H20849/H9275i−/H9275/H20850=2/H9251KM S/H9275./H20849D43/H20850 Checking the constraints,/H20858 i/H9253i2/H9261 mi/H20849/H92612+/H9275i2/H20850=2/H9261 /H9266/H20885d/H9275Jp/H20849/H9275/H20850 /H9275/H20849/H92612+/H92752/H20850 =2/H9261 /H9266/H9251S 2KM/H20885d/H92751 /H92612+/H92752=/H9251S 2KM, /H20849D44/H20850 /H20858 imi/H9275i2/H9261 /H92612+/H9275i2=2/H9261 /H9266/H20885d/H9275Jx/H20849/H9275/H20850 /H9275/H20849/H92612+/H92752/H20850 =2/H9261 /H92662/H9251KM S/H20885d/H92751 /H92612+/H92752=2/H9251KM S. /H20849D45/H20850 Finally, /H20858 i/H9253i2 mi=2 /H9266/H20885d/H9275/H9251S 2KM=/H9251S /H9266KM/H9275c, /H20849D46/H20850 /H20858 imi/H9275i2=2 /H9266/H20885d/H92752/H9251KM S=4/H9251KM S/H9266/H9275c,/H20849D47/H20850 where /H9275cis the critical frequency of the environmental exci- tations. Therefore, /H9261M/H11349/H208491 M+2/H9251S /H9266KM/H9275c/H20850/H20849/H20841b/H20841+8/H9251KM S/H9266/H9275c/H20850. Hence, one fi- nally finds that the sufficient condition of the high tempera- ture is T/H11271Tc, where the critical temperature Tcis defined as Tc/H11013/H6036 2kB/H20881/H208731 M+2/H9251S /H9266KM/H9275c/H20874/H20873/H20841b/H20841+8/H9251KM S/H9266/H9275c/H20874. /H20849D48/H20850 Now, we check if the above condition is satisfied in ex- perimental situations. Ignoring /H20841b/H20841, the critical temperature becomes /H20881/H208491+2/H9251S /H9266K/H9275c/H208502/H9251K S/H9266/H9275c. Since the environmental excita- tion is caused by magnetization dynamics, one can note thatthere is no need to consider the environmental excitationwith frequencies far exceeding the frequency scale of mag-netization dynamics. 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Rev. Lett. 98, 037204 /H208492007/H20850. 41For permalloy, /H20841e/H92530K/H9261//H9262B/H20841/H11011109A/cm2, which is about an or- der larger than the current density of /H11011108A/cm2used in many experiments /H20849Refs. 38–40/H20850. 42Y . Le Maho, J.-V . Kim, and G. Tatara, Phys. Rev. B 79, 174404 /H208492009/H20850. 43T. Kim, J. Ieda, and S. Maekawa, arXiv:0901.3066 /H20849unpub- lished/H20850. 44V . W. Döring, Z. Naturforsch. A 3A, 373/H208491948/H20850. 45We thank M. Stiles for pointing out this point. 46To solve this system, one of the constaints Eq. /H2084920a/H20850is general- ized to hold even for /H9261=0. That is, /H20858i/H9253i=0. See, for a detail, Appendix C. 47To consider a force on Eq. /H208496a/H20850, the potential should be general- ized to depend on the momentum. 48Forv=0, the terminal velocity of the DW vanishes indepen- dently of its the initial velocity since the environmental mass ismuch larger than the DW mass. With v/H110220, one can perform the Galilean transformation to make /H20855x˙i/H208490/H20850/H20856=0 instead of /H20855x˙i/H208490/H20850/H20856 =v. Since the system is Galilean invariant, one expect that the DW also stops in this frame, just as v=0. It implies that the terminal velocity of the DW in the lab frame is also v. 49V . Kamberský, Czech. J. Phys., Sect. B 26, 1366 /H208491976/H20850; Can. J. Phys. 48, 2906 /H208491970/H20850;Czech. J. Phys., Sect. B 34, 1111 /H208491984/H20850. 50K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 /H208492007/H20850. 51C.-Y . You, I. M. Sung, and B.-K. Joe, Appl. Phys. Lett. 89, 222513 /H208492006/H20850; C.-Y . You and S.-S. Ha, ibid. 91, 022507 /H208492007/H20850. 52This section summarizes the work by Kim et al./H20849Ref. 43/H20850. 53J. M. Winter, Phys. Rev. 124, 452/H208491961/H20850. 54By the same argument, Eq. /H2084932/H20850is obtained under an arbitrary potential V/H20849Q/H20850. Since the system was in equilibrium before ap- plying current, we assume V/H11032/H20849Q/H20850=0. At high temperature limit, /H9273moves in very short /H20849imaginary /H20850time interval. Therefore, we can take quadratic approximation and V/H20849Q/H20850to be the form of bQ2. 55See, for example, G. Strang, Linear Algebra and its Applications /H20849Thomson, USA, 1988 /H20850, Chap. 7. 56A. Mourachkine, O. V . Yazyev, C. Ducati, and J.-Ph. Ansermet, Nano Lett. 8, 3683 /H208492008/H20850. 57C. Boone, J. A. Katine, J. R. Childress, J. Zhu, X. Cheng, and I. N. Krivorotov, Phys. Rev. B 79, 140404 /H20849R/H20850/H208492009/H20850.KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850 134431-16

2010-10-04

Current-induced domain wall motion in magnetic nanowires is affected by thermal fluctuation. In order to account for this effect, the Landau-Lifshitz-Gilbert equation includes a thermal fluctuation field and literature often utilizes the fluctuation-dissipation theorem to characterize statistical properties of the thermal fluctuation field. However, the theorem is not applicable to the system under finite current since it is not in equilibrium. To examine the effect of finite current on the thermal fluctuation, we adopt the influence functional formalism developed by Feynman and Vernon, which is known to be a useful tool to analyze effects of dissipation and thermal fluctuation. For this purpose, we construct a quantum mechanical effective Hamiltonian describing current-induced domain wall motion by generalizing the Caldeira-Leggett description of quantum dissipation. We find that even for the current-induced domain wall motion, the statistical properties of the thermal noise is still described by the fluctuation-dissipation theorem if the current density is sufficiently lower than the intrinsic critical current density and thus the domain wall tilting angle is sufficiently lower than pi/4. The relation between our result and a recent result, which also addresses the thermal fluctuation, is discussed. We also find interesting physical meanings of the Gilbert damping alpha and the nonadiabaticy parameter beta; while alpha characterizes the coupling strength between the magnetization dynamics (the domain wall motion in this paper) and the thermal reservoir (or environment), beta characterizes the coupling strength between the spin current and the thermal reservoir.

Thermal fluctuation field for current-induced domain wall motion

1010.0478v2

Localized spin waves in isolated kskyrmions Levente Rózsa,1,Julian Hagemeister,1Elena Y. Vedmedenko,1and Roland Wiesendanger1 1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany (Dated: October 16, 2018) Thelocalizedmagnonmodesofisolated kskyrmionsonafield-polarizedbackgroundareanalyzed basedontheLandau–Lifshitz–Gilbertequationwithinthetermsofanatomisticclassicalspinmodel, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion orderka higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. Itisshownthatatlowfields 2and 3skyrmionsaredestroyedviaaburstinstability connected to a breathing mode, while 1skyrmions undergo an elliptic instability. At high fields all kskyrmions collapse due to the instability of a breathing mode. The effective damping parameters of the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge in the case of the lowest-lying modes at the burst and collapse instabilities, but not at the elliptic instability. It is shown that the breathing modes of kskyrmions may become overdamped at higher Gilbert damping values. I. INTRODUCTION Magneticskyrmionsarelocalizedparticle-likespincon- figurations [1], which have become the focus of intense research activities over the last years due to their promis- ing applications in spintronic devices [2–5]. While their particle-like properties make them suitable to be used as bits of information, the collective excitations of the spins constituting the magnetic skyrmion, known as spin waves or magnons, open possible applications in the field of magnonics [6]. These spin wave modes were first investigated theo- retically [7–10] and experimentally [11–13] in skyrmion lattice phases, where the interactions between the skyrmions lead to the formation of magnon bands. If a skyrmion is confined in a finite-sized nanoelement, it will possess discrete excitation frequencies [14–17]. Al- though such geometries have also been successfully ap- plied to the time-resolved imaging of the dynamical mo- tion of magnetic bubble domains [18, 19], in such a case it is not possible to distinguish between the excitations of the particle-like object itself and spin waves forming at the edges of the sample [14]. In order to rule out boundary effects, the excitations of isolated skyrmions have to be investigated, as was performed theoretically in Refs. [20–23]. It was suggested recently [24] that the experimentally determined excitation frequencies in the Ir/Fe/Co/Pt multilayer system may be identified as spin wave modes of isolated skyrmions, rather than as magnons stemming from an ordered skyrmion lattice. In most investigations skyrmions correspond to sim- ple domains with the magnetization in their core point- ing opposite to the collinear background. However, it was shown already in Ref. [25] that the Dzyaloshinsky– Moriya interaction [26, 27] responsible for their stabiliza- tion may also lead to the formation of structures where the direction of the magnetization rotates multiple times rozsa.levente@physnet.uni-hamburg.debetween the center of the structure and the collinear re- gion. Such target states or kskyrmions, where kis the number of sign changes of the out-of-plane magnetization when moving along the radial direction, have also been investigated in constricted geometries [28–32]. The ex- perimental observation of localized spin structures with multiple rotations has been mainly restricted to systems with negligible Dzyaloshinsky–Moriya interaction so far [19, 33, 34], where the formation of domain structures is attributed to the magnetostatic dipolar interaction. The collapse of isolated kskyrmions and their cre- ationinnanodotsbyswitchingtheexternalfielddirection was recently investigated in Ref. [35]. It was found that during the creation process the skyrmions display signif- icant size oscillations resembling breathing eigenmodes. In Ref. [25], the stability of kskyrmions was studied in a system with a ferromagnetic ground state, and it was found that applying the external field opposite to the background magnetization leads to a divergence of the skyrmion radius at a critical field value, a so-called burst instability. This instability can be attributed to a sign change of one of the eigenvalues of the energy func- tional expanded around the kskyrmion configuration, intrinsically related to the dynamics of the system. How- ever, the spin wave frequencies of isolated kskyrmions remain unexplored. Besides the excitation frequencies themselves, the life- time of spin waves is also of crucial importance in magnonics applications. This is primarily influenced by the Gilbert damping parameter [36], the value of which can be determined experimentally based on resonance lineshapes measured in the collinear state [11, 19, 24]. It was demonstrated recently [23] that the noncollinear spin structure drastically influences the effective damp- ing parameter acting on the spin waves, leading to mode- dependent and enhanced values compared to the Gilbert damping parameter. This effect was discussed through the example of the 1skyrmion in Ref. [23], but it is also expected to be observable for kskyrmions with higher orderk. Here the localized spin wave frequencies of isolated karXiv:1810.06471v1 [cond-mat.mes-hall] 15 Oct 20182 skyrmions are investigated in a classical atomistic spin model. The parameters in the Hamiltonian represent the Pd/Fe/Ir(111) model-type system, where the properties of skyrmions have been studied in detail both from the experimental [37, 38] and from the theoretical [35, 39– 41] side. The paper is organized as follows. The classical atomistic spin Hamiltonian and the method of calculat- ing the eigenmodes is introduced in Sec. IIA, while the angular momentum and nodal quantum numbers charac- terizing the excitations are defined in Sec. IIB within the framework of the corresponding micromagnetic model. Eigenfrequencies equal to or approaching zero are dis- cussed in Sec. IIC, and the effective damping param- eters are introduced in Sec. IID. The eigenmodes of k skyrmions with k= 1;2;3are compared in Sec. IIIA, the instabilities occurring at low and high field values are dis- cussed in connection to magnons with vanishing frequen- cies in Sec. IIIB, and the effective damping parameters of the different modes are calculated for vanishing and higher values of the Gilbert damping in Secs. IIIC and IIID, respectively. A summary is given in Sec. IV. II. METHODS A. Atomistic model The system is described by the classical atomistic model Hamiltonian H=1 2X hi;jiJSiSj1 2X hi;jiDij(SiSj) X iK(Sz i)2X isBSi; (1) with theSiunit vectors representing the spins in a single-layer triangular lattice; J,Dij, andKde- noting nearest-neighbor Heisenberg and Dzyaloshinsky– Moriya exchange interactions and on-site magnetocrys- talline anisotropy, respectively; while sandBstand for the spin magnetic moment and the external mag- netic field. The numerical values of the parameters are taken from Ref. [35], being J= 5:72meV;D=jDijj= 1:52meV;K= 0:4meV, ands= 3B, describing the Pd/Fe/Ir(111) system. The energy parameters were de- termined based on measuring the field-dependence of 1 skyrmion profiles in the system by spin-polarized scan- ning tunneling microscopy in Ref. [38]. During the calculations the external field Bis ori- ented along the out-of-plane zdirection. The equilib- riumkskyrmion structures are determined from a rea- sonable initial configuration by iteratively rotating the spinsSitowards the direction of the effective magnetic fieldBeff i=1 s@H @Si. The iteration is performed un- til the torque acting on the spins, Ti=SiBeff i, becomes smaller at every lattice site than a predefinedvalue, generally chosen to be 108meV=B. The calcula- tions are performed on a lattice with periodic boundary conditions, with system sizes up to 256256for the largestkskyrmions in order to avoid edge effects and enable the accurate modeling of isolated skyrmions. Once the equilibrium configuration S(0) iis determined, the spins are rotated to a local coordinate system ~Si= RiSiusing the rotational matrices Ri. In the local coor- dinatesystemtheequilibriumspindirectionsarepointing along the local zaxis, ~S(0) i= (0;0;1). The Hamiltonian in Eq. (1) is expanded up to second-order terms in the small variables ~Sx i;~Sy ias (cf. Ref. [23]) HH0+1 2 ~S?T HSW~S? =H0+1 2~Sx~SyA1A2 Ay 2A3~Sx ~Sy :(2) Thematrix products areunderstoodtorunoverlattice site indices i, with the matrix components reading A1;ij=~Jxx ij+ij X k~Jzz ik2~Kxx i+ 2~Kzz i+s~Bz i! ;(3) A2;ij=~Jxy ijij2~Kxy i; (4) A3;ij=~Jyy ij+ij X k~Jzz ik2~Kyy i+ 2~Kzz i+s~Bz i! :(5) The energy terms in the Hamiltonian are ro- tated to the local coordinate system via ~Jij= Ri[JIDij]RT j;~Ki=RiKRT j;and ~Bi=RiB, whereIis the 33identity matrix, Dijis the ma- trix describing the vector product with Dij, andKis the anisotropy matrix with the only nonzero element be- ingKzz=K. The spin wave frequencies are obtained from the lin- earized Landau–Lifshitz–Gilbert equation [36, 42] @t~S?= 0 s(iy)HSW~S?=DSW~S?;(6) withy= 0iIs iIs0 the Pauli matrix in Cartesian components and acting as the identity matrix Isin the lattice site summations. The symbol 0denotes the gyro- magnetic ratio =ge 2mdivided by a factor of 1+2, with gthe electron gfactor,ethe elementary charge, mthe electron’s mass, and the Gilbert damping parameter. Equation (6) is rewritten as an eigenvalue equation by assuming the time dependence ~S?(t) =ei!qt~S? qand performing the replacement @t!i!q. Since thekskyrmions represent local energy minima, HSWin Eq. (2) is a positive semidefinite matrix. For = 0the!qfrequenciesof DSWarerealandtheyalways occurin!qpairsonthesubspacewhere HSWisstrictly positive, for details see, e.g., Ref. [23]. In the following, we will only treat the solutions with Re !q>0, but their3 Re!q<0pairs are also necessary for constructing real- valued eigenvectors of Eq. (6). The zero eigenvalues are discussed in Sec. IIC. As is known from previous calculations for 1 skyrmions [21–23], the localized excitation modes of k skyrmions are found below the ferromagnetic resonance frequency!FMR = s(2K+sB). During the numerical solution of Eq. (6) these lowest-lying eigenmodes of the sparse matrix DSWare determined, as implemented in themontecrystal atomistic spin simulation program [43]. B. Micromagnetic model The atomistic model described in the previous Sec- tion enables the treatment of noncollinear spin structures where the direction of the spins significantly differs be- tween neighboring lattice sites. This is especially impor- tant when discussing the collapse of kskyrmions on the lattice as was performed in Ref. [35]. Here we will dis- cuss the micromagnetic model which on the one hand is applicable only if the characteristic length scale of non- collinear structures is significantly larger than the lattice constant, but on the other hand enables a simple classi- fication of the spin wave modes. The free energy functional of the micromagnetic model is defined as H=Z AX =x;y;z(rS)2+K(Sz)2MBSz +D(Sz@xSxSx@xSz+Sz@ySySy@ySz)dr; (7) where for the Pd/Fe/Ir(111) system the following pa- rameter values were used: A= 2:0pJ/m is the ex- change stiffness,D=3:9mJ/m2is the Dzyaloshinsky– Moriya interaction describing right-handed rotation [39], K=2:5MJ/m3is the easy-axis anisotropy, and M= 1:1MA/m is the saturation magnetization. The equilibrium spin structure S(0)= (sin  0cos  0;sin  0sin  0;cos  0)ofkskyrmions will be cylindrically symmetric, given by 0(r;') ='+ due to the right-handed rotational sense and 0(r;') =  0(r), which is the solution of the Euler–Lagrange equation A @2 r0+1 r@r01 r2sin  0cos  0 +jDj1 rsin20 +Ksin  0cos  01 2MBsin  0= 0: (8) The skyrmion order kis encapsulated in the bound- ary conditions 0(0) =k;0(1) = 0. Equation (8) is solved numerically in a finite interval r2[0;R]signifi- cantly larger than the equilibrium kskyrmion size. A first approximation to the spin structure is constructed based on the corresponding initial value problem usingthe shooting method [25], then iteratively optimizing the structure using a finite-difference discretization. The spin wave Hamiltonian may be determined anal- ogously to Eq. (2), by using the local coordinate system  =  0+~Sx; =  0+1 sin  0~Sy. ThematricesinEqs.(3)- (5) are replaced by the operators A1=2A r21 r2cos 2 0 2jDj1 rsin 2 0 2Kcos 2 0+MBcos  0; (9) A2= 4A1 r2cos  0@'2jDj1 rsin  0@'; (10) A3=2A r2+ (@r0)21 r2cos20 2jDj @r0+1 rsin  0cos  0 2Kcos20+MBcos  0: (11) Due to the cylindrical symmetry of the structure, the solutions of Eq. (6) are sought in the form ~S?(r;';t ) = ei!n;mteim'~S? n;m(r), performing the replacements @t! i!n;mand@'!im. For each angular momentum quantum number m, an infinite number of solutions in- dexed bynmay be found, but only a few of these are located below !FMR = M(2K+MB), hence repre- senting localized spin wave modes of the kskyrmions. The different nquantum numbers typically denote solu- tions with different numbers of nodes, analogously to the quantum-mechanical eigenstates of a particle in a box. Because of the property HSW(m) =H SW(m)and HSWbeing self-adjoint, the eigenvalues of HSW(m) andHSW(m)coincide, leading to a double degeneracy apart from the m= 0modes. The!qeigenvalue pairs ofDSWdiscussed in Sec. IIA for the atomistic model at = 0in this case can be written as !n;m=!n;m. However, considering only the modes with Re !n;m>0, one has!n;m6=!n;mindicating nonreciprocity or an energy difference between clockwise ( m < 0) and coun- terclockwise ( m> 0) rotating modes [17, 23]. For finding the eigenvectors and eigenvalues of the micromagnetic model, Eq. (6) is solved using a finite- difference method on the r2[0;R]interval. For treat- ing the Laplacian r2in Eqs. (9) and (11) the improved discretization scheme suggested in Ref. [44] was applied, which enables a more accurate treatment of modes with eigenvalues converging to zero in the infinite and contin- uous micromagnetic limit. The spin wave modes of the atomistic model discussed in Sec. IIA were assigned the (n;m)quantum numbers, which are strictly speaking only applicable in the mi- cromagnetic limit with perfect cylindrical symmetry, by visualizingthe real-spacestructureofthe numericallyob- tained eigenvectors.4 C. Goldstone modes and instabilities Since the translation of the kskyrmions on the collinear background in the plane costs no energy, the spin wave Hamiltonian HSWpossesses two eigenvectors belonging to zero eigenvalue, representing the Goldstone modes of the system. Within the micromagnetic descrip- tion of Sec. IIB, these may be expressed analytically as [21–23]  ~Sx;~Sy =ei' @r0;i1 rsin  0 ;(12)  ~Sx;~Sy =ei' @r0;i1 rsin  0 :(13) Equations (12) and (13) represent eigenvectors of the dynamical matrix DSWas well. From Eqs. (2) and (6) it follows that the eigenvectors of HSWandDSWbelong- ing to zero eigenvalue must coincide, HSW~S?=0, DSW~S?=0, because (iy)in Eq. (6) is an in- vertible matrix. Because from the solutions of the equa- tion of motion (6) we will only keep the ones satisfying Re!n;m>0, the eigenvectors from Eqs. (12) and (13) will be denoted as the single spin wave mode !0;1= 0. Since the eigenvectors and eigenvalues are determined numerically in a finite system by using a discretization procedure, the Goldstone modes will possess a small fi- nite frequency. However, these will not be presented in Sec. IIIA together with the other frequencies since they represent a numerical artifact. For the 1and 3skyrmions the !0;1eigenmode has a positive fre- quency and an eigenvector clearly distinguishable from that of the !0;1translational mode. However, for the 2skyrmion both the !0;1and the!0;1eigenfrequen- cies ofDSWare very close to zero, and the correspond- ing eigenvectors converge to Eqs. (12) and (13) as the discretization is refined and the system size is increased. This can occur because DSWis not self-adjoint and its eigenvectors are generally not orthogonal. In contrast, the eigenvectors of HSWremain orthogonal, with only a single pair of them taking the form of Eqs. (12) and (13). In contrast to the Goldstone modes with always zero energy, the sign change of another eigenvalue of HSW indicates that the isolated kskyrmion is transformed from a stable local energy minimum into an unstable saddle point, leading to its disappearance from the sys- tem. Such instabilities were determined by calculating the lowest-lying eigenvalues of HSWin Eq. (2). Due to the connection between the HSWandDSWmatrices expressed in Eq. (6), at least one of the precession fre- quencies!qwill also approach zero at such an instability point. D. Effective damping parameters For finite values of the Gilbert damping , the spin waves in the system will decay over time as the systemrelaxes to the equilibrium state during the time evolu- tion described by the Landau–Lifshitz–Gilbert equation. The speed of the relaxation can be characterized by the effective damping parameter, which for a given mode q is defined as q;eff=Im!q Re!q: (14) As discussed in detail in Ref. [23], q;effis mode- dependent and can be significantly higher than the Gilbert damping parameter due to the elliptic polar- ization of spin waves, which can primarily be attributed to the noncollinear spin structure of the kskyrmions. For1,q;effmay be expressed as q;eff =X i~S(0);x q;i2 +~S(0);y q;i2 X i2Imh ~S(0);x q;i~S(0);y q;ii;(15) wheretheeigenvectorsinEq.(15)arecalculatedat = 0 from Eq. (6). Equation (15) may also be expressed by the axes of the polarization ellipse of the spins in mode q, see Ref. [23] for details. Forhighervaluesof , thecomplexfrequencies !qhave to be determined from Eq. (6), while the effective damp- ing parameters can be calculated from Eq. (14). Also for finite values of for each frequency with Re !q>0there exists a pair with Re !q0<0such that!q0=! q[23]. The spin waves will be circularly polarized if A1=A3 andAy 2=A2in Eq. (2), in which case the dependence of!qonmay simply be expressed by the undamped frequency!(0) qas Re!q() =1 1 +2!(0) q; (16) jIm!q()j= 1 +2!(0) q: (17) These relations are known for uniaxial ferromagnets; see, e.g., Ref. [45]. In the elliptically polarized modes of noncollinear structures, such as kskyrmions, a devia- tion from Eqs. (16)-(17) is expected. III. RESULTS A. Eigenmodes The frequencies of the localized spin wave modes of the 1,2, and 3skyrmion, calculated from the atomistic model for= 0as described in Sec. IIA, are shown in Fig. 1. For the 1skyrmion six localized modes can be observed below the FMR frequency of the field-polarized background in Fig. 1(a), four of which are clockwise ro- tating modes ( m < 0), one is a gyration mode rotating counterclockwise ( m= 1), while the final one is a breath- ing mode (m= 0). The excitation frequencies show good5 0.7 0.8 0.9 1.0 1.1 1.20255075100125150175 (a) 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175 (b) 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.150255075100125150175 (c) FIG. 1. Frequencies of localized spin wave modes at = 0for (a) the 1, (b) the 2, and (c) the 3skyrmion. Selected spin wave modes are visualized in contour plots of the out-of-plane spin component and denoted by open symbols connected by lines in the figure, the remaining modes are denoted by con- nected dots.quantitative agreement with the ones calculated from the micromagnetic model for the same system in Ref. [23]. Compared to Ref. [21], the additional appearance of the eigenmodes with m= 1;4;5can be attributed to the finite value of the anisotropy parameter Kin the present case. Increasing the anisotropy value makes it possible to stabilize the skyrmions at lower field values, down to zerofieldatthecriticalvalueinthemicromagneticmodel jKcj=2D2 16A, where the transition from the spin spiral to the ferromagnetic ground state occurs at zero exter- nal field [46]. Since the excitation frequencies decrease at lower field values as shown in Fig. 1(a), this favors the appearance of further modes. Simultaneously, the FMR frequency increases with K, meaning that modes with higher frequencies become observable for larger uni- axial anisotropy. For each angular momentum quantum numberm, only a single mode ( n= 0) appears. In the case of the 2skyrmion an increased number of eigenmodes may be seen in Fig. 1(b). This can mainly be attributed to the appearance of spin waves with higher angular momentum quantum numbers both for clockwise (up tom=17) and counterclockwise (up to m= 12) rotational directions. Furthermore, in this case modes withn= 1node in the eigenfunction can be observed as well. The same trend continues in the case of 3 skyrmionsinFig.1(c), thelargenumberofinternaleigen- modes can be attributed to angular momentum quantum numbers ranging from m=22tom= 16, as well as to spin wave eigenvectors with up to n= 2nodes. The dif- ferent rotational directions and numbers of nodes are il- lustrated in Supplemental Videos 1-4 [47] via the square- shaped modes ( n= 0;1,m=4) of the 3skyrmion at B= 0:825T. The increase of possible angular momentum quantum numbers for higher skyrmion order kas well as for de- creasing magnetic field Bmay be qualitatively explained by an increase in the skyrmion size. Modes with a given value ofmindicate a total of jmjmodulation periods along the perimeter of the skyrmion; for larger skyrmion sizes this corresponds to a modulation on a longer length scale, which has a smaller cost in exchange energy. The breathing modes of the 3skyrmion with dif- ferent numbers of nodes are visualized in Fig. 2 at B= 1T. The results shown in Fig. 2 are obtained from the micromagnetic model in Sec. IIB, which is in good quantitative agreement with the atomistic calcu- lations at the given field. All the eigenmodes display three peaks of various heights, while they decay expo- nentially outside the 3skyrmion. As can be seen in Fig. 2, the peaks are localized roughly around the re- gions where the spins are lying in-plane, indicated by the domain walls (DW) between pairs of dashed lines. The widths of the domain walls were determined by ap- proximating the 3skyrmion profile with linear func- tions close to the inflection points rj;0;j;j= 1;2;3 where the spins are lying in-plane, and calculating where these linear functions intersect integer multiples of in6 0 10 20 30 40 50-2-023 -0.100.000.10 FIG. 2. Comparison between the 3skyrmion profile (left vertical axis) and the eigenvectors of the breathing modes (m= 0) with different numbers of nodes n= 0;1;2(right vertical axis). The calculations were performed using the mi- cromagnetic model described in Sec. IIB at B= 1T, the lattice constant is a= 0:271nm. Double arrows between ver- tical dashed lines indicate the extensions of the domain walls in the structure. 0. Thus, the domain walls are located between the in- nerRin;j=rj+[@r0(rj)]1[(4j)0;j]and outer Rout;j=rj+ [@r0(rj)]1[(3j)0;j]radii. Such a description was used to calculate the skyrmion radius in, e.g., Ref. [46], and it was also applied for calculating the widths of planar domain walls [48]. The nodes of the eigenmodes are located roughly be- tween these domain walls, meaning that typically excita- tion modes with n= 0;:::;k1nodes may be observed inkskyrmions, in agreement with the results in Fig. 1. A higher number of nodes would require splitting a single peak into multiple peaks, the energy cost of which gen- erally exceeds the FMR frequency, thereby making these modes unobservable. The sign changes in the ~Sx n;meigen- vectors mean that the different modes can be imagined as the domain walls breathing in the same phase or in opposite phase, as can be seen in Supplemental Videos 5-7 [47]. Note that eigenmodes with higher nquantum numbers may also be observed for skyrmions confined in nanodots [14–16] where the peaks of the eigenmodes may also be localized at the edge of the sample, in contrast to the present case where isolated kskyrmions are dis- cussed on an infinite collinear background. Itisalsoworthnotingthatthelowest-lyingnonzerogy- ration mode is n= 0;m= 1for the 1and3skyrmions, while it isn= 1;m= 1for the 2skyrmion, see Fig. 1. As already mentioned in Sec. IIC, numerical calculations for the 2skyrmion indicate both in the atomistic and themicromagneticcasethatbyincreasingthesystemsize or refining the discretization the eigenvectors of both the n= 0;m=1and then= 0;m= 1modes ofDSW in Eq. (6) converge to the same eigenvectors in Eqs. (12) and(13)and 0eigenvalue, whichcorrespondtothetrans- lational Goldstone mode in the infinite system. This dif-ference can probably be attributed to the deviation in the value of the topological charge, being finite for 1 and3skyrmions but zero for the 2skyrmion [35]. B. Instabilities Skyrmions with different order kdeviate in their low- field behavior. Since the considered Pd/Fe/Ir(111) sys- tem has a spin spiral ground state [38], decreasing the magnetic field value will make the formation of domain wallsenergeticallypreferableinthesystem. Inthecaseof the1skyrmion this means that the lowest-lying eigen- mode ofHSWin Eq. (2), which is an elliptic mode with m=2, changes sign from positive to negative, occur- ring between B= 0:650T andB= 0:625T in the present system. This is indicated in Fig. 1(a) by the fact that the frequency of the n= 0;m=2eigenmode of DSWin Eq. (6) converges to zero. This leads to an elongation of the skyrmion into a spin spiral segment which gradually fills the ferromagnetic background, a so-called strip-out or elliptic instability already discussed in previous publi- cations [21, 46]. In contrast, for the 2and3skyrmions the lowest-lying eigenmode of HSWis a breathing mode withm= 0, which tends to zero between B= 0:800T andB= 0:775Tforbothskyrmions. Thisisindicatedby the lowest-lying n= 0;m= 0mode ofDSWin Fig. 1(b) for the 2skyrmion, which is the second lowest after the n= 0;m= 1mode for the 3skyrmion in Fig. 1(c). This means that the radius of the outer two rings of 2and 3skyrmions diverges at a finite field value, leading to a burst instability. Such a type of instability was already shown to occur in Ref. [25] in the case of a ferromagnetic ground state at negative field values, in which case it also affects 1skyrmions. At the burst instability, modes with n= 0and all angular momentum quantum numbers mappear to ap- proach zero because of the drastic increase in skyrmion radius decreasing the frequency of these modes as dis- cussed in Sec. IIIA. A similar effect was observed for the 1skyrmion in Ref. [22] when the critical value of the Dzyaloshinsky–Moriya interaction, jDcj=4 p AjKj, was approached at zero external field from the direction of the ferromagnetic ground state. In contrast, the ellip- tic instability only seems to affect the n= 0;m=2 mode, while other mvalues and the nonreciprocity are apparently weakly influenced. In the atomistic model, skyrmions collapse when their characteristic size becomes comparable to the lattice constant. For the 1,2, and 3skyrmions the col- lapse of the innermost ring occurs at Bc;14:495T, Bc;21:175T, andBc;31:155T, respectively [35]. As can be seen in Figs. 1(b), 1(c), and 3, this instabil- ity is again signaled by the n= 0;m= 0eigenfrequency going to zero, but in contrast to the burst instability, the other excitation frequencies keep increasing with the field in this regime. Figure 3 demonstrates that close to the collapse field the excitation frequency may be well7 4.45 4.46 4.47 4.48 4.49 4.50020406080100 FIG. 3. Frequency of the breathing mode n= 0;m = 0of the 1skyrmion close to the collapse field. Calculation data are shown by open symbols, red line denotes the power-law fitf0;0=Af(Bc;1B)f. approximated by the power law f0;0=Af(Bc;1B)f, withAf= 175:6GHz Tf,Bc;1= 4:4957T, andf= 0:23. C. Effective damping parameters in the limit of low The effective damping parameters n;m;effwere first calculated from the eigenvectors obtained at = 0fol- lowing Eq. (15). The results for the 1,2, and 3 skyrmions are summarized in Fig. 4. As discussed in Ref. [23], the n;m;effvalues are always larger than the Gilbert damping , and they tend to decrease with in- creasing angular momentum quantum number jmjand magnetic field B. The spin wave possessing the high- est effective damping is the n= 0;m= 0breathing mode both for the 1and 2skyrmion, but it is the n= 0;m= 1gyration mode for the 3skyrmion for a large part of the external field range where the struc- ture is stable. Excitation pairs with quantum num- bersn;mtend to decay with similar n;m;effvalues to each other, with n;jmj;eff< n;jmj;eff, where clockwise modes (m < 0) have lower frequencies and higher effec- tive damping due to the nonreciprocity. The effective damping parameters drastically increase and for the lowest-lying modes apparently diverge close to the burst instability, while no such sign of nonan- alytical behavior can be observed in the case of the 1skyrmion with the elliptic instability. For the same n;mmode, the effective damping parameter tends to in- crease with skyrmion order kaway from the critical field regimes; for example, for the n= 0;m= 0mode at B= 1:00T one finds 0;0;eff;1= 2:04,0;0;eff;2= 5:87, and0;0;eff;3= 10:09. Close to the collapse field, the effective damping pa- rameter of the n= 0;m= 0breathing mode tends to 0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.5 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15125102050100FIG.4. Effectivedamping parameters calculatedaccordingto Eq. (15) for the eigenmodes of the (a) 1, (b) 2, and (c) 3 skyrmions, plotted on a logarithmic scale. The corresponding excitation frequencies are shown in Fig. 1. diverge as shown in Figs. 4(b), 4(c), and 5 for the 2, 3, and 1skyrmions, respectively. Similarly to the eigenfrequency converging to zero in Fig. 3, the criti- cal behavior of the effective damping may be approxi- mated by a power-law fit 0;0;eff=A(Bc;1B) as shown in Fig. 5, this time with a negative exponent8 4.45 4.46 4.47 4.48 4.49 4.50024681012 FIG. 5. Effective damping parameter 0;0;effof the breathing moden= 0;m= 0of the 1skyrmion close to the collapse field. The corresponding excitation frequencies are shown in Fig. 3. Calculation data are shown by open symbols, red line denotes the power-law fit 0;0;eff=A(Bc;1B). due to the divergence. The fitting yields the parameters A= 0:96T,Bc;1= 4:4957T, and= 0:23. Natu- rally, the critical field values agree between the two fits, but interestingly one also finds f=up to two digits precision. Rearranging Eq. (14) yields 0;0;eff Re!0;0=1 jIm!0;0j; (18) where the left-hand side is proportional to (Bc;1B)fwhich is approximately constant due to the exponents canceling. This indicates that while Re!0;0diverges close to the collapse field, jIm!0;0j=remains almost constant at low values. D. Damping for higher values Due tothe divergences oftheeffective damping param- eters found at the burst instability and collapse fields, it is worthwhile to investigate the consequences of using a finitevalue in Eq. (6), in contrast to relying on Eq. (15) which is determined from the eigenvectors at = 0. The dependence of the real and imaginary parts of the !0;0 breathingmodefrequencyofthe 1skyrmionisdisplayed in Fig. 6, at a field value of B= 1T far from the el- liptic and collapse instabilities. As shown in Fig. 6(a), unlike circularly polarized modes described by Eq. (16) where Re!qdecreases smoothly and equals half of the undamped value at = 1, the Re!0;0value for the ellip- tically polarized eigenmode displays a much faster decay and reaches exactly zero at around 0:58. According toEq.(14), thisindicatesthatthecorrespondingeffective damping parameter 0;0;effdiverges at this point. Since the real part of the frequency disappears, the !q0=! qrelation connecting Re !q>0and Re!q0<0 0.0 0.2 0.4 0.6 0.8 1.00102030405060 0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6FIG. 6. (a) Frequency f0;0=Re!0;0=2and (b) inverse lifetimejIm!0;0jof then= 0;m = 0breathing mode of the 1skyrmion at B= 1T as a function of the Gilbert damping parameter . The solutions of Eq. (6) for the ellip- tically polarized eigenmode of the 1skyrmion are compared to Eqs. (16)-(17) which are only valid for circularly polarized modes. solutions of Eq. (6) discussed in Sec. IID no longer holds, and two different purely imaginary eigenfrequencies are found in this regime as shown in Fig. 6(b). This is analo- gous to overdamping in a classical linear harmonic oscil- lator, meaningthatthepurelyprecessionalfirst-orderdif- ferential equation describing circularly polarized modes is transformed into two coupled first-order differential equations [23] with an effective mass term for the breath- ing mode of kskyrmions. This implies that when per- forming spin dynamics simulations based on the Landau– Lifshitz–Gilbert equation, the value of the Gilbert damp- ing parameter has to be chosen carefully if the fastest relaxation to the equilibrium spin structure is required. The high effective damping of the breathing mode in the 1limit (cf. Fig. 4(a)) ensures that the inverse lifetime of the elliptically polarized excitations remains larger for a wide range of values in Fig. 6(b) than what would be expected for circularly polarized modes based9 0.85 0.90 0.95 1.00 1.05 1.10 1.1505101520 0.85 0.90 0.95 1.00 1.05 1.10 1.150.000.020.040.060.080.10 FIG. 7. (a) Frequency f0;0=Re!0;0=2and (b) inverse lifetimejIm!0;0jof then= 0;m= 0breathing mode of the 2skyrmion at = 0:1as a function of the external magnetic fieldB. The solutions of Eq. (6) for the elliptically polarized eigenmode of the 2skyrmion are compared to Eqs. (16)-(17) which are only valid for circularly polarized modes. on Eq. (17). Note that contrary to Sec. IIIB, Re !0;0be- comingzeroinFig.6(a)doesnotindicateaninstabilityof the system, since stability is determined by the eigenval- ues of the matrix HSWin Eq. (2) which are independent of. Since the disappearance of Re !0;0and the bifurcation of Im!0;0occurs as the excitation frequency becomes smaller, it is expected that such an effect may also be ob- servedatafixed valueastheexternalfieldisdecreased. This is illustrated for the n= 0;m= 0breathing mode of the 2skyrmion in Fig. 7 at = 0:1. For this interme- diate value of the damping, the breathing mode becomes overdamped around B= 0:875T, which is significantly higher than the burst instability between B= 0:775T andB= 0:800T (cf. Fig. 1(b) and the circularly polar- ized approximation in Fig. 7(a)). This means that the lowest-lying breathing mode of the 2skyrmion cannot be excited below this external field value. In Fig. 7(b) it can be observed that contrary to the circularly polarizedapproximation Eq. (17) following the field dependence of the frequency, for the actual elliptically polarized eigen- modejIm!0;0jisalmostconstantforallfieldvaluesabove thebifurcationpoint. Althoughasimilarobservationwas made at the end of Sec. IIIC as the system approached the collapse field at = 0, it is to be emphasized again that no instability occurs where Re !0;0disappears in Fig. 7(a). IV. CONCLUSION In summary, the localized spin wave modes of k skyrmions were investigated in an atomistic spin model, with parameters based on the Pd/Fe/Ir(111) system. It was found that the number of observable modes increases with skyrmion order k, firstly because of excitations with higher angular momentum quantum numbers mforming along the larger perimeter of the skyrmion, secondly be- cause of nodes appearing between the multiple domain walls. It was found that the 2and3skyrmions un- dergo a burst instability at low fields, in contrast to the elliptic instability of the 1skyrmion. At high field val- ues the innermost ring of the structure collapses in all cases, connected to an instability of a breathing mode. The effective damping parameters of the excitation modes were determined, and it was found that for the samen;mmode they tend to increase with skyrmion orderk. The effective damping parameter of the n= 0;m= 0breathing mode diverges at the burst and collapse instabilities, but no such effect was observed in case of the elliptic instability. For higher values of the Gilbert damping parameter a deviation from the behavior of circularly polarized modes has been found, with the breathing modes becoming overdamped. It was demonstrated that such an overdamping may be observ- able in 2and3skyrmions for intermediate values of the damping significantly above the burst instability field where the structures themselves disappear from the sys- tem. The results presented here may motivate further ex- perimental and theoretical studies on kskyrmions, of- fering a wider selection of localized excitations compared to the 1skyrmion, thereby opening further possibilities in magnonics applications. ACKNOWLEDGMENTS The authors would like to thank A. Siemens for fruit- ful discussions. Financial support for this work from the Alexander von Humboldt Foundation, from the Deutsche Forschungsgemeinschaft via SFB 668, from the European Union via the Horizon 2020 research and innovation pro- gram under Grant Agreement No. 665095 (MAGicSky), and from the National Research, Development and Inno- vation Office of Hungary under Project No. K115575 is gratefully acknowledged.10 [1] A. N. Bogdanov and D. A. Yablonski ˘i, Sov. Phys. JETP 68, 101 (1989). [2] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013). [3] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. B. Jungfleisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L. Wang, Y. Zhou, A. Hoffmann, and S. G. E. te Velthuis, Nat. Phys. 13, 162 (2017). [4] P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von Bergmann, and R. Wiesendanger, Nat. Nanotechnol. 12, 123 (2017). [5] F. Büttner, I. Lemesh, M. Schneider, B. Pfau, C. M. 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2018-10-15

The localized magnon modes of isolated $k\pi$ skyrmions on a field-polarized background are analyzed based on the Landau-Lifshitz-Gilbert equation within the terms of an atomistic classical spin model, with system parameters based on the Pd/Fe biatomic layer on Ir(111). For increasing skyrmion order $k$ a higher number of excitation modes are found, including modes with nodes in the radial eigenfunctions. It is shown that at low fields $2\pi$ and $3\pi$ skyrmions are destroyed via a burst instability connected to a breathing mode, while $1\pi$ skyrmions undergo an elliptic instability. At high fields all $k\pi$ skyrmions collapse due to the instability of a breathing mode. The effective damping parameters of the spin waves are calculated in the low Gilbert damping limit, and they are found to diverge in the case of the lowest-lying modes at the burst and collapse instabilities, but not at the elliptic instability. It is shown that the breathing modes of $k\pi$ skyrmions may become overdamped at higher Gilbert damping values.

Localized spin waves in isolated $kπ$ skyrmions

1810.06471v1

Molecular Hybridization Induced Antidamping and Sizable Enhanced Spin-to-Charge Conversion in Co 20Fe60B20/β-W/C 60Heterostructures Antarjami Sahoo 1, Aritra Mukhopadhyaya 2, Swayang Priya Mahanta 1, Md. Ehesan Ali 2, Subhankar Bedanta 1,3 1 Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical Sciences, National Institute of Science Education and Research (NISER), An OCC of Homi Bhabha National Institute (HBNI), Jatni 752050, Odisha, India 2 Institute of Nano Science and Technology, Knowledge City, Sector-81, Mohali, Punjab 140306, India and 3 Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and Research (NISER), An OCC of Homi Bhabha National Institute (HBNI), Jatni, Odisha 752050, India Md. Ehesan Ali∗and Subhankar Bedanta† Development of power efficient spintronics devices has been the compelling need in the post-CMOS technology era. The effective tunability of spin-orbit-coupling (SOC) in bulk and at the interfaces of hybrid materials stacking is a prerequisite for scaling down the dimension and power consumption of these devices. In this work, we demonstrate the strong chemisorption of C 60molecules when grown on the high SOC β-W layer. The parent CFB/ β-W bilayer exhibits large spin-to-charge intercon- version efficiency, which can be ascribed to the interfacial SOC observed at the Ferromagnet/Heavy metal interface. Further, the adsorption of C 60molecules on β-W reduces the effective Gilbert damping by ∼15% in the CFB/ β-W/C 60heterostructures. The anti-damping is accompanied by a gigantic ∼115% enhancement in the spin-pumping induced output voltage owing to the molecular hybridization. The non-collinear Density Functional Theory calculations confirm the long-range en- hancement of SOC of β-W upon the chemisorption of C 60molecules, which in turn can also enhance the SOC at the CFB/ β-W interface in CFB/ β-W/C 60heterostructures. The combined amplifica- tion of bulk as well interfacial SOC upon molecular hybridization stabilizes the anti-damping and enhanced spin-to-charge conversion, which can pave the way for the fabrication of power efficient spintronics devices. I. INTRODUCTION Spintronic logic and memory devices have proven to be one of the most suitable research domains to meet the ultra-low power consumption demand in the post- Complementary Metal Oxide Semiconductor (CMOS) technology era. Especially, with the advent of artificial intelligence and the Internet of Things (IoT), the further scaling down of CMOS technology can reach its physi- cal limits in size, speed, and static energy consumption. The conceptualized spin orbit torque magnetic random access memory (SOT-MRAM) devices which take the ad- vantage of spin Hall effect (SHE) can bring down the energy consumption to femto Joule from the pico Joule scale [1, 2]. The SHE based magnetization switching mechanism in SOT-MRAMs also offers much improved endurance owing to the separation in data writing and reading paths. Though these potentials of SOT-MRAMs have attracted major foundries, several challenges need to be addressed before the commercialization of SOT- MRAMs [1, 2]. The increase of writing efficiency to re- duce power consumption is one of those aspects which requires significant consideration. In this context, the spin Hall angle, θSH(JS⁄JC) of the nonmagnetic layer present in the SOT-MRAMs, where J Cand J Sare the charge and spin current densities, respectively, plays a ∗ehesan.ali@inst.ac.in †sbedanta@niser.ac.incritical role in determining the writing efficiency [3]. The efficient charge to spin interconversion can lead to the faster switching of magnetization of the adjacent mag- netic layer via SHE. Hence, various types of heavy metals (HMs), like Pt, Ta, W, Ir etc. have been investigated in the past two decades to reduce the power consumption of future spintronic devices [4–6]. On a similar note, the Rashba-Edelstein effect (REE) occurring at the interfaces with spatial inversion symmetry breaking and high spin orbit coupling (SOC) has also the potential for the man- ifestation of efficient charge to spin interconversion[7–9]. Hence, the combination of SHE and REE can be the most suitable alternative for the development of power efficient spintronics application. Among all the heavy metals, highly resistive ( ρβ−W∼ 100−300µΩ cm) metastable β-W possesses the largest θSH∼-0.3 to -0.4 [10–14], which makes it a strong can- didate for SOT-MRAM devices. Usually, additional re- active gases, like O 2, N2, and F are employed to stabilize the A15 crystal structure of β-W [11] and consequently, a larger θSHis realized. For example, Demasius et al., have been able to achieve θSH∼-0.5 by incorporating the oxy- gen into the tungsten thin films [12]. Interface engineer- ing also acts as a powerful tool for enhancing the writing efficiency in β-W based SOT-MRAM devices [15–17]. For instance, the presence of an interfacial atomically thin α-W layer in CoFeB/ α-W/β-W trilayer suppresses the spin backflow current, resulting in a 45% increase in the spin mixing conductance [15]. Further, the REE evolved at the W/Pt interface owing to the charge accumulationarXiv:2401.00486v1 [cond-mat.mtrl-sci] 31 Dec 20232 generates an additional spin orbit field on the adjacent ferromagnet (FM) NiFe (Py) layer [18]. The coexistence of SHE and REE has also been reported in CoFeB/ β-Ta and NiFe/Pt bilayers, where the interfacial SOC arising at the FM/HM interface plays a vital role in the spin- to-charge interconversion phenomena [19, 20]. More in- terestingly, a recent theoretical work has predicted the interfacial SOC mediated spin Hall angle of Pt can be 25 times larger than the bulk value in NiFe/Pt heterostruc- ture [21]. The interfacial SOC mediated spin accumula- tion has also been reported to occur at the Rashba-like β-Ta/Py interface without flowing the DC current [22]. The spin pumping induced by the ferromagnetic reso- nance results in non-equilibrium spin accumulation at the interface which consequently reduces the effective Gilbert damping of the β-Ta/Py bilayer. The reduction in ef- fective damping, also termed as antidamping, is similar to the interfacial Rashba like SOT, observed in various HM/FM heterostructures [22]. The anti-damping phe- nomena without the requirement of DC current depends on several factors, like SOC of HM, strength of built in electric field at the interface, interface quality etc. Hence, the interface engineering via tuning the interfacial SOC inβ-W based HM/FM heterostructures can be the path forward for developing power efficient SOT-MRAM de- vices. Till the date, most of the interface engineering re- search have been focused on employing an additional metallic or oxide layer in the HM/FM system for the enhancement of spin-to-charge interconversion efficiency. Whereas, the organic semiconductors (OSCs) can also be incorporated in the HM/FM system to fabricate hy- brid power efficient spintronic devices owing to their strong interfacial hybridization and charge transfer na- ture at metal/OSC interface [23]. Recently, the SOC of Pt has been found to be enhanced due to the on- surface physical adsorption of C 60(fullerene) molecules in YIG/Pt/C 60trilayer [24]. However, the θSHof Pt is usually found to be smaller compared to β-W and it is important to investigate the magnetization dynamics and spin to charge conversion phenomena in FM/ β-W/C 60 heterostructures. Hence, in this article, we report the effect of molecular hybridization at β-W/C 60interface on magnetization dynamics and spin-to-charge conver- sion phenomena in Co 20Fe60B20(CFB)/ β-W/C 60het- erostructures. The molecular hybridization reduces the effective Gilbert damping and also enhances the spin-to- charge conversion efficiency owing to the enhanced SOC ofβ-W and consequent strengthening of possible Rashba- like interaction at the CFB/ β-W interface. The strong chemisorption at the β-W/C 60interface and evolution of enhanced SOC of β-W upon the molecular hybridization have also been confirmed by the first principle density functional theory (DFT) based calculations.II. EXPERIMENTAL AND COMPUTATIONAL METHODS Four different types of heterostructures with CFB (7 nm)/ β-W (2.5, 5 nm) (Figure 1 (a)) and CFB (7 nm)/ β- W (2.5, 5 nm)/C 60(13 nm) (Figure 1 (b)) stackings were fabricated on Si/SiO 2(300 nm) substrates for the inves- tigation of magnetization dynamics and spin pumping phenomena. In addition, the CFB (7 nm)/ β-W (10, 13 nm) heterostructures were also fabricated to reaffirm the stabilization of β-W. The heterostructure stackings and their nomenclatures are mentioned in Table I. The CFB andβ-W layers were grown by DC magnetron sputter- ing, while the Effusion cell equipped in a separate cham- ber (Manufactured by EXCEL Instruments, India) was used for the growth of the C 60over layers in the CFWC series. While preparing the CFWC1 and CFWC2, the samples were transferred in-situ into the chamber with Effusion cell in a vacuum of ∼10−8mbar for the de- position of C 60. Before the fabrication of heterostruc- tures, thin films of CFB, β-W, and C 60were prepared for thickness calibration and study of magnetic and electrical properties. The base pressure of the sputtering chamber and chamber with Effusion cells were usually maintained at∼4×10−8mbar and ∼6×10−9mbar, respectively. The structural characterizations of individual thin films and heterostructures were performed by x-ray diffrac- tion (XRD), x-ray reflectivity (XRR), and Raman spec- trometer. The magneto-optic Kerr effect (MOKE) based microscope and superconducting quantum interference device based vibrating sample magnetometer (SQUID- VSM) were employed for the static magnetization char- acterization and magnetic domain imaging. The mag- netization dynamics was investigated by a lock-in based ferromagnetic resonance (FMR) spectrometer manufac- tured by NanOsc, Sweden. The heterostructures were kept in a flip-chip manner on the co-planner waveguide (CPW). The FMR spectra were recorded in the 4-17 GHz range for all the samples. The FMR spectrometer set-up is also equipped with an additional nano voltmeter using which spin-to-charge conversion phenomena of all the de- vices were measured via inverse spin Hall effect (ISHE) with 5-22 dBm RF power. The contacts were given at the two opposite ends of 3 mm ×2 mm devices using silver paste to measure the ISHE induced voltage drop across the samples. The details of the ISHE measure- ment set-up are mentioned elsewhere [25, 26]. Density functional theory (DFT)-based electronic structure calculations were performed in the Vienna Ab- initio simulation package (VASP) [27, 28] to understand the interface’s chemical bonding and surface reconstruc- tions. The plane wave basis sets expand the valance electronic states, and the core electrons are treated with the pseudopotentials. The core-valance interac- tions are considered with the Projected Augmented Wave method. The exchange-correlation potentials are treated with Perdew, Bruke and Ernzerof (PBE) [29] functional which inherits the Generalized Gradient Approximation3 TABLE I. Details of the heterostructures and their nomenclatures Sl. No. Stacking Nomenclature 1 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm) CFW1 2 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm) CFW2 3 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm)/C 60(13 nm) CFWC1 4 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm)/C 60(13 nm) CFWC2 (GGA). This functional produces a reliable understand- ing of similar kinds of interfaces. The convergences in the self-consistent field iterations were ensured with a plane- wave cutoff energy of 500 eV and a tolerance of 10−6 eV/cycle. A D3 dispersion correction term, devised by Grimme, accounts for the long-range interaction terms was employed in the calculations. The optimized unit cell parameter obtained from the aforementioned meth- ods for the cubic A15 crystal of the β-W is 5.014 ˚A, which resembles the experimental parameter of 5.036 ˚A. A 5×2×1 repetition is used to construct the (210) surface unit cell of the β-W to model the surface supercell. The lower two atomic layers were fixed at the bulk, and the remaining three layers were allowed to relax during the geometry optimization. The surface layer of β-W con- tains the C 60molecules. To understand the effect of the spin-orbit coupling interactions, we have performed the non-collinear DFT calculations as implemented in VASP. The E SOC calculated from these calculations quantifies the strength of the SOC term in the Hamiltonian. III. RESULTS AND DISCUSSION The grazing incidence x-ray diffraction (GIXRD) was performed for all the heterostructures. The XRD pat- terns of CFB/ β-W heterostructures with different thick- nesses of β-W are shown in Figure 1 (c). The presence of (200), (210) and (211) Bragg’s peaks of W at 35.5◦, 39.8◦ and 43.5◦indicate the stabilization of metastable βphase of W (A-15 crystal structure) [5, 30]. In addition, we have also observed the (320) and (321) Bragg’s peaks of W, which further suggests the growth of polycrystalline β-W. The relative intensity of (320) and (321) Bragg’s peaks of W is lower compared to (200), (210) and (211) Bragg’s peaks, consistent with previous reports [30]. The Bragg’s peaks are more prominent for heterostructures with thicker W layers as diffraction intensity increases with the increase in W thickness. The XRD patterns for CFWC1 and CFWC2 are similar to that for CFW1 and CFW2, respectively, as the thickness of β-W are same. Here, we have not used the reactive gases like, O 2and N2for the growth of β-W unlike some previous report [11]. The resistivity of W films with thicknesses 2.5, 5, 10 nm were measured by standard four probe methods. The resistivity decreases with increase in thickness of W and were found in between ∼300-100 µΩ-cm, further confirming the growth of βphase of W [5, 30]. We donot also observe the (110), (200), (210) Bragg’s peaks for the bcc α-W in the XRD patterns and the α-W would have also exhibited one order less resistivity compared to what we have observed [30]. The stabilization of pure β phase of W is quite important for future SOT device fab- rication and hence, we can expect a high spin-to-charge conversion efficiency in our CFB/ β-W heterostructures owing to high SOC of β-W [10, 30]. The XRR measurements were performed for all the samples in both the CFW and CFWC series to confirm the desired thickness of individual layers and to investi- gate the interface quality. Figure S1 (Supporting Infor- mation) shows the XRR patterns of all the heterostruc- tures considered for the present study. The experimental data were fitted using GenX software and the simulated patterns are also shown in Figure S1 (red curves). The presence of Kiessig oscillations for all the films infer the absence of a high degree of interfacial disorder and dis- locations. The relative peak positions and intensity of the simulated patterns agree quite well with the experi- mentally observed low angel XRR data. The fit provides the anticipated thickness of individual layer in each sam- ple as mentioned in Table I. The interface roughness for all the heterostructures were found in between 0.2-0.5 nm, further inferring the high quality growth of both the series of samples. Figure S2 (Supporting Information) displays the Raman spectra of 13 nm C 60film grown on Si/SiO 2(300 nm) substrate with the same growth con- dition as in the heterostructures. The presence of A g(2) and H g(8) Raman modes of C 60around ∼1460 cm−1 and 1566 cm−1, respectively confirms the growth of C 60 film [31, 32]. In addition, the Raman mode around ∼ 495 cm−1corresponding to A g(1) mode of C 60is also ob- served in the Raman spectrum. The anticipated thick- ness of C 60in the C 60thin film, CFWC1 and CFWC2 has also been confirmed from the XRR measurements. The Raman spectrum of our C 60film grown by effusion cells are quite similar to those prepared by different so- lution methods in HCl or N 2atmosphere [31, 32]. The saturation magnetization and the magnetic domain im- ages of all the heterostructures are found to be similar (see Supporting Information) as the bottom CFB layer is same for all the heterostructures. The magnetization relaxation and propagation of spin angular momentum in the CFB thin film and the het- erostructures in both the CFW and CFWC series were studied to explore the effect of high resistive β-W and β-W/C 60bilayer by in-plane FMR technique. The het-4 FIG. 1. Schematics of (a) Si/SiO 2/CFB/ β-W and (b) Si/SiO 2/CFB/ β-W/C 60heterostructures, (c) GIXRD patterns of Si/SiO 2/CFB/ β-W heterostructures with various thicknesses of β-W. erostructures are placed in a flip-chip manner on CPW as shown in the schematics in Figure S4 (a) (Support- ing Information). Figure S4 (c) shows the typical FMR spectra of CFW1 and CFWC1 heterostructures measuredin the 4-17 GHz range. All the FMR spectra were fit- ted to the derivative of symmetric and antisymmetric Lorentzian function to evaluate the resonance field ( Hres) and linewidth (∆ H) [33]: FMRSignal =K14(∆H)(H−Hres) [(∆H)2+ 4(H−Hres)2]2−K2(∆H)2−4(H−Hres)2 [(∆H)2+ 4(H−Hres)2]2+Offset, (1) where K 1and K 2are the antisymmetric and symmetric absorption coefficients, respectively. The extracted Hres and ∆ Hvalues at different resonance frequencies ( f) of all the heterostructures are shown in Figure 2 (a-b). The fvsHresof different samples in the CFW and CFWC series are plotted in Figure 2 (a). The fvsHresplots are fitted by using equation 2 [33]: f=γ 2πq (HK+Hres)(HK+Hres+ 4πMeff),(2) where 4πMeff= 4πMS+2KS MStFM and H K, KS, and t FMare the anisotropy field, perpen- dicular surface anisotropy constant, and the thicknessof FM, respectively. Here, γis the gyromagnetic ra- tio and 4 πMeffrepresents the effective magnetization. The 4 πMeffextracted from the fitting gives similar val- ues as compared with the saturation magnetization value (4πMS) calculated from the SQUID-VSM. Further, the effective Gilbert damping constant ( αeff) and hence, the magnetization relaxation mechanism are studied from the resonance frequency dependent FMR linewidth be- havior. The ∆ Hvsfplots are shown in Figure 2 (b). The linear dependency of ∆ Honfindicates the mag- netic damping is mainly governed by intrinsic mechanism via electron-magnon scattering rather than the extrinsic two magnon scattering. The ∆ Hvsfplots are fitted by5 FIG. 2. (a) Frequency ( f) versus resonance field ( Hres) and (b) linewidth (∆ H) versus frequency ( f) behaviour for various heterostructures. The solid lines are the best fits to equation 2 and 3. the following linear equation [33] to evaluate the αeff. ∆H= ∆H0+4παeff γf, (3) where the ∆ H0is the inhomogeneous linewidth broad- ening. The αeffvalues for all the heterostructures and CFB thin film obtained from the fitting are shown in Ta- ble II. The αeffvalue for CFW series ( ∼0.0075 ±0.0001 for CFW1 and ∼0.0080 ±0.0001 for CFW2) are found to be larger compared to that of the CFB thin film (∼0.0059 ±0.0001). The enhancement of αeffindicates the possible evolution of spin pumping mechanism in the CFB/ β-W bilayers. Interestingly, the αeffdecreases to ∼0.0065 ±0.0001 upon the deposition of C 60molecules on CFB/ β-W bilayers in CFWC series. The signifi- cant change in αefffor the CFB/ β-W/C 60heterostruc- tures compared to CFB/ β-W bilayers infers the modi- fication of physical properties of β-W layer in CFB/ β- W/C 60. The deposition of C 60molecules can lead to the metal/molecule hybridization at the β-W/C 60interface, which in turn can alter the properties of β-W. The DFT based first principle calculations were per- formed to elucidate further the molecular hybridization at the β-W/C 60interface and its consequences on the magnetization dynamics of CFB/ β-W/C 60heterostruc- tures. The extended simulation supercell for the C 60on β-W(210) are shown in Figure 3 (a). The C 60molecule is observed as strongly chemisorbed onto the β-W (210) surface with an adsorption energy of -253.5 kcal/mol. The adsorption energy is quite high as compared to the other substrates. For example, the adsorption energy for Co/C 60was found to be -90 kcal/mol [34] while for the Pt/C 60interface it is reported to be -115 kcal/mol [35]. The chemisorption in case of β-W/C 60is quite strong and induces distortion to the spherical shape of the ad- sorbed C 60. The distance between two carbon atoms from two opposite hexagons of adsorbed C 60is shorter along one direction compared to the other measured in the plane (left panel of Figure 3 (a)). The diameter of C 60molecules decreases by 0.3 ˚Awhen it is measured perpen- dicular to the β-W (210) surface (right panel of Figure 3 (a)). This distortion can be attributed to the W-C bond formation due to the strong chemisorption at the β-W/C 60interface. This chemisorption strongly alters the electronic structure of the β-W and C 60molecule (Figure 3 (b)). The pzorbital, which accommodates the π-electrons of the C 60, hybridizes with the d-orbitals of theβ-W atom and forms the hybridised interfacial states. The out-of-plane d-orbitals ( dxz,dyzanddz2orbitals) are strongly hybridized with the pzorbital of the carbon atom over a large energy window near the Fermi energy level (Figure 3 and Figure S5 (Supporting Information)). The sharp peaks observed in the DOS of free C 60layer gets significantly broadened, flattened, and shifted for β- W/C 60stacking. The strong metallo-organic hybridiza- tion also modifies the PDOS of various d-orbitals of β- W. The various d-orbitals become flattened and spread over larger energy spectrum around the Fermi level upon molecular hybridization. The formation of the W-C bond also costs a transfer of 3.25e−from the interfacial layer of theβ-W to C 60molecule (Figure 3 (c)). This is relatively higher compared to the previously reported the 0.25e− transfer from Pt (111) and 3e−transfer from Cu (111) to the adjacent C 60molecule, inferring the metallo-organic hybridization is quite stronger in case of β-W/C 60in- terface [35].Hence, the molecular hybridization of β-W is expected to alter its physical properties with greater effect and can be considered as an important tool to op- timize the spintronics device performances. The modified electronic structure was found to carry a long-range effect on the strength of the spin-orbit cou- pling. The E SOC of bare 2.5 nm β-W and 2.5 nm β- W covered with C 60molecules, and the variation of the ESOC(∆E SOC) due to β-W/C 60hybridization are shown in Figure 4. The interfacial W atoms involved in the hy- bridization with C 60show a decrease in the E SOC. The rest of the W atoms from the surface layer exhibit an increase in the E SOC. The lower atomic layers of W6 FIG. 3. (a) The extended simulation supercell for the C 60onβ-W(210) substrate. The left panel shows the top view of the surface supercell (along the z-axis), and the right panel shows the side view of the same. The pink balls of larger size and cyan balls of smaller size represent the tungsten and carbon atoms, respectively. The yellow bonds highlight the part of the C 60 which takes part in the interface formation. The double-headed dotted arrows quantify the diameter of the C 60spheres in two directions. (b-c) The modification in the electronic structure due to chemisorption of the C 60molecule on the β-W. (b) The atom projected orbital resolved partial density of states of β-W(210), C 60, and β-W(210)/C 60, and (c) The electron density redistribution due to chemisorption. The red and green iso-surfaces depict electron density depletion and accumulation of the electron density at the interface, respectively. The bi-coloured arrow depicts the direction of the electron transfer process. also show an increment in the E SOC. The W layer, far- thest from the β-W/C 60interface (nearer to the CFB/ β- W interface), exhibits the most increased E SOC. Hence, the hybridization at the β-W/C 60interface increases the overall spin-orbit coupling strength of the β-W layer. More importantly, the SOC at the CFB/ β-W interface is enhanced for CFB/ β-W/C 60stacking compared to the CFB/ β-W bilayer. The enhanced bulk SOC of β-W and the interfacial SOC at CFB/ β-W interface can facilitate an efficient spin to charge conversion in CFB/ β-W/C 60 heterostructures. The decrease in damping, usually know as anti- damping, has been observed previously in FM/HM bilay-ers [22, 26, 30]. In those systems, the effective damping values become lower than the αeffof the FM layer and this phenomenon has been attributed to the formation of Rashba like interfacial states [22, 30]. Similar type of evolution of Rashba like states at the CFB/ β-W inter- face can be expected due to structural inversion asym- metry and large SOC of β-W. The spin accumulation at the CFB/ β-W interface can lead to evolution of the non-equilibrium spin states. The non-equilibrium spin states along with the enhanced SOC at CFB/ β-W inter- face due to molecular hybridization as confirmed from the DFT calculations can generate an additional charge cur- rent due to IREE and can also induce the antidamping7 FIG. 4. The effect of the chemisorption of the C 60molecule at the β-W(210) surface on the E SOCof various atomic sites. The percentage change in the E SOC (∆E SOC) is calculated in terms of the change in the E SOC of the bare β-W(210) substrate. Layer 5 is the interfacial layer that interacts with the C 60, and layer 1 is the opposite to the β-W/C 60interface layer. torque on the magnetization of FM layer. The antidamp- ing torque can make the magnetization precession rela- tively slower and thus decreasing the αeffof the CFB/ β- W/C 60heterostructures compared to the CFB/ β-W bi- layer. The control of Gilbert damping of FMs by inter- facing with adjacent non-magnetic metal/organic bilay- ers can also provide an alternative to the search for low damping magnetic materials. Especially, the low cost and abundant availability of carbon based organic molecules can be commercially beneficial in optimizing the mag- netic damping for spintronic applications. Further, the Gilbert damping modulation can also control the effec- tive spin mixing conductance ( g(↑↓) eff) of the heterostruc- tures which also plays a vital role for efficient spin current transport across the interface. Hence, the g(↑↓) effof all the heterostructures was calculated from the damping con- stant measurement by equation 4 [33]: g(↑↓) eff=4πMstCFB gµB(αCFB/NM −αCFB), (4) where g, µBandtCFB are the Land´ e g factor (2.1), Bohr’s magnetron, and thickness of CFB layer, respec- tively. αCFB/NM is the damping constant of bilayer ortri-layers and αCFB is the damping constant of the ref- erence CFB thin film. The g(↑↓) efffor CFW1 and CFW2 (Table II) are relatively higher compared to the previ- ous reports on FM/ β-W bilayers. Especially, the g(↑↓) effof CFW2 is one order higher than that reported for Py/ β-W bilayer (1.63 ×1018m−2) [30], and 2 order higher com- pared to that of the YIG/ β-W (5.98 ×1017m−2) [14]. This indicates the absence of any significant amount of spin back flow from β-W layer and high SOC strength of parent β-W layer in our system. However, the g(↑↓) eff values decrease for the CFWC1 and CFWC2 tri-layers owing to anti-damping phenomena. The ISHE measurements were performed for all the heterostructures in CFW and CFWC series to gain more insights about the effect of molecular hybridization in CFB/ β-W/C 60on the magnetization dynamics and spin to charge conversion efficiency. Figure 5 shows the typi- cal field dependent DC voltage ( Vdc) measured across the CFB (7 nm)/ β-W (5 nm)/C 60(13 nm) heterostructure under FMR conditions. In order to separate the symmet- ric (VSY M) and asymmetric ( VASY M ) components, the VdcvsHplots were fitted with the following Lorentzian function: Vdc=VSY M(∆H)2 (∆H)2+ (H−Hres)2+VASY M(∆H)(H−Hres) (∆H)2+ (H−Hres)2(5) The extracted field dependent VSY M andVASY M are also plotted in Figure 5. Similar type of field depen-8 FIG. 5. VMEAS ,VSY M andVASY M versus Hfor CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructure with ϕ∼(a) 180◦and (b) 0◦measured at 15 dBm RF power. The red curve is Lorentzian fit with equation 5 to VdcvsHplot. TABLE II. Effective Gilbert damping, spin mixing conduc- tance, and symmetric component of measured DC voltage for different heterostructures. Heterostructures αeff(±0.0001) g(↑↓) eff(1019m−2)VSY M(µV) CFB 0.0059 - - CFW1 0.0075 0.87 1.08 CFW2 0.0080 1.13 1.25 CFWC1 0.0064 0.27 2.32 CFWC2 0.0065 0.32 1.78 dent VMEAS ,VSY M, and VASY M are also observed for other samples in both CFW and CFWC series. The VSY M is mainly contributed by the spin pumping voltage (VISHE ) and the spin rectification effects arising from the anisotropic magnetoresistance (AMR) [ VAMR] [33]. Whereas, the asymmetric component of the measured voltage arises solely due to anomalous Hall effect and AMR [33]. The sign of VSY M is reversed when ϕ(angel between the perpendicular direction to the applied mag- netic field ( H) and direction of voltage measurement) is changed from 0◦to 180◦(Figure 5), confirming the pres- ence of ISHE in our heterostructures. The field depen- dent VSY M for all the four heterostructures are plotted in Figure 6 (a-b). Interestingly, the VSY M value at the resonance field for CFB/ β-W/C 60trilayers is found to be increased compared to that for CFB/ β-W bilayers. Theincrement is ∼115% for β-W thickness 2.5 nm, while it becomes ∼20% for β-W thickness 5 nm. The gigantic enhancement of VSY M for CFB (7)/ β-W(2.5)/C 60(13) infers the modification of SOC of β-W when capped with organic C 60molecules and the presence of an additional spin to charge conversion effect in the heterostructures. The power dependent spin-to-charge conversion measure- ments were also performed to further confirm the en- hancement of VSY M. The spin pumping induced voltage increases linearly with the RF power as shown in Fig- ure 6 (c) for both CFW1 and CFWC1. The VSY M at different RF power is found to be increased for CFWC1 compared to CFW1, which further confirms the molec- ular hybridization induced enhanced spin-to-charge con- version. As the thickness, magnetic properties of bot- tom CFB layer is same for all the heterostructures, the contribution of VAMR is expected to be same for CFB (7 nm)/ β-W(2.5 nm)/C 60(13 nm) and CFB (7 nm)/ β- W(2.5 nm). Hence, the sizable increase in the measured voltage can be attributed to the enhanced SOC of β-W due to molecular hybridization and additional charge cur- rent flowing at the CFB/ β-W interface due to IREE as shown in the Figure 6 (d). In order to understand the en- hanced spin-to-charge conversion phenomena further, we also calculated the θSHof the heterostructures by using equations 6 and 7 [14, 33]: Js=g(↑↓) effγ2h2 rfℏ[γ4πMs+p (γ4πMs)2+ 4ω2] 8πα2 eff[(γ4πMs)2+ 4ω2]×(2e ℏ), (6)9 FIG. 6. VSY M versus applied magnetic field with ϕ∼180◦for (a) CFB (7)/ β-W(2.5) [CFW1] and CFB (7)/ β-W(2.5)/C 60 (13) [CFWC1] and (b) CFB (7)/ β-W(5) [CFW2] and CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructures measured at 15 dBm RF power, (c) Power dependent VSY M for CFW1 and CFWC1 (The solid line is the linear fit), (d) Schematic showing the spin-to-charge conversion phenomena in CFB/ β-W/C 60heterostructures. VISHE =wyLρNM tNMθSHλNMtanh(tNM 2λNM)Js, (7) where the ρNMis the resistivity of the β-W measured by four-probe technique and Lis the length of sample. The RF field ( hrf) and the width of the CPW transmission line ( wy) in our measurements are 0.5 Oe (at 15 dBm RF power) and 200 µm, respectively. The λNMfor the β-W has been taken as ∼3 nm from the literature [36]. Angel dependent ISHE measurements were performed to sepa- rate the AMR contribution from the VSY M. The contri- bution of VAMR was found to be one order smaller com- pared to V ISHE . For example, the VAMR and V ISHE for CFW2 heterostructure are found to be ∼0.15µV and ∼ 1.25µV, respectively (See Supporting Information). The ρNMfor 5 nm β-W is found to be 250 µΩ cm. Hence, the θSHfor CFB (7 nm)/ β-W (5 nm) bilayer estimated using equations 6 and 7 is found to be ∼-0.6±0.01. A similar type of calculation for CFB (7 nm)/ β-W (2.5 nm) bilayer estimates the θSHto be∼-0.67±0.01. The observed θSHvalue is larger compared to that reported in the literature [10–12]. The high SOC of our β-W and higher spin mix- ing conductance could be responsible for this enhanced θSH. Further, the interfacial SOC at CFB/ β-W interface can also induce an additive spin-to-charge conversion ef- fect, contributing to the enhancement of θSH. Such type of interfacial SOC mediated enhanced spin-to-charge con- version has been reported previously for NiFe/Pt and CFB/ β-Ta [19, 20]. Here, it is important to note that it is difficult to disentangle the IREE and ISHE effect in these type of FM/HM systems. On the other hand, the g(↑↓) efffor CFWC1 and CFWC2 decreases by 70 % due to the anti-damping phenomena and hence, the reduction inJsaccording to equation 6. However, the VISHE for the CFWC1 and CFWC2 are found to be larger than CFW1 and CFW2, respectively (Figure 6). This leads to theθSHvalue >1, calculated using the equation 6 and10 7 for CFB/ β-W/C 60heterostructures. This type of gi- gantic enhancement of θSHcannot be explained by mere bulk ISHE in β-W. The enhanced θSHcan be partly at- tributed to the enhanced bulk SOC of β-W upon molec- ular hybridization as predicted by the DFT calculations. Further, our DFT calculations also predict the enhance- ment of SOC of β-W layer closer to the CFB/ β-W inter- face due to the molecular hybridization in the CFB/ β- W/C 60heterostructures. The larger interfacial SOC and inversion symmetry breaking at the CFB/ β-W interface makes the scenario favorable for realizing an enhanced interfacial charge current due to the IREE as depicted in Figure 6 (d). Hence, the combination of bulk and interfa- cial SOC enhancement owing to the strong chemisorption of C 60onβ-W can attribute to the sizable increase in the θSHin CFB/ β-W/C 60heterostructures. The enhanced output DC voltage due to the spin pumping upon the C 60deposition on β-W is also con- sistent with the reduced effective damping value as dis- cussed earlier. The enhanced SOC of β-W and the struc- tural inversion asymmetry at the CFB/ β-W interface can stabilize the Rashba like states at FM/HM inter- face [19, 20]. The IREE mediated spin to charge con- version has received considerable interest after it was discovered at the Ag/Bi interface [7]. Till the date, most of the IREE effects have been experimentally re- alized at the all inorganic metal/metal, metal/oxide or oxide/oxide interfaces [9]. Our experiments and theoret- ical calculations show that the molecular hybridization at the HM/OSC interface can also help in strengthen- ing the Rashba spin-orbit coupling at the FM/HM in- terface. The Rashba interaction leads to the spin split- ting of bands, whose magnitude is dependent on the SOC strength at the interface. Upon the molecular hybridiza- tion, the SOC strength of β-W is further enhanced. This could have lead for a larger Rashba coefficient αRand hence, a relatively larger IREE at the FM/HM inter- face. The simultaneous observation of ISHE and IREE by engineering the HM interface with OSC can help in reducing the power consumption of future SOT-MRAM devices. As the CFB/ β-W stacking is employed for fab- rication of spin Hall nano oscillators (SHNOs) [37], theincorporation organic molecules can also significantly en- hance their efficiency. Hence, the HM/C 60interface can reduce the power consumption for data storage as well as facilitate in performing efficient spin logic operations. IV. CONCLUSION In conclusion, we present that a strong interfacial SOC can lead to the larger spin Hall angle in CFB/ β-W bi- layer. The thermally evaporated organic C 60molecules on CFB/ β-W bilayer leads to a strong chemisorption at theβ-W/C 60interface. The experimental and theoreti- cal calculations confirm that the molecular hybridization enhances the bulk as well as interfacial SOC in CFB/ β- W/C 60heterostructures. The strengthening of techno- logically important SOC manifests an anti-damping phe- nomena and gigantic ∼115% increase in spin-pumping induced output voltage for CFB/ β-W/C 60stacking. The control of magnetization dynamics and output efficiency in spintronics devices by the molecular hybridization can be a viable alternative to the other interface engineering and surface alloying techniques. The stabilization of the anti-damping and enhanced spin-to-charge conversion by tuning the bulk as well interfacial SOC via employing the cost effective, abundant organic molecule can pave the way for the fabrication of next generation power effi- cient spintronics devices. V. ACKNOWLEDGEMENT We acknowledge the Department of Atomic En- ergy (DAE), the Department of Science and Technol- ogy (DST) of the Government of India, and SERB project CRG/2021/001245. A.S. acknowledges the DST- National Postdoctoral Fellowship in Nano Science and Technology. We are also thankful to the Center for Inter- disciplinary Sciences, NISER for providing Raman spec- troscopy measurement facility. [1] Z. Guo, J. Yin, Y. Bai, D. Zhu, K. Shi, G. Wang, K. Cao, and W. 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2023-12-31

Development of power efficient spintronics devices has been the compelling need in the post-CMOS technology era. The effective tunability of spin-orbit-coupling (SOC) in bulk and at the interfaces of hybrid materials stacking is a prerequisite for scaling down the dimension and power consumption of these devices. In this work, we demonstrate the strong chemisorption of C60 molecules when grown on the high SOC $\beta$-W layer. The parent CFB/$\beta$-W bilayer exhibits large spin-to-charge interconversion efficiency, which can be ascribed to the interfacial SOC observed at the Ferromagnet/Heavy metal interface. Further, the adsorption of C60 molecules on $\beta$-W reduces the effective Gilbert damping by $\sim$15% in the CFB/$\beta$-W/C60 heterostructures. The anti-damping is accompanied by a gigantic $\sim$115% enhancement in the spin-pumping induced output voltage owing to the molecular hybridization. The non-collinear Density Functional Theory calculations confirm the long-range enhancement of SOC of $\beta$-W upon the chemisorption of C60 molecules, which in turn can also enhance the SOC at the CFB/$\beta$-W interface in CFB/$\beta$-W/C60 heterostructures. The combined amplification of bulk as well interfacial SOC upon molecular hybridization stabilizes the anti-damping and enhanced spin-to-charge conversion, which can pave the way for the fabrication of power efficient spintronics devices.

Molecular Hybridization Induced Antidamping and Sizable Enhanced Spin-to-Charge Conversion in Co20Fe60B20/$β$-W/C60 Heterostructures

2401.00486v1

Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 1 Magnified Damping under Rashba Spin Orbit Coupling Seng Ghee Tan† 1,2 ,Mansoor B.A.Jalil1,2 (1) Data Storage Institute, Agency for Science, Technology and Research (A*STAR) 2 Fusionopolis Way , #08-01 DSI , Innovis , Singapore 138634 (2) Department of Electrical Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576 Abstract The spin orbit coupling spin torque consists of the field -like [REF: S.G. Tan et al., arXiv:0705.3502, (2007). ] and the damping -like terms [REF: H. Kurebayas hi et al., Nature Nanotechnology 9, 211 (2014). ] that have been widely studied for applications in magnetic memory. We focus , in this article, not on the spin orbit effect producing the above spin torques, but on its magnifying the damping constant of all field like spin torques. As first order precession leads to second order damping, the Rashba constant is naturally co -opted, producing a magnified field -like dam ping effect. The Landau -Liftshitz -Gilbert equations are written separately for the local magnet ization and the itinerant spin, allowing the progression of magnetization to be self -consistently locked to the spin. PACS: 03.65.Vf, 73.63. -b, 73.43. -f † Correspondence author: Seng Ghee Tan Email: Tan_Seng_Ghee@dsi.a -star.edu.sg Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 2 1. Introduction In spintronic and magnetic physics, magnetization switching and spin torque [1] have been well -studied. The advent of the Rashba spin-orbit coupling ( RSOC) [2,3] due to inversion asymmetry at the i nterface of the ferromagnetic/heavy atom (FM/HA) heterostructure introduces new spin torque to the FM magnetization. The field -like [4-6] and the damping -like [7] SOC spin torque had been theoretically derived based on the gauge physics and the Pancharatna m-Berry’s phase , as well as experimentally verified and resolved . The numerous observation s of spin -orbit generation of spin torque [8-10], are all related to the experimental resolutions [6,7] of their field -like and damping -like nature, thus ushering in the possibility of spin -orbit based magnetic memory. While the damping -like spin torque due to Kurebayashi et al. [7] is dissipative in nature, the field -like due to Tan et al. [4,11], is non-dissipative , and precession causing . Recent studies have even mo re clearly demonstrated the physics and application promises of both the field -like and the damping - like SOC spin torque [12-14]. Besides , similar SOC spin torque have also been studied theoretically in FM/3D -Rashba [15] and FM -topological -insulator materi al [16,17] , and experimentally shown [18, 19 ] in topological insulator materials. The dissipative physics of all field -like magnetic torque terms have been derived in second -order manifestation in a manner introduced by Gilbert in the 1950 ’s. Conven tional stud y of magnetization dynamics is based on a Gilbert damping constant which is incorporated manually into the Landau -Lifshit z-Gilbert (LLG) equation. In this paper, we will focus our attention not so much on the spin-orbit effect producing the SOC spin torque, as on the spin -orbit effect magnifying the damping constant of all field -like spin torques. As field -like spin torques, regardless of origin s, generate first-order precession , the Rashba constant will be co -opted in to the second -order damping effect, producing a mag nified damping constant . On the other hand, c onventional incorporation of the dissipative damping physics into the LLG would fail t o account for the spin-orbit magnification of the damping strength . It would therefore be necessary to deriv e the LLG equations from a Hamiltonian which describes electron due to the local FM magnetization (𝒎), and those itinerant (𝒔) and injected from external parts . We present a set of modified LLG equation s for the 𝒎 and the 𝒔. This will be necessary for a more precise modeling of the 𝒎 trajectory that simultaneously tracks the 𝒔 trajectory. In summary, the two central themes of this Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 3 paper is our presentation of a self-consistent set of LLG equations under the Rashba SOC and the derivation of the Ra shba -magnified d amping constant in the second -order damping - like spin torque . 2. Theory of Magnified Damping The system under consideration is a FM/HA hetero -structure with inversion asymmetry provided by the interface. F ree electron denoted by 𝒔, is injected in an in -plane manner into the device . The FM equilibrium electron is denoted by 𝒎. One considers the external source -drain bias to inject electron of free-electron nature 𝒔 into the FM with kinetic, scattering , magnetic, and spin -orbit energ ies. The Hamiltonian is 𝐻𝑓=𝑝2 2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴 +𝜇0𝑴.𝑯𝒂𝒏𝒊+(2(𝜆+𝜆′) ℏ)(𝒔+𝒎).(𝒑×𝑬𝒕) −𝑖(𝜆+𝜆′)(𝒔+𝒎).(∇×𝑬𝒕) (1) where 𝒔,𝒎 have the unit s of angular momentum i.e. 𝑛ℏ 2 , while 𝑴=(𝑔𝑠𝜇𝐵 ℏ)𝒎 has the unit of magnetic moment , and 𝜇𝐵=𝑒ℏ 2𝑚 is the Bohr magneton . Note that (2𝜆 ℏ) is the vacuum SOC constant, while (2𝜆′ ℏ=2𝜂𝑅 ℏ2𝐸𝑖𝑛𝑣) is the Rashba SOC constant. The SOC part of the Hamiltonian illustrates th e simultaneous presence of vacuum and Rashba SOC. The proportion of the number of electron subject to each coupling would depend on the degree of hybridization. But s ince 𝜆′≫𝜆, the above can be written with just the Rashba SOC effect. Care is taken t o ensure 𝜆,𝜆′ share the same dimension of 𝑇𝑒𝑠𝑙 𝑎−1, and 𝑬𝒕 is the total electric field , 𝐽𝑠𝑑 is the s -d coupling constant, 𝑉𝑖𝑚𝑝𝑠 denotes the spin flip scattering potential, 𝑯𝒂𝒏𝒊 denotes the aniso tropy field of the FM material. On the other hand, one needs to be aware that the above is an e xpanded SOC expression that c omprises a momentum part as well as a curvature part [20]. One can then consider the physics of the electric curvature as related to the time dynamic of the spin moment , which bears a similar origin to the Faraday effect. In the modern context of Rashba physics [21], one considers electron spin to lock to the orbital angular momentum 𝑳 due to intrinsic spin orbit coupling at the atomic level. Due to broken Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 4 inversion symmetry , electric field (𝑬𝒊𝒏𝒗) points perpendicular to the plane of the FM/HA host . Because of hybridization, the 𝒔,𝑳,𝒑 of an electron is coupled in a complic ated way by the electric field. In a simple way, one first considers 𝑳 to be coupled as 𝐻=(2𝜆 ℏ)𝑳.(𝒑× 𝑬𝒊𝒏𝒗). As spin 𝒔 is coupled via atomic spin orbit locking to 𝑳, an effective coupling of 𝒔 to 𝑬𝒊𝒏𝒗 can be expected to occur with strength as determined by the atomic electric field. We will now take things a step further to make an assumption that 𝒔 is also coupled via 𝑳 to other sources of electric f ields e.g. those arising from spin dynamic (𝒅 𝑴 𝒅𝒕,𝒅 𝑺 𝒅𝒕), in the same way that it is coupled to 𝑬𝒊𝒏𝒗 . The actual extent of coupling will , however, be an experimental parameter that measures the efficiency of Rashba coupling to 𝑬𝒊𝒏𝒗 as opposed to electric fields (𝑬𝒎 ,𝑬𝒔) arising due to spin dynamic . The total electric field in the system is now 𝑬𝒕=𝑬𝒊𝒏𝒗 +𝑬𝒎+𝑬𝒔 , where 𝑬𝒎 ,𝑬𝒔 arise due to 𝒅 𝑴 𝒅𝒕,𝒅 𝑺 𝒅𝒕, respectively. On the momentum part of the Hamiltonian 2 𝜆′𝒔.(𝒌×𝑬𝒕), we only need to consider that 𝑬𝒕=𝑬𝒊𝒏𝒗 as one can, for simplicity, consider 𝑬𝒎 and 𝑬𝒔 to simply vanish on average. Thus in this renewed treatment, the momentum part is : 2𝜆′ ℏ𝒔.(𝒑×𝑬𝒊𝒏𝒗)=𝜂𝑅𝝈.(𝒌×𝒆𝒊𝒏𝒗) (2) where 𝜂𝑅=𝜆′ℏ𝐸𝑖𝑛𝑣 is the Rashba constant that has been vastly measured in many material systems with experimental values ranging from 0.1 to 2 𝑒𝑉𝐴̇. On the curvature part, one considers 𝑬𝒕=𝑬𝒎+𝑬𝒔 without the 𝑬𝒊𝒏𝒗 as 𝑬𝒊𝒏𝒗 is spatially uniform and thus would have zero curvature. In summary, the theory of this paper has it that the time -dynamic of the spin in a Rashba system produces a curvature part o f 𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕). Without the Rashba effect, this energy term would just take on the vacuum constant of (2𝜆 ℏ) instead of the magnified (2𝜆′ ℏ). The key physics is that in a Rashba FM/HA system , curvature 𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕) is satisfied by the first-order precession due to 𝒅𝑴 𝒅𝒕,𝒅𝑺 𝒅𝒕 which provide the electric field curvature in the form of −𝜇0(1+𝜒𝑚)𝑑 𝑴 𝑑𝑡=∇×𝑬𝒎 , and −𝜇0(1+ 𝜒𝑠)𝑑𝑺 𝑑𝑡=∇×𝑬𝒔 , where we remind reader again that 𝑴,𝑺 have the unit of magnetic moment. This results in spin becoming couple d to its own time dynamic, producing a spin -Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 5 orbit second -order damping -like spin torque. The electric field effect is illustrated in Fig. 1 below: Fig.1 . Magnetic precession under the effect of electric fields due to inv ersion asymmetry, self -dynamic of 𝑑𝑴 𝑑𝑡 and the spin dynamic of 𝑑𝑺 𝑑𝑡 . Projecting 𝑑𝑀 to the heterostructure surface, one could visualize the emergence of an induced electric field in the form of 𝛻𝑋𝐸 in such orientation as to satisfy the law of electromagnetism. One notes that the LLG equation is normally derived by letting 𝑺 satisfy the physical requirements of spin transport . One example of these requirements is assumed and discussed in REF 1 , with definitions contained therein : 𝑺(𝒓,𝑡)=𝑆0𝒏+𝜹𝑺 𝑱(𝒓,𝑡)=−𝜇𝐵𝑃 𝑒 𝑱𝒆⊗𝒏−𝐷0∇𝜹𝑺 (3) where 𝒏 is the unit vector of 𝑴, and 𝐷0 is the spin diffusion constant. Thus 𝑺=𝑺𝟎+𝜹𝑺 would be the total spin density that contains , respectively, the equilibrium, the non- equilibrium adiabatic, non-adiabatic , and Rashba field -like terms , i.e. 𝜹𝑺=𝜹𝑺𝒂+𝜹𝑺𝒏𝒂+ 𝜹𝑺𝑹. One notes that 𝑺𝟎 is the equilibrium part of 𝒔 that is aligned to 𝒎, meaning 𝒔𝟎 could exist in the absence of external field and current in the system. The conditions to satisfy are represented explicitly by the equations of: 𝑑𝑀 𝐸 𝑓𝑖𝑒𝑙𝑑 𝑑𝑢𝑒 𝑡𝑜 𝑖𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 𝛻𝑋𝐸 𝑑𝑀 Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 6 𝜕𝜹𝑺 𝜕𝑡=0, 𝐷0∇2𝜹𝑺=0,−𝜇𝐵𝑃 𝑒 𝛁.𝑱𝒆𝑴 𝑀𝑠=0, 𝑠0𝑴(𝒓,𝑡) 𝑡𝑓𝑀𝑠=0 (4) In the steady state treatment where 𝜕 𝜹𝑺 𝜕𝑡=0, one recover s the adiabatic component of 𝜹𝑺𝒂=𝒏×𝒋𝒆.𝛁𝒏 , and the non -adiabatic component of 𝜹𝑺𝒏𝒂=𝒋𝒆.𝛁𝒏. We also take the opportunity here to reconcile this with the gauge physics of spin torque, in which case , the spin potential 𝐴𝜇𝑠𝑚=𝑒 [𝛼 𝑈𝐸𝑖𝜎𝑗𝜀𝑖𝑗𝜇𝑈†+𝑖ℏ 𝑒𝑈𝜕𝜇𝑈†] would correspond , respectively, to 𝜹𝑺𝑹+ 𝜹𝑺𝒂. In fact, t he emergent spin p otential [22, 23] can be considered to encapsulate the physics of electron interaction with the local magnetization under the effect of SOC [4, 5 , 24-26]. Here we caution that 𝜹𝒔𝑹 is restricted to the field -like spin -orbit effect only . However, in this paper , 𝑺 is defined to satisfy the transport equations in Eq.(4) except for 𝜕𝜹𝑺 𝜕𝑡=0. Keeping the dynamic property of 𝑺 here allows a self -consistent equation set 𝑑𝑺 𝑑𝑡,𝑑𝑴 𝑑𝑡 to be introduced . The energy as experienced by the 𝑺,𝑴 electron are, respectively, 𝐻𝑓𝑠=𝑺.𝛿𝐻𝑓 𝛿𝑺 , 𝐻𝑓𝑚=𝑴.𝛿𝐻𝑓 𝛿𝑴 (5) with caution that 𝐻𝑓𝑠≠𝐻𝑓𝑚 . Upon rearrangement, the 𝒔,𝒎 centric energ ies are, respectively, 𝐻𝑓𝑠=(𝑝2 2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴+𝑺.𝑩𝑹−𝒊𝜆′𝒔.(∇×𝑬𝒕)) 𝐻𝑓𝑚=(𝐽𝑠𝑑𝑴.𝑺+𝜇0𝑴.𝑯𝒂−𝒊𝜆′𝒎.(∇×𝑬𝒕) ) (6) where 2𝜆′ ℏ𝒔.(𝒑×𝑬𝒕)=𝑺.𝑩𝑹, while 2𝜆′ ℏ𝒎.(𝒑×𝑬𝒕) vanishes . We particularly note that there have been recent discussions on the field -like [4,6,11 ] spin orbit torque as well as the damping [7] version. With 𝑑𝒔 𝑑𝑡=−𝟏 𝒊ℏ[𝒔,𝐻𝑓𝑠] ,𝑑𝒎 𝑑𝑡=−𝟏 𝒊ℏ[𝒎,𝐻𝑓𝑚], one would now have four dissipative torque t erms experienced by electron 𝒔,𝒎 as shown below : Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 7 ( 𝝉𝑺𝑺 𝝉𝑺𝑴 𝝉𝑴𝑺 𝝉𝑴𝑴)=𝑖𝜆′𝜇0(𝒔×(1+𝜒𝑠−1)𝑑𝑺 𝑑𝑡𝒔×(1+𝜒𝑚−1)𝑑𝑴 𝑑𝑡 𝒎×(1+𝜒𝑠−1)𝑑𝑺 𝑑𝑡𝒎×(1+𝜒𝑚−1)𝑑𝑴 𝑑𝑡) (7) To be consistent with conventional necessity to preserve magnetization norm in the physics of the LLG equation, we will drop the off-diagonal terms which are norm -breaking (non - conservation) . This is in order to keep the LLG equation in its conventional norm -conserving form, simplifying physics and calculation therefrom. Nonetheless, the non -conserving parts represent new dynamic physics that can be analysed in the future with techniques other than the familiar LLG equations. The self -consistent pair of spin torque equations in their open forms are: 𝜕𝑺 𝜕𝑡=−(𝑺× 𝑩𝑹+𝑺 𝑡𝑓)−1 𝑒𝛻𝑎(𝑗𝑎𝒔 𝑺)−(𝑺×𝑴 𝑚𝑡𝑒𝑥)−𝝉𝑺𝑺 𝜕𝑴 𝜕𝑡=−𝛾𝑴×𝜇0𝑯𝒂−𝑴×𝑺 𝑚 𝑡𝑒𝑥−𝝉𝑴𝑴 (8) where 𝐽𝑠𝑑=1 𝑚𝑡𝑒𝑥 has been applied, 𝛾 is the gyromagnetic ratio, 𝜒𝑚 is the susceptibility. For the stud y of Rashba -magnified damping i n this paper, we only need to keep the most relevant term which is 𝝉𝑴𝑴=𝑖 𝜂𝑅 ℏ𝐸𝑖𝑛𝑣𝜇0(1+𝜒𝑚−1) 𝒎×𝑑 𝑴 𝑑𝑡. In the phenomenological physics of Gilbert, the first-order precession leads inevitably to the second -order dissipative terms via 𝒔.𝒅𝑺 𝒅𝒕 ,𝒎.𝒅𝑴 𝒅𝒕. But in this paper, the general SOC physics had been expanded as shown in earlier sections, so that the dissipative terms are to naturally arise fr om such expansion. The advantage of the non -phenomenological approach is that, as said earlier, the Rashba constant will be co -opted into the second -order damping effect, resulting in the magnification of the damping constant associated with all field -like spin torque. Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 8 3. Conclusion The im portant result in this paper is that the damping constants have been magnified by the Rashba effect. This would not be possible if the damping constant was incorporated manually by standard means of Gilbert. As the Rashba constant is larger than the vacuum SOC constant as can be deduced from Table 1 and shown below 𝛼𝑅=𝛼𝜆′ 𝜆 , (9) magnetization dynamics in FM/HA hetero -structure with inversion asymmetry (interface, or bulk) might have to be modelled with the new equations. It is important to remind that all previously measured 𝜂𝑅 has had 𝐸𝑖𝑛𝑣 captured in the measured value. But w hat is needed in our study is the coupling of 𝑺 to a dynamic electric field, and that requires the value of just the coupling strength (𝜆′). As most measurement is carr ied out for 𝜂𝑅, the exact knowledge of 𝐸𝑖𝑛𝑣 corresponding to a specific 𝜂𝑅 will have a direct impact on the actual value of 𝜆′. We will, nonetheless, provide a quick, possibly exaggerated estimate. Noting that 𝜆=𝑒ℏ 4𝑚2𝑐2 and 𝜆′=𝜂𝑅 ℏ𝐸𝑖𝑛𝑣, and taking one measured value of 𝜂𝑅=1×10−10𝑒𝑉𝑚 , corresponding to a 𝐸𝑖𝑛𝑣=1010𝑉/𝑚, the magnification of 𝛼 works out to 104 times in magnitude , which may seem unrealistically strong . The caveat lies in the exact correspondence of 𝜂𝑅 to 𝐸𝑖𝑛𝑣, which remains to be determined experimentally. For example, if an experimentally determined 𝜂𝑅 actual ly corresponds to a much larger 𝐸𝑖𝑛𝑣, that would mean that 𝜆′=𝜂𝑅 ℏ𝐸𝑖𝑛𝑣 which magnifies the damping constant through 𝛼𝑅=𝛼𝜆′ 𝜆 might actually be much lower than pres ent estimate. Therefore, it is worth remembering, for simplicity sake that 𝛼𝑅 actually depends on the ratio of 𝜂𝑅 𝐸𝑖𝑛𝑣 but not 𝜂𝑅. It has also been assumed that 𝑳 couples to 𝑬𝒔,𝑬𝒎 with the same efficiency that it couples to 𝑬𝒊𝒏𝒗. This is still uncertain as th e Rashba constant with respect to 𝑬𝒔,𝑬𝒎 might actually be lower than those 𝜂𝑅 values that have been experimentally measured mostly with respect to 𝑬𝒊𝒏𝒗. Last, we note that as damping constant has been magnified here, and as increasingly high -precision, live monitoring of simult aneous 𝒔,𝒎 evolution is no longer redundant in smaller devices, care has been taken Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 9 to present the LLG equations in the form of a self-consistent pair of dynamic equations involving 𝑴 and 𝑺. This will be necessary for the accurate modeling of the simultaneous trajectory of both 𝑴 and 𝑺. Table 1. Summary of damping torque and damping con stant with and without Rashba effects. Hamiltonian Torque Damping constant 1. 𝐻=(2𝜆 ℏ)𝒔.(𝒑×𝑬𝒕) 𝜆=𝑒ℏ 4𝑚2𝑐2 𝜕𝒎 𝑑𝑡=𝑖𝜆𝜇0𝒎×(1+𝜒𝑚−1)𝜕𝑴 𝑑𝑡 𝛼=𝑖𝜆 2𝜇0𝑀𝑠(1+𝜒𝑚−1) 2. 𝐻𝑅=(2𝜆′ ℏ)𝒔.(𝒑×𝑬𝒕) 𝜆′=𝜂𝑅 ℏ𝐸𝑖𝑛𝑣 𝜕𝒎 𝑑𝑡=𝑖𝜆′𝜇0𝒎×(1+𝜒𝑚−1)𝑑𝑴 𝑑𝑡 𝛼𝑅=𝑖𝜆′ 2𝜇0𝑀𝑠(1+𝜒𝑚−1) REFERENCES [1] S. Zhang & Z. Li, “Roles of Non -equilibrium conduction electrons on the magnetization dynamics of ferromagnets”, Phys. Rev. Letts 93, 127204 (2004). [2] F.T. Vasko, “Spin splitting in the spectrum of two -dimensional electrons due to the surface potential”, Pis’ma Zh. Eksp. Teor. Fiz. 30, 574 (1979) [ JETP Lett. , 30, 541]. [3] Y.A. Bychkov & E.I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy”, Pis’ma Zh. Eksp. Teor. Fiz. , 39, 66 (1984) [ JETP Lett. , 39, 78]. [4] S. G. Tan, M. B. A. Jalil, and Xiong -Jun Liu, ”Local spin dynamic arising from the non -perturbative SU(2) gauge field of the spin orbit effect”, arXiv:0705.3502, (2007). [5] K. Obata and G. Tatara, “Current -induced domain wall motion in Rashba spin -orbit system”, Phys. Rev. B 77, 214429 (2008). [6] Jun Yeon Kim et al., “Layer thickness depende nce of the current -induced effective field vector in Ta/CoFeB/MgO”, Nature Materials 12, 240 (2013). [7] H. Kurebayashi et al., “An antidamping spin –orbit torque originating from the Berry curvature”, Nature Nanotechnology 9, 211 (2014). [8] Ioan Mihai M iron et al., “Current -driven spin torque induced by the Rashba effect in a ferromagnetic metal layer”, Nat. Mater. 9, 230 (2010). [9] Luqiao Liu et al., “ Spin -Torque Switching with the Giant Spin Hall Effect of Tantalum”, Science 336, 555 (2012). [10] Xin Fan et al., “Observation of the nonlocal spin -orbital effective field”, Nature Communications 4, 1799 (2013). [11] S. G. Tan, M. B. A. Jalil, T. Fujita, and X. J. Liu, “Spin dynamics under local gauge fields in chiral spin–orbit coupling systems”, Ann. Phy s. (NY) 326, 207 (2011). Magnified Damping under Rashba Spin Orbit Coupling November 1, 2015 10 [12] T.I. Mahdi Jamali et al., “Spin -Orbit Torques in Co/Pd Multilayer Nanowires”, Phys. Rev. Letts. 111, 246602 (2013) [13] Xuepeng Qiu et al., “Angular and temperature dependence of current induced spin -orbit effective fields in Ta/CoFeB/MgO nanowires”, Scientific Report 4, 4491 (2014). [14] Junyeon Kim, Jaivardhan Sinha, Seiji Mitani, Masamitsu Hayashi, Saburo Takahashi, Sadamichi Maekawa, Michihiko Yamanouchi, and Hideo Ohno, “Anomalous temperature dependence of current -induced torques in CoFeB/MgO heterostructures with Ta -based underlayers “, Physical Rev. B 89, 174424 (2014) . [15] Kazuhiro Tsutsui & Shuichi Murakami, “Spin -torque efficiency enhanced by Rashba spin splitting in three dimensions”, Phys. Rev. B 86, 115201 (2012) . [16] T. Yokoyama, J. D. Zhang, and N. Nagaosa, “Theoretical study of the dynamics of magnetization on the topological surface”, Phys. Rev. B 81, 241410(R) (2010). [17] Ji Chen, Mansoor Bin Abdul Jalil, and Seng Ghee Tan, “Current -Induced Spin Torque on Magnetization Textures Coupled to the Topological Surface States of Three -Dimensional Topological Insulators”, J. Phys. Soc. Japan 83, 064710 (2014). [18] Yabin Fan et al., “Magnetization switching through giant spin –orbit torque in a magnetically doped t opological insulator heterostructure”, Nature Material 13, 700 (2014). [19] A. R. Mellnik at al., “Spin -transfer torque generated by a topological insulator”, Nature 511, 449 (2014). [20] M.C. Hickey, J. S. Moodera, “Origin of Intrinsic Gilbert Damping”, Phys. Rev. Letts. 102, 137601 (2009). [21] Jin -Hong Park, Choong H. Kim, Hyun -Woo Lee, and Jung Hoon Han, “Orbital chirality and Rashba interaction in magnetic bands”, Phys. Rev. B 87, 041301(R) (2013) [22] Naoto Nagaosa & Yoshinori Tokura, “Emergent ele ctromagnetism in solids”, P hys. Scr. T146 014020 (2012). [23] Mansoor B. A. Jalil & Seng Ghee Tan, “Robustness of topological Hall effect of nontrivial spin textures”, Scientific Reports 4, 5123 (2014). [24] Y.B. Bazaliy, B.A. Jones, S. -C. Zhang, “Modific ation of the Landau -Lifshitz equation in the presence of a spin -polarized current in colossal - and giant -magnetoresistive materials”, Phys. Rev. B 57 (1998) 3213(R). [25] Takashi Fujita, MBA Jalil, SG Tan, Shuichi Murakami, “Gauge Field in Spintronics”, J. Appl. Phys. [Appl. Phys. Rev.] 110, 121301 (2011). [26] Seng Ghee Tan & Mansoor B.A. Jalil “Introduction to the Physics of Nanoelectronic”, Woodhead Publishing Limited, Cambridge, U.K . Chapter -5 (2012).

2015-11-13

The spin orbit coupling spin torque consists of the field-like [REF: S.G. Tan et al., arXiv:0705.3502, (2007).] and the damping-like terms [REF: H. Kurebayashi et al., Nature Nanotechnology 9, 211 (2014).] that have been widely studied for applications in magnetic memory. We focus, in this article, not on the spin orbit effect producing the above spin torques, but on its magnifying the damping constant of all field like spin torques. As first order precession leads to second order damping, the Rashba constant is naturally co-opted, producing a magnified field-like damping effect. The Landau-Liftshitz-Gilbert equations are written separately for the local magnetization and the itinerant spin, allowing the progression of magnetization to be self-consistently locked to the spin.

Magnified Damping under Rashba Spin Orbit Coupling

1511.04227v1

arXiv:1805.11468v1 [cond-mat.mtrl-sci] 29 May 2018APS/123-QED Gilbert damping in non-collinear magnetic systems S. Mankovsky, S. Wimmer, H. Ebert Department of Chemistry/Phys. Chemistry, LMU Munich, Butenandtstrasse 11, D-81377 Munich, Germany (Dated: May 30, 2018) The modification of the magnetization dissipation or Gilber t damping caused by an inhomoge- neous magnetic structure and expressed in terms of a wave vec tor dependent tensor α(/vector q) is in- vestigated by means of linear response theory. A correspond ing expression for α(/vector q) in terms of the electronic Green function has been developed giving in p articular the leading contributions to the Gilbert damping linear and quadratic in q. Numerical results for realistic systems are pre- sented that have been obtained by implementing the scheme wi thin the framework of the fully relativistic KKR (Korringa-Kohn-Rostoker) band structur e method. Using the multilayered system (Cu/Fe 1−xCox/Pt)nas an example for systems without inversion symmetry we demo nstrate the occurrence of non-vanishing linear contributions. For the alloy system bcc Fe 1−xCoxhaving inver- sion symmetry, on the other hand, only the quadratic contrib ution is non-zero. As it is shown, this quadratic contribution does not vanish even if the spin-orb it coupling is suppressed, i.e. it is a direct consequence of the non-collinear spin configuration. PACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds I. INTRODUCTION The magnetization dissipation in magnetic materi- als is conventionally characterized by means of the Gilbert damping (GD) tensor αthat enters the Landau- Lifshitz-Gilbert (LLG) equation [1]. This positive- definite second-rank tensor depends in general on the magnetization direction. It is well established that in the case of spatially uniformly magnetized ferromagnetic (FM) metals two regimes of slow magnetization dynam- ics can be distinguished, which are governed by differ- ent mechanisms of dissipation [2–4]: a conductivity-like behaviour occuring in the limiting case of ordered com- pounds that may be connected to the Fermi breathing mechanism and a resistivity-likebehaviourshown by ma- terials with appreciable structural, chemical or tempera- ture induced disorder and connected to a spin-flip scat- teringmechanism. Animportantissueisthatbothmech- anisms are determined by the spin-orbit coupling in the system (see e.g. [2, 4, 5]). During the last years, it was demonstrated by variousauthors that first-principles cal- culationsforthe GD parameterforcollinearferromagntic materials allow to cover both regimes without use of any phenomenological parameters. In fact, in spite of the dif- ferences concerning the formulation for the damping pa- rameter and the corresponding implementaion [6–8], the numerical results are in generalin rathergood agreement with each other as well as with experiment. In the case of a pronounced non-collinear magnetic texture, e.g. in the case of domain walls or topologi- cally nontrivial magnetic configurations like skyrmions, the description of the magnetization dissipation assum- ing a spatial-invariant tensor αis incomplete, and a non- local character of GD tensor in such systems has to be taken into account [9–11]. This implies that the dissipa- tive torque on the magnetization should be representedby the expression of the following general form [12]: τGD= ˆm(/vector r,t)×/integraldisplay d3r′α(/vector r−/vector r′)∂ ∂tˆm(/vector r′,t).(1) In the case of a magnetic texture varying slowly in space, however, an expansion of the damping parameter in terms of the magnetization density and its gradients [11] is nevertheless appropriate: αij=αij+αkl ijmkml+αklp ijmk∂ ∂rlmp(2) +αklpq ij∂ ∂rkml∂ ∂rpmq+... , where the first term αijstands for the conventional isotropic GD and the second term αkl ijmkmlis associated with the magneto-crystalline anisotropy (MCA). The third so-called chiral term αklp ijmk∂ ∂rlmpis non-vanishing in non-centrosymmetric systems. The important role of this contribution to the damping was demonstrated ex- perimentally when investigating the field-driven domain wall(DW)motioninasymmetricPt/Co/Pttrilayers[13]. As an alternative to the expansion in Eq. (2) one can discuss the Fourier transform α(/vector q) of the damping pa- rametercharacterizinginhomogeneousmagneticsystems, which enter the spin dynamics equation ∂ ∂t/vector m(/vector q) =−γ/vector m(/vector q)×/vectorH−/vector m(/vector q)×α(/vector q)∂ ∂t/vector m(/vector q).(3) In this formulation the term linear in qis the first chiral term appearing in the expansion of α(/vector q) in powers of q. Furthermore, it is important to note that it is directly connected to the αklp ijmk∂ ∂rlmpterm in Eq. (2). By applying a gauge field theory, the origin of the non-collinear corrections to the GD can be ascribed to the emergent electromagnetic field created in the time- dependent magnetic texture [14, 15]. Such an emergent2 electromagneticfieldgivesrisetoaspincurrentwhosedi- vergence characterizes the change of the angular momen- tum in the system. This allows to discuss the impact of non-collinearity on the GD via a spin-pumping formula- tion[9,14,16]. Somedetailsofthephysicsbehind thisef- fect depend on the specific propertiesofthe materialcon- sidered. Accordingly, different models for magnetisation dissipation were discussed in the literature [9, 12, 14, 17– 19]. Non-centrosymmetric two-dimensional systems for which the Rashba-like spin-orbit coupling plays an im- portant role havereceived special interest in this context. They have been discussed in particular by Akosa et al. [19], in order to explain the origin of chiral GD in the presence of a chiral magnetic structure. The fourth term on the r.h.s. of Eq. (2) corresponds to a quadratic term of an expansion of α(/vector q) with re- spect to q. It was investigated for bulk systems with non-magnetic [20] and magnetic [9] impurity atoms, for which the authors have shown on the basis of model con- sideration that it can give a significant correction to the homogeneous GD in the case of weak metallic ferromag- nets. In striking contrast to the uniform part of the GD this contribution does not require a non-vanishing spin- orbit interaction. To our knowledge, only very few ab-initio investiga- tions on the Gilbert damping in non-collinear magnetic systems along the lines sketched above have been re- ported so far in the literature. Yuan et al. [21] calcu- lated the in-plane and out-of-plane damping parameters in terms of the scattering matrix for permalloy in the presence of N´ eel and Bloch domain walls. Freimuth et al. [22], discuss the properties of a q-dependent Gilbert damping α(/vector q) calculated for the one-dimensional Rashba modelinthepresenceofthe N´ eel-typenon-collinearmag- netic exchange field, demonstrating different GD for left- handed and right-handed DWs. Here we extend the for- malism developed before to deal with the GD in ferro- magnets [6], to get access to non-collinear system. The formalism based on linear response theory allows to ex- pand the GD parameters with respect to a modulation of the magnetization expressed in terms of a wave vector /vector q. Correspondingnumerical results will be presented and discussed. II. GILBERT DAMPING FOR NON-COLLINEAR MAGNETIZATION In the following we focus on the intrinsic contribution to the Gilbert damping, excluding spin current induced magnetizationdissipationwhich occursin the presenceof an external electric field. For the considerations on the magnetization dissipation an adiabatic variation of the magnetization in the time and space domain is assumed. Moreover, it is assumed that the magnitude of the local magnetic moments is unchanged during a change of the magnetization, i.e. the exchange field should be strong enough to separate transverse and longitudinal parts ofthe magnetic susceptibility. With these restrictions, the non-local Gilbert damping can be determined in terms of the spin susceptibility tensor χαβ(/vector q,ω) =i1 V∞/integraldisplay 0dt∝angbracketleftˆSα(/vector q,t)ˆSα(−/vector q,0)∝angbracketright0ei(ω−δ)t,(4) whereˆSα(/vector q,t) is the /vector q- andt-dependent spin operator and reduced units havebeen used ( /planckover2pi1= 1). With this, the Fourier transformationofthe real-spaceGilbert damping can be represented by the expression [23, 24] ααβ(/vector q) =γ M0Vlim ω→0∂ℑ[χ−1]αβ(/vector q,ω) ∂ω.(5) Hereγ=gµBis the gyromagneticratio, M0=µtotµB/V is the equilibrium magnetization and Vis the volume of the system. In order to avoid the calculation of the dy- namical magnetic susceptibility tensor χ(/vector q,ω), which is the Fourier transformed of the real space susceptibility χ(/vector r−/vector r′,ω), it is convenient to represent χ(/vector q,ω) in Eq. (5), in terms of a correlation function of time deriva- tives ofˆS. As˙ˆScorresponds to the torque /vectorT, that may include non-dissipative and dissipative parts, one may consider instead the torque-torque correlation function π(/vector q,ω) [24–27]. Assuming the magnetization direction parallelto ˆ zone obtains the expression for the Gilbert damping α(/vector q) α(/vector q) =γ M0Vlim ω→0∂ℑ[ǫ·π(/vector q,ω)·ǫ] ∂ω. (6) whereǫ=/bracketleftbigg 0 1 −1 0/bracketrightbigg is the transverse Levi-Civita tensor. Thisimpliesthefollowingrelationshipofthe αtensorele- ments with the elements of the torque-torque correlation tensorπ:αxx∼ −πyyandαyy∼ −πxx[24]. Using Kubo’s linear response theory in the Matsubara representation and taking into account the translational symmetry of a solid the torque-torque correlation func- tionπαβ(/vector q,ω) can be expressed by (see, e.g. [28]): παβ(/vector q,iωn) =1 β/summationdisplay pm∝angbracketleftTαG(/vectork+/vector q,iωn+ipm) TβG(/vectork,ipm)∝angbracketrightc,(7) whereG(/vectork,ip) is the Matsubara Green function and ∝angbracketleft...∝angbracketrightc indicates a configurational average required in the pres- ence of any disorder (chemical, structural or magnetic) in the system. Using a Lehman representation for the Green function [28] G(/vectork,ipm) =/integraldisplay+∞ −∞dE πℑG+(/vectork,E) ipm−E(8) withG+(/vectork,E) the retarded Green function and using the relation 1 β/summationdisplay pm1 ipm+iωn−E11 ipm−E2=f(E2)−f(E1) iωn+E2−E13 for the sum over the Matsubara poles in Eq. (7), the torque-torq ue correlation function is obtained as: παβ(/vector q,iωn) =1 ΩBZ/integraldisplay d3k+∞/integraldisplay −∞dE1 π+∞/integraldisplay −∞dE2 πTr/angbracketleftbigg TαℑG(/vectork,E1)TβℑG(/vectork,E2)f(E2)−f(E1) iωn+E2−E1/angbracketrightbigg c. (9) Perfoming finally the analytical continuation iωn→ω+iδone arrives at the expression Γαβ(/vector q,ω) =−π ΩBZ/integraldisplay d3k+∞/integraldisplay −∞dE1 π+∞/integraldisplay −∞dE2 πTr/angbracketleftbigg TαℑG(/vectork+/vector q,E1)TβℑG(/vectork,E2)/angbracketrightbigg c(f(E2)−f(E1))δ(ω+E2−E1) =−π ΩBZ/integraldisplay d3k+∞/integraldisplay −∞dE πTr/angbracketleftbigg TαℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg c(f(E)−f(E+ω)) (10) for the imaginary part of the correlation function with Γ αβ(/vector q,ω) =−πℑπαβ(/vector q,ω). Accordingly one gets for the diagonal elements of Gilbert damping tensor the expression ααα(/vector q) =γ M0Vlim ω→0∂[ǫ·Γ(/vector q,ω)·ǫ] ∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle αα =γπ M0Vlim ω→0∂ ∂ω1 ΩBZ/integraldisplay d3k+∞/integraldisplay −∞dE π2(f(E+ω)−f(E))Tr/angbracketleftbigg TβℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg c =γ M0V1 ΩBZ/integraldisplay d3k+∞/integraldisplay −∞dE πδ(E−EF)Tr/angbracketleftbigg TβℑG(/vectork+/vector q,E)TβℑG(/vectork,E)/angbracketrightbigg c =1 4[ααα(/vector q,G+,G+)+ααα(/vector q,G−,G−)−ααα(/vector q,G+,G−)−ααα(/vector q,G−,G+)], (11) where the index βof the torque operator Tβis related to the index αaccording to Eq. 6, and the auxiliary functions ααα(/vector q,G±,G±) =γ M0Vπ1 ΩBZ/integraldisplay d3kTr/angbracketleftbigg TβG±(/vectork+/vector q,EF)TβG±(/vectork,EF)/angbracketrightbigg c(12) expressed in terms of the retarded and advanced Green function s,G+andG−, respectively. To account properly for the impact of spin-orbit coupling when dealin g with Eqs. (11) and (12) a description of the electronic structure based on the fully relativistic Dirac formalis m is used. Working within the framework of local spin density formalism (LSDA) this implies for the Hamiltonian the form [2 9]: ˆHD=c/vectorα·/vector p+βmc2+V(/vector r)+β/vectorσ·ˆ/vector mBxc(/vector r). (13) Hereαiandβare the standard Dirac matrices, /vectorσdenotes the vector of relativistic Pauli matrices, /vector pis the relativistic momentum operator [30] and the functions V(/vector r) and/vectorBxc=/vectorσ·ˆ/vector mBxc(/vector r) are the spin-averaged and spin-dependent parts, respectively, of the LSDA potential [31] with ˆ/vector mgiving the orientation of the magnetisation. With the Dirac Hamiltonian given by Eq. (13), the torque operator ma y be written as /vectorT=β[/vector σ׈/vector m]Bxc(/vector r). Furthermore, the Green functions entering Eqs. (11) and (12) a re determined using the spin-polarized relativistic version of multiple scattering theory [29, 32] with the real space re presentation of the retarded Green function given by: G+(/vector r,/vector r′,E) =/summationdisplay ΛΛ′Zn Λ(/vector r,E)τnm ΛΛ′(E)Zm× Λ′(/vector r′,E) −δnm/summationdisplay Λ/bracketleftbig Zn Λ(/vector r,E)Jn× Λ′(/vector r′,E)Θ(r′ n−rn) +Jn Λ(/vector r,E)Zn× Λ′(/vector r′,E)Θ(rn−r′ n)/bracketrightbig . (14)4 Here/vector r,/vector r′refertoatomiccellscenteredatsites nandm, respectively,where Zn Λ(/vector r,E) =ZΛ(/vector rn,E) =ZΛ(/vector r−/vectorRn,E) isa function centered at the corresponding lattice vector /vectorRn. The four-component wave functions Zn Λ(/vector r,E) (Jn Λ(/vector r,E)) are regular (irregular) solutions to the single-site Dirac equation labeled by the combined quantum numbers Λ = ( κ,µ), withκandµbeing the spin-orbit and magnetic quantum numbers [30]. Finally, τnm ΛΛ′(E) is the so-called scattering path operator that transfers an electronic wave coming in at site minto a wave going out from site nwith all possible intermediate scattering events accounted for. Using matrix notation with respect to Λ, this leads to the following exp ression for the auxilary damping parameters in Eq. (12): ααα(/vector q,G±,G±) =γ M0Vπ1 ΩBZ/integraldisplay d3kTr/angbracketleftbigg Tβτ(/vectork+/vector q,E± F)Tβτ(/vectork,E± F)/angbracketrightbigg c. (15) In the case of a uniform magnetization, i.e. for q= 0 one obviously gets an expression for the Gilbert damping tensor as it was worked out before [7]. Assuming small wave vectors, the te rmτ(/vectork+/vector q,E± F) can be expanded w.r.t. /vector qleading to the series τ(/vectork+/vector q,EF) =τ(/vectork,E)+/summationdisplay µ∂τ(/vectork,E) ∂kµqα+1 2/summationdisplay µν∂τ(/vectork,E) ∂kµ∂kνqµqν+... (16) that results in a corresponding expansion for the Gilbert damping: α(/vector q) =α+/summationdisplay µαµqµ+1 2/summationdisplay µναµνqµqν+... (17) with the following expansion coefficients: α0±± αα=g πµtot1 ΩBZTrace/integraldisplay d3k/angbracketleftbigg Tβτ(/vectork,E± F)Tβτ(/vectork,E± F)/angbracketrightbigg c(18) αµ±± αα=g πµtot1 ΩBZTrace/integraldisplay d3k/angbracketleftbigg Tβ∂τ(/vectork,E± F) ∂kµTβτ(/vectork,E± F)/angbracketrightbigg c(19) αµν±± αα=g πµtot1 2ΩBZTrace/integraldisplay d3k/angbracketleftbigg Tβ∂2τ(/vectork,E± F) ∂kµ∂kνTβτ(/vectork,E± F)/angbracketrightbigg c, (20) and with the g-factor 2(1+ µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, and the total magnetic moment µtot=µspin+µorb. The numerically cumbersome term in Eq. (20), that involves the sec ond order derivative of the matrix of /vectork-dependent scattering path operator τ(/vectork,E), can be reformulated by means of an integration by parts: 1 ΩBZ/integraldisplay d3kTβτ(/vectork,EF)Tβ∂2τ(/vectork,EF) ∂kµ∂kν=/bracketleftBigg/integraldisplay /integraldisplay dkβdkγTi βτ(/vectork,E)Tj β∂τ(/vectork,E) ∂kβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleKα 2 −Kα 2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright =0 −/integraldisplay /integraldisplay /integraldisplay dkαdkβdkγTβ∂τ(/vectork,EF) ∂kµTβ∂τ(/vectork,EF) ∂kν/bracketrightBigg =−1 ΩBZ/integraldisplay d3kTβ∂τ(/vectork,EF) ∂kµTβ∂τ(/vectork,EF) ∂kν leading to the much more convenient expression: αµν±± αα=−g 2πµtot/integraldisplay d3kTr/angbracketleftbigg Tβ∂τ(/vectork,E± F) ∂kµTβ∂τ(/vectork,E± F) ∂kν/angbracketrightbigg c. (21) III. RESULTS AND DISCUSSIONS The scheme presented above to deal with the Gilbert damping in non-collinear systems has been implementedwithin the SPR-KKR program package [33]. To exam-5 ine the importance of the chiral correction to the Gilbert damping a first application of Eq. (19) has been made for the multilayer system (Cu/Fe 1−xCox/Pt)nseen as a non-centrosymmetricmodelsystem. Thecalculatedzero- order (uniform) GD parameter αxxand the correspond- ing first-order (chiral) αx xxcorrection term for /vector q∝bardblˆxare plotted in Fig. 1 top and bottom, respectively, as a func- tion of the Fe concentration x. Both terms, αxxandαx xx, 0 0.2 0.4 0.6 0.8 100.20.4αxx 0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.) FIG. 1: The Gilbert damping parameters αxx(top) and αx xx(bottom) calculated for the model multilayer system (Cu/Fe 1−xCox/Pt)nusing Eqs. (18) and (19), respectively. increase approaching the pure limits w.r.t. the Fe 1−xCox alloy subsystem. In the case of the uniform parame- terαxx, this increase is associated with the dominating breathing Fermi-surface damping mechanism. This im- plies that the modification of the Fermi surface (FS) in- duced by the spin-orbit coupling (SOC) follows the mag- netization direction that slowly varies with time. An ad- ditional contribution to the GD, having a similar origin, occurs for the non-centrosymmertic systems with heli- magnetic structure. In this case, the features of the elec- tronicstructure governedby the lackofinversionsymme- try result in a FS modification dependent on the helicity of the magnetic structure. This implies a chiral contri- bution to the GD which can be associated with the term proportional to the gradient of the magnetization. Ob- viously, this additional modification of the FS and the associated mechanism for the GD does not show up for a uniform ferromagnet. As αis caused by the SOC one can expect that it vanishes for vanishing SOC. This was indeed demonstrated before [5]. The same holds also for αxthat is cased by SOC as well. Another system considered is the ferromagnetic alloy system bcc Fe 1−xCox. As this system has inversion sym- metry the first-order term αµshould vanish. This expec- tation could also be confirmed by calculations that ac-count for the SOC. The next non-vanishing term of the expansion of the GD is the term ∝q2. The correspond- ing second-order term αxx xxis plotted in Fig. 2 (bottom) together with the zero-order term αxx(top). The bot- 0 0.1 0.2 0.3 0.4 0.500.511.52αxx× 103 0 0.1 0.2 0.3 0.4 0.5xCo012αxxxx ((a.u.)2)Fe1-xCox FIG. 2: The Gilbert damping terms αxx(top) and αxx xx(bot- tom) calculated for bcc Fe 1−xCox. tom panel shows in addition results for αxx xxthat have been obtained by calculations with the SOC suppressed. As one notes the results for the full SOC and for SOC suppressed are very close to each other. The small dif- ference between the curves for that reason have to be as- cribed to the hybridization of the spin-up and spin-down subsystems due to SOC. As discussed in the literature [9, 17, 20] a non-collinear magnetic texture has a corre- sponding consequence but a much stronger impact here. In contrastto the GDin uniform FM systemswhereSOC isrequiredto breakthe totalspin conservationin the sys- tem,αxx xxis associated with the spin-pumping effect that can be ascribed to an emergent electric field created in the non-uniform magnetic system. In this case magnetic dissipation occurs due to the misalignment of the elec- tron spin following the dynamic magnetic profile and the magnetization orientation at each atomic site, leading to the dephasing of electron spins [16] IV. SUMMARY To summarize, expressions for corrections to the GD ofhomogeneoussystems werederived which areexpected to contribute in the case of non-collinear magnetic sys- tems. The expression for the GD parameter α(/vector q) seen as a function of the wave vector /vector qis expanded in powers ofq. In the limit of weakly varying magnetic textures, this leads to the standard uniform term, α, and the first- and second-order corrections, αµandαµν, respectively. Model calculations confirmed that a non-vanishing value6 forαµcan be expected for systems without inversion symmetry. In addition, SOC has been identified as the major source for this term. The second-order term, on the other hand, may also show up for systems with inver- sion symmetry. In this case it was demonstrated by nu- merical work, that SOC plays only a minor role for αµν, while the non-collinearity of the magnetization plays the central role.V. ACKNOWLEDGEMENT Financial support by the DFG via SFB 1277 (Emer- gente relativistische Effekte in der Kondensierten Ma- terie) is gratefully acknowledged. [1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443 (2004). [2] V. Kambersky, Can. J. Phys. 48, 2906 (1970), URL http://www.nrcresearchpress.com/doi/abs/10.1139/p70 -361. [3] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006). [4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007), URL http://link.aps.org/doi/10.1103/PhysRevLett.99.0272 04. [5] S. Mankovsky, D. K¨ odderitzsch, G. Woltersdorf, and H. Ebert, Phys. Rev. B 87, 014430 (2013), URL http://link.aps.org/doi/10.1103/PhysRevB.87.014430 . [6] A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010), URL http://link.aps.org/doi/10.1103/PhysRevLett.105.236 601. [7] H. Ebert, S. Mankovsky, D. K¨ odderitzsch, and P. J. Kelly, Phys. Rev. Lett. 107, 066603 (2011), http://arxiv.org/abs/1102.4551v1, URL http://link.aps.org/doi/10.1103/PhysRevLett.107.066 603. [8] I. Turek, J. Kudrnovsk´ y, and V. Drchal, Phys. Rev. B 92, 214407 (2015), URL http://link.aps.org/doi/10.1103/PhysRevB.92.214407 . [9] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vi- gnale, Phys. Rev. B 79, 094415 (2009), URL https://link.aps.org/doi/10.1103/PhysRevB.79.094415 . [10] K. M. D. Hals, K. Flensberg, and M. S. Rud- ner, Phys. Rev. B 92, 094403 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.094403 . [11] K. M. D. Hals and A. Brataas, Phys. Rev. B 89, 064426 (2014), URL https://link.aps.org/doi/10.1103/PhysRevB.89.064426 . [12] N. Umetsu, D. Miura, and A. Sakuma, Phys. Rev. B 91, 174440 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.91.174440 . [13] E. Jue, C. K. Safeer, M. Drouard, A. Lopez, P. Balint, L. Buda-Prejbeanu, O. Boulle, S. Auffret, A. Schuhl, A. Manchon, et al., Nature Materials 15, 272 (2015), URLhttp://dx.doi.org/10.1038/nmat4518 . [14] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.102.08 6601. [15] G. Tatara and N. Nakabayashi, Journal of Applied Physics 115, 172609 (2014), http://dx.doi.org/10.1063/1.4870919, URL http://dx.doi.org/10.1063/1.4870919 . [16] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77, 134407 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.77.134407 . [17] J. Zang, M. Mostovoy, J. H. Han, and N. Na- gaosa, Phys. Rev. Lett. 107, 136804 (2011), URL https://link.aps.org/doi/10.1103/PhysRevLett.107.13 6804.[18] T. Chiba, G. E. W. Bauer, and S. Taka- hashi, Phys. Rev. B 92, 054407 (2015), URL https://link.aps.org/doi/10.1103/PhysRevB.92.054407 . [19] C. A. Akosa, I. M. Miron, G. Gaudin, and A. Man- chon, Phys. Rev. B 93, 214429 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.214429 . [20] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B 78, 020404 (2008), URL https://link.aps.org/doi/10.1103/PhysRevB.78.020404 . [21] Z. Yuan, K. M. D. Hals, Y. Liu, A. A. Starikov, A. Brataas, and P. J. Kelly, Phys. Rev. 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2018-05-29

The modification of the magnetization dissipation or Gilbert damping caused by an inhomogeneous magnetic structure and expressed in terms of a wave vector dependent tensor $\underline{\alpha}(\vec{q})$ is investigated by means of linear response theory. A corresponding expression for $\underline{\alpha}(\vec{q})$ in terms of the electronic Green function has been developed giving in particular the leading contributions to the Gilbert damping linear and quadratic in $q$. Numerical results for realistic systems are presented that have been obtained by implementing the scheme within the framework of the fully relativistic KKR (Korringa-Kohn-Rostoker) band structure method. Using the multilayered system (Cu/Fe$_{1-x}$Co$_x$/Pt)$_n$ as an example for systems without inversion symmetry we demonstrate the occurrence of non-vanishing linear contributions. For the alloy system bcc Fe$_{1-x}$Co$_x$ having inversion symmetry, on the other hand, only the quadratic contribution is non-zero. As it is shown, this quadratic contribution does not vanish even if the spin-orbit coupling is suppressed, i.e.\ it is a direct consequence of the non-collinear spin configuration.

Gilbert damping in non-collinear magnetic system

1805.11468v1

arXiv:0705.0406v1 [cond-mat.mtrl-sci] 3 May 2007Planar spin-transfer device with a dynamic polarizer. Ya. B. Bazaliy,1D. Olaosebikan,2and B. A Jones1 1IBM Almaden Research Center, 650 Harry Road, San Jose, CA 951 20 2Department of Physics, Cornell University, Ithaca, NY 1485 3 (Dated: July, 2006) In planar nano-magnetic devices magnetization direction i s kept close to a given plane by the large easy-plane magnetic anisotropy, for example by the shape an isotropy in a thin film. In this case magnetization shows effectively in-plane dynamics with onl y one angle required for its description. Moreover, the motion can become overdamped even for small va lues of Gilbert damping. We derive the equations of effective in-plane dynamics in the pr esence of spin-transfer torques. The simplifications achieved in the overdamped regime allow to s tudy systems with several dynamic magnetic pieces (“free layers”). A transition from a spin-t ransfer device with a static polarizer to a device with two equivalent magnets is observed. When the si ze difference between the magnets is less than critical, the device does not exhibit switching , but goes directly into the “windmill” precession state. PACS numbers: 72.25.Pn, 72.25.Mk, 85.75.-d I. INTRODUCTION The prediction1,2and first experimental observations3,4,5,6,7,8of spin-transfer torques opened a new field in magnetism which studies non-equilibrium magnetic interactions induced by electric current. Since such interactions are relatively significant only in very small structures, the topic is a part of nano-magnetism. The current-induced switching of magnetic devices achieved through spin-transfer torques is a candidate for being used as a writing process in magnetic random access memory (MRAM) devices. The MRAM memory cell is a typical example of a spintronic device in which the electron spin is used to achieve useful logic, memory or other operations normally performed by electronic circuits. To produce the spin-transfer torques, electric currents have to flow through the spatially non-uniform mag- netic configurations in which the variation of magneti- zation can be either continuous or abrupt. The first case is usually experimentally realized in magnetic domain walls.3,9,10,11Here we will be focusing on the second case realized in the artificially grown nano-structures. Such spin-transfer devices contain severalmagnetic pieces sep- arated by non-magnetic metal spacers allowing for arbi- traryanglesbetweenthemagneticmomentsofthepieces. Magnetizationmyvarywithineachpieceaswell, butthat variation is usually much smaller and vanishes as the size of piece is reduced, or for larger values of spin-stiffness of magnetic material. The typical examples of a system with discrete variation of magnetization are the “nano- pillar” devices8(Fig.1A). Their behavior can be reason- ablywellapproximatedbyassumingthatmagneticpieces are mono-domain, each described by a single magnetiza- tion vector /vectorM(t) =Ms/vector n(t) where /vector nis the unit vector andMsis the saturation magnetization. The evolution of/vector n(t) is governedby the Landau-Lifshitz-Gilbert (LLG) equation with spin-transfer terms.2,12 It is often the case that magnetic pieces in a spin-transfer device have a strong easy-plane anisotropy. For example, in nano-pillars both the polarizer and the free magnetic layer are disks with the diameter much larger than the thickness. Consequently, the shape anisotropy makes the plane of the disk an easy magnetic plane. In the planar devices13built from thin film layers (Fig. 1B) the shape anisotropy produces the same effect. When the easy-plane anisotropy energy is much larger then all otherenergies,thedeviationsof /vector n(t)fromthein-planedi- rection are very small. An approximation based on such smallness is possible and providesan effective description of the magnetic dynamics in terms of the direction of the projection of /vector n(t) on the easy plane, i.e. in terms of one azimuthalangle. Inthispaperwederivetheequationsfor effective in-plane motion in the presence of spin-transfer effect and discuss their use by considering several exam- ples. In the absence of spin-transfer effects the large easy- plane anisotropy creates a regime of overdamped mo- tion even for the small values of Gilbert damping con- stantα≪1.14In that regime the equations simplify further. Here the overdamped regime is discussed in the presence of electric current. The reduction of the number of equations allows for a simple consideration of a spin-transfer device with two dynamic magnetic pieces. We show how an asymmetry in the sizes of these pieces createsa transition between the polarizer-analyzer (“fixed layer - free layer”) operation regime2,8,12,15and the regime of nearly identical pieces where current leads ns jj j snA B FIG. 1: Planar spin-transfer devices2 not to switching, but directly to the Slonczewski “wind- mill” dynamic state.2Finally, we point out the limita- tions of the overdamped approximation in the presence of the spin-transfer torques. II. DYNAMIC EQUATIONS IN THE LIMIT OF A LARGE EASY-PLANE ANISOTROPY Magnetizationdynamicsin thepresenceofelectriccur- rent is governed by the LLG equation with the spin- transfer term.2,12For each of the magnets in the device shown on Fig. 1A ˙/vector n=γ Ms/bracketleftbigg −δE δ/vector n×/vector n/bracketrightbigg +u[/vector n×[/vector s×/vector n]]+α[/vector n×˙/vector n] (1) where/vector s(t) is the unit vector along the instantaneous magnetization of the other magnet and the spin-transfer magnitude u=g(P)γ(¯h/2) VMsI e(2) is proportional to the electric current I. Hereeis the (negative) electroncharge, so uis positivewhen electrons flow into the magnet. Due to the inverse proportionality to the volume V, the larger magnets become less sensi- tive to the current and can serve as spin-polarizers with a fixed magnetization direction. As for the other pa- rameters, γis the gyromagnetic ratio, g(P,(/vector n·/vector s)) is the Slonczewski spin polarization factor2which depends on manysystemparameters,16,17andαis the Gilbert damp- ing which also depends on /vector nand/vector swhen spin pumping18 is taken into account. We will restrict our treatment to the constant gandαto focus on the effects specific to the strong easy plane anisotropy. In terms of the polar angles ( θ,φ) the LLG equation (1) has the form ˙θ+α˙φsinθ=−γ Msinθ∂E ∂φ+u(/vector s·/vector eθ) ˙φsinθ−α˙θ=γ M∂E ∂θ+u(/vector s·/vector eφ) (3) where the tangent unit vectors /vector eθand/vector eφare defined in Appendix A. We will consider a model for which the energy of a magnet is given by E=K⊥cos2θ 2+Er(φ) (4) withK⊥being the easy-plane constant, Erbeing the “residual”in-plane anisotropy energy and z-axis directed perpendicular to the easy plane. The limit of a strong easy-planeanisotropyisachievedwhenthe maximalvari- ation of the residual energy is small compared to the easy-plane energy, ∆ Er≪K⊥. In this case θ=π/2+δθ withδθ≪1.To estimate δθ, consider the motion of magnetization initially lying in-plane offthe minimum of Erand neglect for the moment the spin-transfer terms in Eq.(3). Mag- netization starts movingand a certain deviation from the easy plane is developed. For the estimate, assume that the energy is conserved during this motion (the presence of damping will only decrease δθ). Then |δθ| ∼/radicalbigg ∆Er K⊥≪1 (5) Wecannowlinearizetherighthandsidesofequations(3) in smallδθ. On top of that, some terms on the left hand sides of (3) turn out to be small and can be discarded. Indeed, taking into account the smallness of αone gets the estimates ˙θ∼ −γ Ms∂Er ∂φ∼ −γ Ms∆Er ˙φ∼γ MsK⊥δθ∼γ Ms/radicalbig K⊥∆Er Consequently ˙θ∼˙φ/radicalbig ∆Er/K⊥≪˙φand˙φ≫α˙θ, there- fore the second term on the left hand side of the second equation of the system (3) can be discarded. No simpli- fication happens on the left hand side of the first equa- tion, where ˙θandα˙φcan be of the same order when α<∼/radicalbig ∆Er/K⊥. Putting the spin-transfer terms back we get the form of equations in the limit of large easy-plane anisotropy: ˙δθ+α˙φ=−γ Ms∂E ∂φ+u(/vector s·/vector eθ) ˙φ=γK⊥ Msδθ+u(/vector s·/vector eφ) (6) Expressions for the scalar products in (6) in terms of polar angles are given in Appendix A. The second equation shows that δθcan be expressed through ( φ,˙φ). Small out-of-plane deviation becomes a “slave” of the in-plane motion.14We get Ms γK⊥/parenleftbigg ¨φ−ud(/vector s·/vector eφ) dt/parenrightbigg +αi˙φ=−γ Ms∂Er ∂φ+u(/vector s·/vector eθ) (7) The term with the second time derivative ¨φdecreases with increasing K⊥. As pointed out in Ref. 14, in the absence of spin-transfer this term can be neglected when K⊥>∆Er/α2. Mathematically this corresponds to a transition from an underdamped to an overdamped be- havior of an oscillator as the oscillator mass decreases. With spin-transfer terms the overdamped approxima- tion gives an equation α˙φ−ξd dt(/vector s·/vector eφ) =−γ Ms∂Er ∂φ+u(/vector s·/vector eθ) (8) whereξ=uMs/(γK⊥). The range of this equation’s validity will be discussed in Sec. IV. The scalar products in Eq. (8) have to be expressed through the polar angles3 (θs(t),φs(t)) ofvector /vector s, and linearizedwith respectto δθ (see Appendix, Eq. A4), which is then substituted from Eq. (6). Finally, the equation is linearized with respect to small spin-transfer magnitude u. We get: α˙φ−ξ/parenleftbiggd dt/bracketleftbig sinθssin(φs−φ)/bracketrightbig −sinθscos(φs−φ)˙φ/parenrightbigg =−γ Ms∂Er ∂φ−ucosθs, (9) describing the in-plane overdamped motion of an ana- lyzer with a polarizer pointed in the arbitrary direction. Next, we show how some known results on spin-transfer systems are recovered in the approximation (9). Consider the device shown on Fig. 1A and assume that the first magnet is very large. As explained above, this magnetisnotaffectedbythecurrentandservesasafixed source of spin-polarized electrons for the second magnet called the analyzer, or the “free” layer. The magneti- zation dynamics of the analyzer is described by Eq. (3). The case of static polarizer is extensively studied in the literature. First, consider the case of collinear switching , exper- imentally realized in a nano-pillar device with the ana- lyzer’s and polarizer’s easy axes along the ˆ xdirection: Er= (1/2)K||sin2φ,/vector s= (1,0,0).7Using Eq. (9) with θs=π/2,φs= 0 we get (α+2ξcosφ)˙φ=−γK|| 2Mssin2φ (10) Without the current, there are four possible equilibria of the analyzer. Two stable equilibria are the parallel (φ= 0) and anti-parallel ( φ=π) states. Two perpen- dicular equilibria ( φ=±π/2) are unstable. Lineariz- ing Eq. (10) near equilibria one finds solutions the form δφ(t)∼exp(ωt) with eigenfrequencies ω=−γK|| Ms(α+2ξ),(φ≈0) ω=−γK|| Ms(α−2ξ),(φ≈π) ω=γK|| Msα,(φ≈ ±π/2) The equilibria are stable for ω <0 and unstable other- wise. Thus the parallel state is stable for ξ >−α/2, the antiparallel state is stable for ξ < α/2, and the perpen- dicular states cannot be stabilized by the current. These conclusions agree with the results of Refs. 2,7,12. The stability regions are shown in Fig. 2A. Note how Eq. (10) emphasizes the fact that spin- transfer torque destabilizes the equilibria by making the effective damping constant αeff=α+2ξcosφnegative, whiletheequilibriumpointsremainaminimumoftheen- ergyEr. Any appreciable influence of the current on the position and nature (minimum or maximum) of the equi- librium can only be observed at the current magnitudes 1/αtimes larger than the actual switching current.12ξ α/2−α/2(A) static polarizer −α/[2(1−ε)](B) dynamic polarizer α/2 −α/(2ε)"windmill" precession"windmill" precession ξ FIG. 2: Stability regions for systems with static (A) and dynamic (B) polarizers as a function of applied current, ξ=g(P)(¯h/2VK⊥)I/e∝I. Second, consider the case of magnetic fan .19Here the easy axis of the polarizer is again directed along ˆ x, but the polarizer is perpendicular to the easy plane: /vector s= (0,0,1),θs= 0. This arrangement is known to produce a constant precession of vector /vector n. Eq. (9) gets a form: α˙φ=−γK|| 2Mssin2φ−u (11) for|u|< γK ||/(2Ms) the current deflects the analyzer direction from the easy axis direction. For larger values ofuthere is no time-independent solution. The angles φgrows with time which corresponds to /vector nmaking full rotations. At |u| ≫γK||/(2Ms) the rotation frequency of the magnetic fan is given by ω∼u/α. III. DEVICE WITH TWO DYNAMIC MAGNETS (TWO “FREE LAYERS”) No let us assume that both magnets in Fig. 1A have finitesize. Eachmagnetservesasapolarizerfortheother one. Without approximations, the evolution of two sets of polar angles ( θi,φi),i= 1,2 is described by two LLG systems of equations ˙θ(i)+αi˙φ(i)sinθ(i)=−γ Msisinθ(i)∂E(i) ∂φ(i)+ +uji(/vector n(j)·/vector e(i) θ) (12) ˙φ(i)sinθ(i)−αi˙θ(i)=γ Msi∂E(i) ∂θ(i)+uji(/vector n(j)·/vector e(i) φ) wherejmeanstheindexnotequalto iandnosummation is implied. We now apply the overdamped, large easy-plane anisotropyapproximationtobothmagnets. Equation(9)4 for each magnet is further simplified since for the magnet ithe angle θs=θj=π/2 +δθj,δθj≪1. Expanding (9) in small δθjand using the slave condition (6) for δθj with (/vector s·/vector eφ) = (/vector n(j)·/vector e(i) φ) expanded in both small angles (see Eq. (A5)) we get the system: (αi+2ξjicos(φj−φi))˙φi− (13) −ξji(cos(φj−φi)+1)˙φj=−∂E(i) ∂φi, withξji=ujiMsi/(γK⊥). It was assumed that K⊥is the same for both magnets. The spin-transfer torque parameters u21andu12have opposite signs and their absolute values are different due to different volumes of the magnets, accordingto Eq. (2). We assume V1≥V2and denote u12=u,u21=−ǫu. The larger magnet experiences a relatively smaller spin transfer effect, and the asymmetry parameter satisfies 0≤ǫ≤1. In general, material parameters α1,2,Ms1,2 and magnetic anisotropy energies E(1,2)of the two mag- nets are also different, but here we focus solely on the asymmetry in spin-transfer parameters. Both E(1)and E(2)are assumed to be given by formula (4) with the same direction of in-plane easy axis. The situation can be viewed as a collinear switching setup with dynamic polarizer. Equations (13) specialize to /vextendsingle/vextendsingle/vextendsingle/vextendsingleα−2ǫξC ǫξ(C+1) −ξ(C+1)α+2ξC/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg˙φ1 ˙φ2/bracketrightbigg =−ω0 2/bracketleftbigg sin2φ1 sin2φ2/bracketrightbigg C= cos(φ1−φ2), ω0=γK|| Ms(14) Next, we study the stability of all equilibrium configu- rations ( φ1,φ2) of two magnets. There are four equilib- rium states that are stable without the current: two par- allel states along the easy axis (0 ,0) and (π,π), two an- tiparallelstatesalongthe easyaxis(0 ,π)and(π,0). Four more equilibrium states have magnetization perpendicu- lar to the easy axis and are unstable without the current: (±π/2,±π/2). Once again, since spin-transfer does not depend on the relative direction of current and magneti- zation, the configurations which can be transformed into each other by a rotation of the magnetic space as a whole behaveidentically. Thusitisenoughtoconsiderfourcon- figurations: (0 ,0), (0,π), (π/2,π/2), and ( π/2,−π/2). We linearize equations (14) near each equilibrium and search for the solution in the form δφi∼exp(ωt). The eigenfrequencies are found to be: (0,0) :ω1=−ω0 α, ω2=−−ω0 α+2ξ(1−ǫ) (0,π) :ω1=−ω0 α+2ǫξ, ω2=−ω0 α−2ξ (π 2,π 2) :ω1=ω0 α, ω2=ω0 α−2ξ(1−ǫ) (π 2,−π 2) :ω1=ω0 α+2ǫξ, ω2=ω0 α−2ξThe state is stable when both eigenfrequencies are negative. We conclude that initially unstable states (π/2,±π/2) are never stabilized by the current, while the (0,0) and (0 ,π) state remain stable for (0,0) :ξ >−α 2(1−ǫ) (0,π) :−α 2ǫ< ξ <α 2 These regions of stability are shown schematically in Fig. 2B in comparison with the case of static magnetic polarizer (Fig. 2A) which is recovered at ǫ→0. As the size of the polarizer is reduced, the asymme- try parameter ǫgrows. The stability region of the an- tiparallel state acquires a lower boundary ξ=−α/(2ǫ). Up toǫ= 1/2, this boundary is still below the lower boundary of the parallel configuration stability region. Consequently, the parallel configuration is switched to the antiparallel at a negative current ξ=−α/(2(1−ǫ)). The system then remains in the antiparallel state down toξ=−α/(2ǫ). Below that threshold no stable configu- rations exist, and the system goes into some type of pre- cession state. This dynamic state is related to the “wind- mill” state predicted in Ref. 2 for two identical magnets in the absence of anisotropies. Obviously, here it is mod- ified by the strong easy-plane anisotropy. Theǫ= 1/2 value represents a transition point in the behavior of the system. For 1 /2< ǫ <1, the stability region of the parallel configuration completely covers the one of the antiparallel state. A transition without hys- teresis now happens at ξ=−α/(2(1−ǫ)) between the parallel state and the precession state. If the system is initially in the antiparallel state, it switches to the par- allel state either at a negative current ξ=−α/(2(1−ǫ)) or at a positive current ξ=α/2, and never returns to the antiparallel state after that. IV. CONCLUDING REMARKS We studied thebehaviorofplanarspin-transferdevices with magnetic energy dominated by the large easy-plane anisotropy. The overdamped approximation in the pres- ence of current-induced torque was derived and checked against the cases already discussed in the literature. In the new “dynamic polarizer” case, we found a transition between two regimes with different switching sequences. The large asymmetry regime is similar to the case of static polarizer and shows hysteretic switching between the parallel and antiparallel configurations, while in the small asymmetry regime the magnets do not switch, but go directly into the “windmill” precession state. We saw that the current-induced switching occurs when the effective damping constant vanishes near a par- ticularequilibrium. Thismakestheoverdampedapproxi- mationinapplicableinthe immediatevicinityofthetran- sition and renders Eqs. (14) ill-defined at some points. However, the overall conclusions about the switching5 events will remain the same as long as the interval of inapplicability is small enough. We also find that the overdamped planar approxima- tion does not work well when a saddle point of magnetic energy is stabilized by spin-transfer torque, e.g. during the operation of a spin-flip transistor.20Description of such cases in terms of effective planar equations requires additional investigations. V. ACKNOWLEDGEMENTS We wish to thank Tom Silva, Oleg Tchernyshyov,Oleg Tretiakov, and G. E. W. Bauer for illuminating discus- sions. This work was supported in part by DMEA con- tract No. H94003-04-2-0404, Ya. B. is grateful to KITP Santa Barbara for hospitality and support under NSF grant No. PHY99-07949. D. O. was supported in part by the IBM undergraduate student internship program. APPENDIX A: VECTOR DEFINITIONS z eφ θen φθ xFIG. 3: Definitions of the tangent vectors and polar angles. We use the standard definitions of polar coordinates and tangent vectors (see Fig. 3): /vector n= (sinθcosφ,sinθsinφ,cosθ) /vector eθ= (cosθcosφ,cosθsinφ,−sinθ) (A1) /vector eφ= (−sinφ,cosφ,0) Whenθ=π/2+δθa linearization in δθgives /vector n≈(cosφ,sinφ,−δθ) /vector eθ≈ −(δθcosφ,δθsinφ,1) (A2) /vector eφ≈(−sinφ,cosφ,0) For two unit vectors /vector n(i),i= 1,2 with polar angles (θi,φi) the scalar product expressions are (/vector n(j)·/vector e(i) θ) = sin θjcosθicos(φj−φi)−cosθjsinθi (/vector n(j)·/vector e(i) φ) = sin θjsin(φj−φi) (A3) Linearizing (A3) with respect to small δθifor arbitrary values of θjone gets: (/vector n(j)·/vector e(i) θ)≈ −sinθjδθicos(φj−φi)−cosθj (/vector n(j)·/vector e(i) φ)≈sinθjsin(φj−φi) (A4) Linearization of (A3) with respect to both δθiandδθj gives (/vector n(j)·/vector e(i) θ)≈ −δθicos(φj−φi)+δθj (/vector n(j)·/vector e(i) φ)≈sin(φj−φi) (A5) 1L. Berger, J. Appl. Phys., 49, 2160 (1978); Phys. Rev. B 33, 1572 (1986); J.Appl.Phys. 63, 1663 (1988). 2J.Slonczewski, J.Magn.Magn.Mater. 159, L1 (1996). 3L. Berger, J. Appl. Phys., 71, 2721 (1992) 4M. Tsoi, A.G.M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 (1998). 5J. Z. Sun, J. Magn. Magn. Mater., 202, 157, (1999). 6J.E. Wegrowe, D. Kelly, Y. Jaccard, Ph. Guittienne, and J.-Ph. Ansermet, Europhys.Lett. 45, 626 (1999). 7E.B. Myers, D.C. Ralph, J.A. Katine, R.N. Louie, R.A. Buhrman, Science 285, 867 (1999). 8J.A. Katine, F.J. Albert, R.A. Buhrman, E.B. Myers, and D.C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). 9E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Na- ture,432, 203 (2004). 10M. Klaui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland, G. Faini, U. Rudiger, C. A. F. Vaz, L. Vila, and C. Vouille, Phys. Rev. Lett., 95, 026601 (2005). 11M. Hayashi, L. Thomas, Ya. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett., 96, 197207 (2006).12Ya. B. Bazaliy, B. A. Jones, and Shou-Cheng Zhang, Phys. Rev. B,69, 094421 (2004). 13A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep., 427, 157 (2006). 14C. J. Garica-Cervera, Weinan E, J. Appl. Phys., 90, 370 (2001). 15S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em- ley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph , Nature,425, 380 (2003). 16X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph, Phys. Rev. B, 62, 12317 (2000). 17Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 18Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, Phys. Rev. Lett.88, 117601 (2002). 19X. Wang, G. E. W. Bauer, and A. Hoffmann, Phys. Rev. B,73, 054436 (2006), and references therein. 20A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); X. Wang, G. E. W. Bauer, and T. Ono, Japan. J. Appl. Phys., 45, 3863 (2006).

2007-05-03

In planar nano-magnetic devices magnetization direction is kept close to a given plane by the large easy-plane magnetic anisotropy, for example by the shape anisotropy in a thin film. In this case magnetization shows effectively in-plane dynamics with only one angle required for its description. Moreover, the motion can become overdamped even for small values of Gilbert damping. We derive the equations of effective in-plane dynamics in the presence of spin-transfer torques. The simplifications achieved in the overdamped regime allow to study systems with several dynamic magnetic pieces (``free layers''). A transition from a spin-transfer device with a static polarizer to a device with two equivalent magnets is observed. When the size difference between the magnets is less than critical, the device does not exhibit switching, but goes directly into the ``windmill'' precession state.

Planar spin-transfer device with a dynamic polarizer

0705.0406v1

arXiv:0705.0508v1 [cond-mat.mtrl-sci] 3 May 2007Effective attraction induced by repulsive interaction in a s pin-transfer system Ya. B. Bazaliy Instituut Lorentz, Leiden University, The Netherlands, Department of Physics and Astronomy, University of South Ca rolina, Columbia, SC, and Institute of Magnetism, National Academy of Science, Ukrai ne. (Dated: April, 2006) In magnetic systems with dominating easy-plane anisotropy the magnetization can be described by an effective one dimensional equation for the in-plane ang le. Re-deriving this equation in the presence of spin-transfer torques, we obtain a description that allows for a more intuitive under- standing of spintronic devices’ operation and can serve as a tool for finding new dynamic regimes. A surprising prediction is obtained for a planar “spin-flip t ransistor”: an unstable equilibrium point can be stabilized by a current induced torque that further re pels the system from that point. Stabi- lization by repulsion happens due to the presence of dissipa tive environment and requires a Gilbert damping constant that is large enough to ensure overdamped d ynamics at zero current. PACS numbers: 72.25.Pn, 72.25.Mk, 85.75.-d In physics, there are cases where due to the presence of complex environment a repulsive force can lead to ac- tual attraction of the entities. A well known example is a superconductor, where the Cooper pairs are formed from electrons repelled by the Coulomb forces due to the dynamical elastic environment. Here we report a phe- nomena of effective attraction induced by the repulsive spin-transfer torque in the presence of highly dissipative environment. The spin-transfer effect producing the re- pulsivetorqueis a non-equilibriuminteractionthat arises when a current of electrons flows through a non-collinear magnetic texture [1, 2, 3]. This interaction can become significant in nanoscopic magnets and is nowadays stud- ied experimentally in a variety of systems. Its manifesta- tions - either current induced magnetic switching [4] or magnetic domain wall motion [5] - serve as an underlying mechanism for a number of suggested memory and logic applications. Here we consider a conventional spin-transfer device consisting of a a magnetic polarizer (fixed layer) and a small magnet (free layer) with electric current flowing from one to another (Fig 1). Both layers can be de- scribed by a macro-spin model due to large exchange stiffness. The free layer is influenced by the spin transfer torque,whilethepolarizeristoolargetofeelit. Magnetic dynamics of the free layer is described by the Landau- Lifshitz-Gilbert (LLG) equation with the spin transfer torque term [2, 6]. The solutions of LLG are easy to find for the simplest easy axis magnetic anisotropy of the free layer. There exists a critical current at which the free layer either switches between the two minima of magnetic energy, or goes into a state of permanent precession, powered by the current source [2, 6, 7]. The same basic processes happen in the case of realistic anisotropies, however the complexity of the calculations increases substantially. In a nanopillar device [8] one additionally finds that stabi- lization of magnetic energy maxima is possible (“cantedstates”[6]) andthat multiple precessionmodesexist with transitions between them happening as the current is in- creased [7, 9, 10]. The anisotropy of a nanopillar device is a combination of a magnetic easy plane and magnetic easy axis directed in that plane. Experimentally, the easy plane anisotropy energy is usually much larger than the easy axis energy, i.e. the system is in the regime of a planar spintronic device [11] (Fig. 1). This limit of dominatingeasyplane energyis characterizedby another simplification of the dynamic equations [12, 13], which comes not from the high symmetry of the problem, but from the existence of a small parameter: the ratio of the energy modulation within the plane to the easy plane en- ergy. The deviation of the magnetization from the plane becomes small, making the motion effectively one dimen- sional. In this paper we present a general form of effective planar equation describing a macrospin free layer in the presence of spin transfer torques. Its relationship to the first order expansion in the current magnitude used in Ref. 13 is discussed at the end. We then use this equa- tion tostudy the “spin-fliptransistor”: aplanardevice in which the spin polarizer is perpendicular to the direction favored by the magnetic anisotropy energy. It was pre- dicted [14] that the competition between the anisotropy and spin transfer torques leads to a 90 degrees jump of s A B jnjn s FIG. 1: Planar spin-transfer devices. Hashed parts of the devices are ferromagnetic, white parts are made from a non- magnetic metal.2 the magnetization at the critical current. Whether the jump happens into the parallel or antiparallel state with respect to the polarizer is determined by the direction of the current. Here it is shown that the behavior of the spin-flip tran- sistor is more complicated than expected from the simple picture above. Namely, the current inducing a jump into the parallel direction can also stabilize the antiparallel direction. This conclusion is certainly counter-intuitive because the spin torque repels the magnetization from this already unstable saddle point of the energy. How- ever, a combination of two destabilizing torques manages to result in a stable equilibrium. We will see that this happens due to the dissipation terms and a sufficiently large(but still smallcomparedto unity) Gilbert damping constant is required to observe the phenomena. The magnetization of the free layer M=Mnhas a constant absolute value Mand a direction given by a unit vector n(t). The LLG equation [2, 6] reads: ˙n=γ M/bracketleftbigg −δE δn×n/bracketrightbigg +u(n)[n×[s×n]]+α[n×˙n].(1) Hereγis the gyromagnetic ratio, E(n) is the magnetic energy of the free layer, and αis the Gilbert damping constant. The second term on the right is the spin trans- fer torque, where sis a unit vector along the direction of the polarizer, and the spin transfer strength u(n) is proportional to the electric current I[6, 13]. In general, spin transfer strength is a function of the angle between the polarizer and the free layer u(n) =f[(n·s)]I, with the function f[(n·s)] being material and device specific. Equation (1) can be written in polar angles ( θ(t),φ(t)): ˙θ+α˙φsinθ=−γ Msinθ∂E ∂φ+u(s·eθ)≡Fθ, ˙φsinθ−α˙θ=γ M∂E ∂θ+u(s·eφ)≡Fφ, (2) with tangent vectors eφ= [ˆz×n]/sinθ,eθ= [eφ×n]. The easy plane is chosen at θ=π/2, and the mag- netic energy has the form E= (K⊥/2)cos2θ+Er(θ,φ), whereEris the “residual” energy. In the planar limit, K⊥→ ∞, the energy minima are very close to the easy plane and the low energysolutionsofLLG havethe prop- ertyθ(t) =π/2 +δθwithδθ→0. Equations (2) can then be expanded in small parameters |Er|/K⊥≪1, |u(n)|/K⊥≪1. Assuming time-independent uandswe obtain an effective equation of the in-plane motion 1 ω⊥¨φ+αeff˙φ=−γ M∂Eeff ∂φ, (3) which has has the form of the Newton’s equation of mo- tion for a particle in external potential Eeff(φ) with a variable viscous friction coefficient αeff(φ). The expres- sions for the effective friction and energy are αeff(φ) =α−(Γφ+Γθ)/ω⊥, (4) Γφ= (∂Fφ/∂φ)θ=π/2,Γθ= (∂Fθ/∂θ)θ=π/2,and Eeff(φ) =Er(π/2,φ)+∆E(φ), (5) ∆E=−M γ/integraldisplayφ/bracketleftbigg u(n)(s·eθ)−Γθ ω⊥Fφ/bracketrightbigg θ=π 2dφ′. Equation (3) with definitions (4,5) gives a general de- scription of a planar device in the presence of spin trans- fer torque. At non-zero current the effective friction can become negative (see below), and the effective energy is not necessarily periodic in φ(e.g. in the case of “mag- netic fan”[13, 15]). Physicallythis reflectsthe possibility of extracting energy from the current source, and thus developing a “negative dissipation” in the system. In many planar devices the polarizer direction slies in the easy plane, θs=π/2, with a direction defined by the azimuthal angle φs. At the same time the resid- ual energy has a property ( ∂Er/∂θ)θ=π/2= 0, i.e. does not shift the energy minima away from the plane. We will also use the simplest form f[(n·s)] = const for the spin transfer strength. A more realistic function will not change the result qualitatively and can be easily used if needed. With these restrictions the effective friction and the energy correction get the form: αeff=α+2ucos(φs−φ) ω⊥(6) ∆E=−Mu2 2γω⊥cos2(φs−φ). In a spin-flip transistor the polarizer direction is given byφs=π/2. Following Ref. 14, we consider in-plane anisotropy energy Er(π/2,φ) =−(K||/2)cos2φcorre- sponding to an easy axis. Then the effective friction isαeff=α+ (2usinφ)/ω⊥and effective energy equals (γ/M)Eeff=−[(ω||−u2/ω⊥)/2]cos2φ+ const with ω||=γK||/M. Equilibrium points φ= 0,±π/2,πare the minima and maxima of the effective energy, and do not depend on u. Stability of any equilibrium in one di- mension depends on whether it is a minimum or a maxi- mum ofEeffand on the sign of αeffat the equilibrium point. It is easy to check, that out of four possibilities only an energy minimum with αeff>0 is stable. In the caseofaspin-fliptransistortheenergylandscapechanges above a threshold |u|>√ω||ω⊥: the energy minima at φ= 0,πbecome maxima, and, vice versa, the energy maxima at φ=±π/2 switch to minima. Effective fric- tion atφ= 0,πis positive independent of u, while at φ=±π/2 it changes sign at u=∓αω⊥/2. The behavior of the spin-flip transistor is summarized in a switching diagram Fig. 2 plotted on the plane of the material characteristic αand the experimental parame- teru∼I. For definiteness we will discuss a current with u >0. The effect of the opposite current is completely symmetric. For small values of Gilbert damping one ob- serves stabilization of the φ=π/2 (parallel) equilibrium3 +π/2 −π/20 πα*u ααEeff 0+π/2 π −π −π/2effαEeff 0+π/2 π −π −π/2eff αEeff 0+π/2 π −π −π/2eff abcd FIG. 2: Switching diagram of the spin-flip transistor. In eac h zone one or two arrows show the possible stable directions of the free layer magnetization. Directions of the easy axis an d spin polarizer are defined in the right bottom corner. Angula r dependencies of αeffandEeffare givenin insets. Stable sub- regions “b” and “c” differ in overdamped vs. underdamped approach to the equilibrium. to which the spin torque attracts the magnetization of the free layer, while the opposite (antiparallel) direction remains unstable. This is in accord with the results of Ref. 14. However, when the damping constant is larger than the critical value α∗= 2/radicalbig ω||/ω⊥, a window of sta- bility of the antiparallel equilibrium opens on the dia- gram. Since α≪1, a sufficiently large easy plane energy is required to achieve α∗< α≪1. If one thinks about the stability of the ( θ,φ) = (π/2,−π/2)equilibriumfor u >0intermsofEq.(1), this prediction seems completely unexpected. The anisotropy torques do not stabilize this equilibrium because it is a saddle point of the total magnetic energy E, and the added spin transfer torque repels nfrom this point as well. The whole phenomena may be called “stabilization by repulsion”. To check the accuracy of the planar ap- proximation (3), the result was verified using the LLG equations (2) with no approximations for the axis-and- plane energy E= (K⊥/2)cos2θ−(K||/2)sin2θcos2φ. Calculatingtheeigenvaluesofthelinearizeddynamicma- trices [6] at the equilibrium points ( π/2,±π/2) we ob- tained the same switching diagram and confirmed the stabilization of the antiparallel direction. Typical trajec- toriesn(t) numerically calculated from the LLG equa- tion with no approximations are shown in Fig. 3 to illus-−0.5π −0.5π −0.7π −0.3π −0.7π −0.3π0.54π0.46π 0.5π 0.54π0.46π 0.5π FIG. 3: Typical trajectories of n(t) forω||/ω⊥= 0.01,α= 1.5α∗. The plot labels correspond to the regions in Fig. 2, the current magnitude is given in the units of u/p ω||/ω⊥and we look at the stability of the φ=−π/2 equilibrium: (a) 0.93, unstable (b): 1.08, stabilized with overdamped approach (c ): 1.38, stable, butwith oscillatory approach (d): 1.53, unst able; a stable cycle is formed around the equilibrium. trate the predictions. At u >√ω||ω⊥theφ=−π/2 equilibrium is stabilized. In accord with the predic- tions of Eqs. (3),(6), the wedge of its stability consists of two regions (b) and (c) characterized by overdamped and underdamped dynamics during the approach to the equilibrium. The dividing dashed line is given by u= ω||/α+αω⊥/4. It was checked that small deviations of the polarizer sfrom the ( π/2,π/2) direction do not change the behavior qualitatively. Larger deviations eventually destroy the effect, especially the out-of-plane deviation which produces the “magnetic fan” effect [15] leading to the full-circle rotation of φin the plane. As the current is further increased to u > αω ⊥/2, the antiparallel state looses stability and the trajectory ap- proaches a stable precession cycle (Fig. 3(d)). The exis- tence of the precession state is easy to understand from (3) viewedas anequation fora particlein externalpoten- tial. Just above the stability boundary the effective fric- tionαeff(φ) is negative in a small vicinity of φ=−π/2, and positive elsewhere. Within the αeff<0 region the dissipation is negative and any small deviation from the equilibrium initiates growing oscillations. As their am- plitude exceeds the size of that region, part of the cycle starts to happen with positive dissipation. Eventually the amplitude reaches a value at which the energy gain during the motion in the αeff<0 region is exactly com- pensated by the energy loss in the αeff>0 region: thus a cycle solutionemerges. The effective planardescription allows for the analysis of the further evolution of the cy- clewithtransitionsintodifferentprecessionmodes,which will be a subject of another publication.4 The fact that α > α ∗condition is required for the stabilization means that dissipation terms play a crucial role entangling two types of repulsion to produce a net attraction to the reversed direction. Note that an in- terplay of a strong easy plane anisotropy and dissipa- tion terms produces unexpected effects already in con- ventional ( u= 0) magnetic systems. The effective planar equation (3) at u= 0 was discussed in Ref. 12. It was found that the same threshold α∗represents a bound- ary between the oscillatory and overdamped approaches the equilibrium. Above α∗the familiar precession of a magnetic moment in the anisotropy field is replaced by the dissipative motion directed towards the energy mini- mum. When the easy plane anisotropy is strong enough to ensure α≫α∗, one can drop the second order time derivative term in Eq. (3) and use the resulting first or- der dissipative equation. In the presence of spin transfer, αeff(φ,u) depends on the current and can assume small values even for α≫α∗, thus no general statement about the¨φterm can be made. The simplest easy axis energy expression Er(π/2,φ) = −(K||/2)cos2φhappens to have the same angular de- pendence as ∆ E(φ) given by Eq. (6). Due to this spe- cial property the energy profile flips upside down at u=√ω||ω⊥. For a generic Er(π/2,φ) with minima at φ= 0,πand maxima at φ=±π/2 the nature of equilib- ria will change at different current thresholds. This will make the switching diagram more complicated, but will notaffectthestabilizationbyrepulsionphenomena. Sim- ilarcomplicationswillbe introducedbyageneric f[(n·s)] angular dependence of the spin transfer strength. In Ref. 13 the known switching diagram for the collinear ( φs= 0) devices [6, 9, 10] were reproduced by equation (3) with Eeff=Er(π/2,φ). The ∆ E term (6) was dropped as being second order in small u. This approximation gives a correct result for the following reason. In a collinear device ( γ/M)Eeff= −[(ω||+u2/ω⊥)/2]cos2φ+const and the current never changes the nature of the equilibrium from a maximum to a minimum. Consequently, dropping ∆ Edoes not af- fect the results. As was already noted in Ref. 13, the first order expansion in uis insufficient for the description of a spin-flip transistor, where the full form (6) is required. In summary, we derived a general form of the effec- tive planar equation (3) for a macrospin free layer in the presence of spin transfer torque produced by a fixed spin-polarizerandtime-independent current. Qualitative understanding of the solutions of planar equation is ob- tained by employing the analogy with a one-dimensional mechanical motion of a particle with variable friction co- efficient in an external potential. The resulting predic- tive power is illustrated by the discovery of the stabi- lization by repulsion phenomena in the spin-flip device. Such stabilization relies on the form of the dissipative torquesin the LLG equationand happens onlyfora large enough Gilbert damping constant. The new stable stateand the corresponding precession cycle can be used to engineer novel memory or logic devices, and microwave nano-generators with tunable frequency. To observe the phenomena experimentally, one has to fabricate a device with α > α∗, and initially set it into a parallel or antiparallel state by external magnetic field. Thenthecurrentisturnedonandthefieldisswitchedoff. Both states should be stabilized by a moderate current√ω||ω⊥< u < αω ⊥/2, but cannot yet be distinguished by their magnetoresistive signals. The difference can be observed as the current is increased above the αω⊥/2 threshold: the parallel state will remain a stable equilib- rium, while the antiparallel state will transform into a precession cycle and an oscillating component of magne- toresistance will appear. The author wishes to thank C. W. J. Beenakker, G. E. W. Bauer, and Yu. V. Nazarov for illuminating dis- cussions. Research at Leiden University was supported by the Dutch Science Foundation NWO/FOM. Part of this work was performed at KITP Santa Barbara sup- ported by the NSF grant No. PHY99-07949, and at As- pen PhysicsInstitute duringthe Winterprogramof2007. [1] L. Berger, J. Appl. Phys., 49, 2160 (1978); Phys. Rev. B 33, 1572 (1986); J. Appl. Phys. 63, 1663 (1988). [2] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [3] Ya. B. Bazaliy et al., Phys. Rev. B, 57, R3213 (1998). [4] S. Kaka et al., Nature 437, 389 (2005); M. R. Pufall et al., Phys.Rev. Lett. 97, 087206 (2006); M. L. Schneider et al., Appl. Phys. Lett., 90, 092504 (2007); X. Jiang et al., Phys. Rev. Lett. 97, 217202 (2006); W. Chen et al., Phys. Rev. B, 74, 144408(2006); B. Ozyilmaz et al., Phys. Rev. Lett., 93, 176604 (2004); I. N. Krivorotov et al.Science, 307, 228 (2005); N. C. Emley et al.Phys. Rev. Lett., 96, 247204 (2006); J. C. Sankey, et al., Phys. Rev. Lett., 96, 227601 (2006). [5] G. Beach et al., Phys. Rev. Lett., 97, 057203 (2006); Na- ture Materials, 4, 741 (2005); M. Klaui et al., Phys. Rev. Lett.,95, 026601 (2005); M. Laufenberg et al., Phys. Rev. Lett., 97, 046602 (2006); L. Thomas et al., Science, 315, 1553 (2007); M. Hayashi et al., Phys. Rev.Lett., 98, 037204 (2007); Nature Physics, 3, 21 (2007); Phys. Rev. Lett.,97, 207205 (2006); M. Yamanouchi et al.Nature, 428, 539 (2004); Phys. Rev. Lett., 96, 096601 (2006). [6] Ya. B. Bazaliy et al., Phys. Rev. B, 69, 094421 (2004). [7] J. Z. Sun, Phys. Rev. B 62, 570 (2000). [8] J. A. Katine et al., Phys. Rev. Lett., 84, 3149 (2000). [9] S. I. Kiselev et al., Nature, 425, 380 (2003). [10] J. Xiao et al., Phys. Rev. B, 72, 014446 (2005) [11] A. Brataas et al., Phys. Rep., 427, 157 (2006). [12] C. Garcia-Cervera et al., J. Appl. Phys., 90, 370 (2001). [13] Ya. B. Bazaliy et al., arXiv:0705.0406v1 (2007), to be published in J. Nanoscience and Nanotechnology. [14] A. Brataas et al., Phys. Rev. Lett. 84, 2481 (2000); X. Wang et al., Japan. J. Appl. Phys., 45, 3863 (2006). [15] X. Wang et al., Phys. Rev. B, 73, 054436 (2006).

2007-05-03

In magnetic systems with dominating easy-plane anisotropy the magnetization can be described by an effective one dimensional equation for the in-plane angle. Re-deriving this equation in the presence of spin-transfer torques, we obtain a description that allows for a more intuitive understanding of spintronic devices' operation and can serve as a tool for finding new dynamic regimes. A surprising prediction is obtained for a planar ``spin-flip transistor'': an unstable equilibrium point can be stabilized by a current induced torque that further repels the system from that point. Stabilization by repulsion happens due to the presence of dissipative environment and requires a Gilbert damping constant that is large enough to ensure overdamped dynamics at zero current.

Effective attraction induced by repulsive interaction in a spin-transfer system

0705.0508v1

arXiv:0705.1990v1 [cond-mat.str-el] 14 May 2007Identification of the dominant precession damping mechanis m in Fe, Co, and Ni by first-principles calculations K. Gilmore1,2, Y.U. Idzerda2, and M.D. Stiles1 1National Institute of Standards and Technology, Gaithersb urg, MD 20899-8412 2Physics Department, Montana State University, Bozeman, MT 59717 (Dated: October 23, 2018) The Landau-Lifshitz equation reliably describes magnetiz ation dynamics using a phenomenolog- ical treatment of damping. This paper presents first-princi ples calculations of the damping param- eters for Fe, Co, and Ni that quantitatively agree with exist ing ferromagnetic resonance measure- ments. This agreement establishes the dominant damping mec hanism for these systems and takes a significant step toward predicting and tailoring the dampi ng constants of new materials. Magnetic damping determines the performance of magnetic devices including hard drives, magnetic ran- dom access memories, magnetic logic devices, and mag- netic field sensors. The behavior of these devices can be modeled using the Landau-Lifshitz (LL) equation [1] ˙m=−|γ|m×Heff−λ m2m×(m×Heff),(1) or the essentially equivalent Gilbert (LLG) form [2, 3]. The first term describes precession of the magnetization mabouttheeffectivefield Heffwhereγ=gµ0µB/¯histhe gyromagnetic ratio. The second term is a phenomeno- logical treatment of damping with the adjustable rate λ. TheLL(G) equationadequatelydescribesdynamicsmea- sured by techniques as varied as ferromagnetic resonance (FMR) [4], magneto-optical Kerr effect [5], x-ray absorp- tion spectroscopy [6], and spin-current driven rotation with the addition of a spin-torque term [7, 8]. Access to a range of damping rates in metallic mate- rials is desirable when constructing devices for different applications. Ideally, one would like the ability to de- sign materials with any desired damping rate. Empiri- cally, dopingNiFe alloyswith transitionmetals[9]orrare earths [10] has produced compounds with damping rates in the range of α= 0.01 to 0.8. A recent investigation of adding vanadium to iron resulted in an alloy with a damping rate slightly lower than that for pure iron [11], the systemwiththe lowestpreviouslyknownvalue. How- ever, the damping rate of a new material cannot be pre- dicted because there has not yet been a first-principles calculation of damping that quantitatively agrees with experiment. The challenging pursuit of new materials with specific or lowered damping rates is further com- plicated by the expectation that, as device size contin- ues to be scaled down, material parameters, such as λ, should change [12]. A detailed understanding of the im- portant damping mechanisms in metallic ferromagnets and the ability to predictively calculate damping rates would greatly facilitate the design of new materials ap- propriate for a variety of applications. The temperature dependence of damping in the tran- sition metals has been carefully characterized through measurement of small angle dynamics by FMR [13].While one might na¨ ıvely expect damping to increase monotonically with temperature, as it does for Fe, both Co and Ni also exhibit a dramatic rise in damping at low temperature as the temperature decreases. These ob- servations indicate that two primary mechanisms are in- volved. Subsequent experiments [14, 15] partition these non-monotonic damping curves into a conductivity-like term that decreases with temperature and a resistivity- liketermthatincreaseswithtemperature. Thetwoterms were found to give nearly equal weight to the damping curve of Ni and have temperature dependencies similar to those of the conductivity and resistivity, suggesting two distinct roles for electron-lattice scattering. The torque-correlation model of Kambersky [16] ap- pears to qualitatively match the data. However, like most of the various models presented by Kambersky [16, 17, 18, 19] and others [20], it has not been quan- titatively evaluated in a rigorous fashion. This has left the community to speculate, based on rough estimates or less, astowhichdampingmechanismsareimportant. We resolve this matter in the present work by reporting first- principles calculations of the Landau-Lifshitz damping constant according to Kambersky’s torque-correlation expression. Quantitative comparison of the present cal- culations to the measured FMR values [13] positively identifies this damping pathway as the dominant effect in the transition metal systems. In addition to present- ing these primary conclusions, we also describe the re- lationship between the torque-correlation model and the more widely understood breathing Fermi surface model [18, 21], showing that the results of both models agree quantitatively in the low scattering rate limit. The breathing Fermi surface model of Kambersky pre- dicts λ=g2µ2 B ¯h/summationdisplay n/integraldisplaydk3 (2π)3η(ǫn,k)/parenleftbigg∂ǫn,k ∂θ/parenrightbigg2τ ¯h.(2) This model offers a qualitative explanation for the low temperature conductivity-like contribution to the mea- sureddamping. The modeldescribesdamping ofuniform precession as due to variations ∂ǫn,k/∂θin the energies ǫn,kof the single-particle states with respect to the spin2 direction θ. The states are labeled with a wavevector kand band index n. As the magnetization precesses, thespin-orbitinteractionchangestheenergyofelectronic statespushingsomeoccupiedstatesabovetheFermilevel and some unoccupied states below the Fermi level. Thus, electron-hole pairs are generated near the Fermi level even in the absence of changes in the electronic popula- tions. The ηfunction in Eq. (2) is the negative derivative of the Fermi function and picks out only states near the Fermi level to contribute to the damping. gis the Land´ e g-factor and µBis the Bohr magneton. The electron-hole pairs created by the precession exist for some lifetime τ beforerelaxingthroughlattice scattering. The amountof energy and angular momentum dissipated to the lattice depends on how far from equilibrium the system gets, thus damping by this mechanism increases linearly with the electron lifetime as seen in Eq.2. Since the electron lifetime is expected to decrease as the temperature in- creases, this model predicts that damping diminishes as the temperature is raised. Because the predicted damping rate is linear in the scattering time the damping rate cannot be calculated more accurately than the scattering time is known. For this reason it is not possible to make quantitative com- parisonsbetween calculationsof the breathing Fermi sur- face and measurements. Further, while the breathing Fermi surface model can explain the dramatic temper- ature dependence observed in the conductivity-like por- tion of the data it fails to capture the physics driving the resistivity-like term. This is a significant limitation from a practical perspective because the resistivity-like term dominates damping at room temperature and above and is the only contribution observed in iron [13] and NiFe alloys [22]. For these reasons it is necessary to turn to more complete models of damping. Kambersky’s torque-correlation model predicts λ=g2µ2 B ¯h/summationdisplay n,m/integraldisplaydk3 (2π)3/vextendsingle/vextendsingleΓ− nm(k)/vextendsingle/vextendsingle2Wnm(k) (3) and we will show that it both incorporates the physics of the breathing Fermi surface model and also accounts for theresistivity-like terms. The matrix elements Γ− nm(k) = /angbracketleftn,k|[σ−, Hso]|m,k/angbracketrightmeasure transitions between states in bands nandminduced by the spin-orbit torque. These transitionsconservewavevector kbecausethey de- scribe the annihilation of a uniform precession magnon, which carries no linear momentum. The nature of these scattering events, which are weighted by the spectral overlapWnm(k) = (1/π)/integraltext dω1η(ω1)Ank(ω1)Amk(ω1), will be discussedin moredetail below. The electronspec- tral functions Ankare Lorentzians centered around the band energies ǫnkand broadenedbyinteractionswith the lattice. The width ofthe spectralfunction ¯ h/τprovidesa phenomenological account for the role of electron-lattice scattering in the damping process. The ηfunction is theh/τ(eV) 108109λ (1/s) 0.001α0.001 0.01 0.1 1 Fe 1081091010λ (1/s) 101310141015 1/τ (1/s)0.0010.010.1αNi108109λ (1/s) 0.0010.01αCo FIG. 1: Calculated Landau-Lifshitz damping constant for Fe , Co, and Ni. Thick solid curves give the total damping param- eter while dotted curves give the intraband and dashed lines the interband contributions. The top axis is the full-width - half-maximum of the electron spectral functions. same as in Eq. (2) and enforces the requirement of spec- tral overlap at the Fermi level. Equation (3) captures two different types of scatter- ing events: scattering within a single band, m=n, for which the initial and final states are the same, and scat- tering between two different bands, m/negationslash=n. As explained in [16] the overlap of the spectral functions is propor- tional (inverse) to the electron scattering time for intra- band (interband) scattering. From this observation the qualitative conclusion is made that the intraband contri- butions matchthe conductivity-like terms while the inter- band contributions give the resistivity-like terms. While this seems promising, evaluation of Eq. (3) is more com- putationally intensive than that of the breathing Fermi surface model and until now only a few estimates for Ni and Fe have been made [19].3 TABLE I: Calculated and measured [13] damping parameters. V alues for λ, the Landau-Lifshitz form, are reported in 109s−1, values of α, the Gilbert form, are dimensionless. The last two columns l ist calculated damping due to the intraband contribution from Eq. (3) and from the breathing Fermi surface model [12], respectively. Values for λ/τare given in 1022s−2. Published numbers from [13] and [12] have been multiplied by 4 πto convert from the cgs unit system to SI. αcalcλcalcλmeasλcalc/λmeas(λ/τ)intra(λ/τ)BFS bcc Fe/angbracketleft001/angbracketright0.0013 0.54 0.88 0.61 1.01 0.968 bcc Fe/angbracketleft111/angbracketright0.0013 0.54 – – 1.35 1.29 hcp Co/angbracketleft0001/angbracketright0.0011 0.37 0.9 0.41 0.786 0.704 fcc Ni/angbracketleft111/angbracketright0.017 2.1 2.9 0.72 6.67 6.66 fcc Ni/angbracketleft001/angbracketright0.018 2.2 – – 8.61 8.42 We have performed first-principles calculations of the torque-correlation model Eq. (3) with realistic band structures for Fe, Co, and Ni. Prior to evaluating Eq. (3) the eigenstates and energies of each metal were found us- ing the linear augmented plane wave method [23] in the local spin density approximation (LSDA) [24, 25, 26]. Details of the calculations for these materials are de- scribed in [27]. The exchange field was fixed in the cho- sen equilibrium magnetization direction. Calculations of Eq. (3) presented in this paper are converged to within a standard deviation of 3 %, which required sampling (160)3k-points for Fe, (120)3for Ni, and (100)2k-points inthe basalplaneby57alongthe c-axisforCo. Electron- lattice interactions were treated phenomenologically as a broadeningofthe spectralfunctions. The Fermi distribu- tion was smeared with an artificial temperature. Results did not vary significantly with reasonable choices of this temperature since the broadening of the Fermi distribu- tion was considerably less than that of the bands. The damping rate was calculated for a range of scattering rates (spectral widths) just as damping has been mea- sured over a range of temperatures. The results ofthese calculations arepresented in Fig. 1 and are decomposed into the intraband and interband terms. The downward sloping line in Fig. 1 represents the intraband contribution to damping. Damping con- stants were recently calculated using the breathing Fermi surface model [12, 21] by evaluating the derivative of the electronic energy with respect to the spin direc- tion according to Eq. (2). The results of the breathing Fermi surface prediction are indistinguishable from the intraband terms of the present calculation even though the computational approaches differed significantly; the agreement is quantified in Table I. The breathing Fermi surface model could not be quan- titatively compared to the experimental results because the temperature dependence of the scattering rate has not been determined sufficiently accurately. While the present calculations also require knowledge of the scat- tering rate to determine the damping rate the non-monotonic dependence of damping on the scattering rate produces a unique minimum damping rate. In the same manner that the calculated curves of Fig. 1 have a mini- mumwithrespecttoscatteringrate,themeasureddamp- ing curves exhibit minima with respect to temperature. Whatever the relation between temperature and scatter- ing rate, the calculated minima may be compared di- rectly and quantitatively to the measured minima. Ta- ble I makes this comparison. The agreement between measured and calculated values shows that the torque- correlationmodelaccountsforthedominantcontribution to damping in these systems. Our calculated values are smaller than the measured values. Using measured gvalues instead of setting g= 2 would increase our results by a factor of ( g/2)2, or about 10 % for Fe and 20 % for Co and Ni. Other pos- sible reasons for the difference include a simplified treat- ment ofelectron-latticescatteringin which the scattering rates for all states were assumed equal, the mean-field approximation for the exchange interaction, errors asso- ciatedwith thelocalspindensityapproximation(LSDA), and numerical convergence (discussed below). Other damping mechanisms may also make small contributions [28, 29, 30]. Since the manipulations involved with the equation of motion techniques employed in deriving Eq. (3) obscure the underlying physics we now discuss the two scatter- ing processes and connect the intraband terms to the breathing Fermi surface model. The intraband terms in Eq. (3) describe scattering from one state to itself by the torque operator, which is similar to a spin-flip oper- ator. A spin-flip operation between some state and itself is only non-zero because the spin-orbit interaction mixes small amounts of the opposite spin direction into each state. Since the initial and final states are the same, the operation is naturally spin conserving. The matrix ele- ments do not describe a real transition, but rather pro- vide a measure of the energy of the electron-hole pairs that are generated as the spin direction changes. The electron-hole pairs are subsequently annihilated by a real4 electron-lattice scattering event. To connect the derivatives ∂ǫ/∂θin Eq. (2) and the torque matrix elements in Eq. (3) we imagine first point- ing the magnetization in some direction ˆ z. The only energy that changes with the magnetization direction is the spin-orbit energy Hso. As the spin of a single parti- cle state |/angbracketrightrotates along ˆθabout ˆxits spin-orbit energy is given by ǫ(θ) =/angbracketleft|eiσxθHsoe−iσxθ|/angbracketright. The derivative with respect to θis∂ǫ(θ)/∂θ=i/angbracketleft|eiσxθ[σx, Hso]e−iσxθ|/angbracketright. Evaluating this derivative at the pole ( θ= 0) gives ∂ǫ/∂θ=i/angbracketleft|[σx, Hso]|/angbracketright. Similarly, rotating the spin along ˆθabout ˆyleads to ∂ǫ/∂θ=i/angbracketleft|[σy, Hso]|/angbracketright. The torque matrix elements in Eq. (3) are Γ−=/angbracketleft|[σ−, Hso]|/angbracketright= /angbracketleft|[σx, Hso]|/angbracketright−i/angbracketleft|[σy, Hso]|/angbracketright. Using the relations between the commutators and derivatives just found the torque is Γ−=−i(∂ǫ/∂θ)x−(∂ǫ/∂θ)ywhere the subscripts in- dicate the rotation axis. Squaring the torque matrix elements gives |Γ−|2= (∂ǫ/∂θ)2 x+ (∂ǫ/∂θ)2 y. For high symmetry directions ( ∂ǫ/∂θ)x= (∂ǫ/∂θ)yand we de- duce|Γ−|2= 2(∂ǫ/∂θ)2demonstrating that the intra- band terms of the torque-correlation model describe the same physics as the breathing Fermi surface. The monotonically increasing curves in Fig. 1 indi- cate the interband contribution to damping. Uniform mode magnons, which have negligible energy, may in- duce quasi-elastic transitions between states with differ- ent energies. This occurs when lattice scattering broad- ens bands sufficiently so that they overlap at the Fermi level. Thesewavevectorconservingtransitions, whichare driven by the precessing exchange field, occur primarily between states with significantly different spin character. The process may roughly be thought of as the decay of a uniform precession magnon into a single electron spin- flip excitation. These events occur more frequently as the band overlaps increase. For this reason the interband terms, which qualitatively match the resistivity-like con- tributions in the experimental data, dominate damping at room temperature and above. We have calculated the Landau-Lifshitz damping pa- rameterfortheitinerantferromagnetsFe, Co, andNiasa function ofthe electron-latticescatteringrate. Theintra- band and interband components match qualitatively to conductivity- andresistivity-like terms observed in FMR measurements. A quantitative comparison was made be- tweentheminimaldampingratescalculatedasafunction of scattering rate and measured with respect to temper- ature. This comparison demonstrates that our calcula- tions account for the dominant contribution to damping in these systemsand identify the primarydamping mech- anism. At room temperature and above damping occurs overwhelmingly through the interband transitions. The contribution of these terms depends in part on the band gap spectrum around the Fermi level, which could be adjusted through doping. K.G. and Y.U.I. acknowledge the support of the Officeof Naval Research through grant N00014-03-1-0692 and throughgrantN00014-06-1-1016. We wouldlike tothank R.D. McMichael and T.J. Silva for valuable discussions. [1] L. LandauandE. Lifshitz, Phys. Z.Sowjet. 8, 153(1935). [2] T. L. Gilbert, Armour research foundation project No. A059, supplementary report, unpublished (1956). [3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [4] D. Twisselmann and R. McMichael, J. Appl. Phys. 93, 6903 (2003). [5] T. Gerrits, J. Hohlfeld, O. Gielkens, K. Veenstra, K. Bal , T. Rasing, and H. van den Berg, J. Appl. Phys. 89, 7648 (2001). [6] W. Bailey, L. Cheng, D. Keavney, C. Kao, E. Vescovo, and D. Arena, Phys. Rev. B 70, 172403 (2004). [7] I. Krivorotov, D. Berkov, N. Gorn, N. Emley, J. Sankey, D. Ralph, and R. Buhrman, Phys. Rev. B (2007). [8] M. Stiles and J. Miltat, Spin dynamics in confined mag- netic structures III (Springer, Berlin, 2006). [9] J. Rantschler, R. McMichael, A. Castiello, A. Shapiro, J. W.F. Egelhoff, B. Maranville, D.Pulugurtha, A. Chen, and L. Conners, J. Appl. Phys. 101, 033911 (2007). [10] W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE Trans. Mag. 37, 1749 (2001). [11] C. Scheck,L.Cheng, I.Barsukov, Z.Frait, andW.Bailey , Phys. Rev. Lett. 98, 117601 (2007). [12] D. Steiauf and M. Faehnle, Phys. Rev. B 72, 064450 (2005). [13] S. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974). [14] B. Heinrich, D. Meredith, and J. Cochran, J. Appl. Phys. 50, 7726 (1979). [15] J.F.Cochran and B. Heinrich, IEEE Trans. Magn. 16, 660 (1980). [16] V. Kambersky, Czech. J. Phys. B 26, 1366 (1976). [17] B. Heinrich, D. Fraitova, and V. Kambersky, Phys. Stat. Sol. 23, 501 (1967). [18] V. Kambersky, Can. J. Phys. 48, 2906 (1970). [19] V. Kambersky, Czech. J. Phys. B 34, 1111 (1984). [20] V. Korenman and R. Prange, Phys. Rev. B 6, 2769 (1972). [21] J. Kunes and V. Kambersky, Phys. Rev. B 65, 212411 (2002). [22] S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczews ki, P. Trouilloud, and R. Koch, Phys. Rev. B 66, 214416 (2002). [23] L. Mattheiss and D. Hamann, Phys. Rev. B 33, 823 (1986). [24] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [25] W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965). [26] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972). [27] M. Stiles, S. Halilov, R. Hyman, and A. Zangwill, Phys. Rev. B 64, 104430 (2001). [28] R. McMichael and A. Kunz, J. Appl. Phys. 91, 8650 (2002). [29] E. Rossi, O. G. Heinonen, and A. H. MacDonald, Phys. Rev. B 72, 174412 (2005). [30] Y. Tserkovnyak, G. Fiete, and B. Halperin, Appl. Phys. Lett. 84, 5234 (2004).

2007-05-14

The Landau-Lifshitz equation reliably describes magnetization dynamics using a phenomenological treatment of damping. This paper presents first-principles calculations of the damping parameters for Fe, Co, and Ni that quantitatively agree with existing ferromagnetic resonance measurements. This agreement establishes the dominant damping mechanism for these systems and takes a significant step toward predicting and tailoring the damping constants of new materials.

Identification of the dominant precession damping mechanism in Fe, Co, and Ni by first-principles calculations

0705.1990v1

arXiv:0706.0529v1 [cond-mat.mtrl-sci] 4 Jun 2007Generation of microwave radiation in planar spin-transfer devices Ya. B. Bazaliy Instituut Lorentz, Leiden University, The Netherlands, Department of Physics and Astronomy, University of South Ca rolina, Columbia, SC, and Institute of Magnetism, National Academy of Science, Ukrai ne. (Dated: May, 2006) Currentinducedprecession states inspin-transfer device sarestudiedinthecaseoflarge easyplane anisotropy (present in most experimental setups). It is sho wn that the effective one-dimensional pla- nar description provides a simple qualitative understandi ng of the emergence and evolution of such states. Switching boundaries are found analytically for th e collinear device and the spin-flip tran- sistor. The latter can generate microwave oscillations at z ero external magnetic field without either special functional form of spin-transfer torque, or “field- like” terms, if Gilbert constant corresponds to the overdamped planar regime. PACS numbers: 85.75.-d, 75.40.Gb, 72.25.Ba, 72.25.Mk Spin-polarized currents are able to change the mag- netic configuration of nanostructures through the spin- transfer effect proposed more than a decade ago [1, 2]. Intensive research is currently directed at understanding the basic physics of this non-equilibrium interaction and designing magnetic nanodevices with all-electric control. Initial spin-transfer experiments emphasized the cur- rent induced switching between two static configurations [3]. Presently, the research focus is broadening to in- clude the states with continuous magnetization preces- sion powered by the energy of the current source [2, 4, 5]. Spin-transfer devices with precession states (PS) serve as nano-generators of microwave oscillations with remark- able properties, e.g. current tunable frequency and ex- tremelynarrowlinewidth [6, 7, 8, 9]. Aparticularissueof technological importance is the search for systems sup- porting PS at zero magnetic field. Here several strate- gies are pursued: (i) engineering unusual angle depen- dence of spin-transfer torque [10, 11, 12], (ii) relying on the presence of the “field-like” component of the spin torque [13], (iii) choosing the “magnetic fan” geometry [14, 15, 16, 17]. PS are more difficult to describe then the fixed equi- libria: the amplitude of precession can be large and non- linear effects are strong. As a result, information about them if often obtained from numeric simulations. Here s A B jnjn s FIG. 1: Planar spin-transfer devices. Hashed parts of the devices are ferromagnetic, white parts are made from a non- magnetic metal.westudy PS in planardevices [18] usingthe effective one- dimensional approximation [19, 20, 21] which is relevant for the majority of experimental setups. It is shown that planarapproximationprovidesaveryintuitivepictureal- lowing to predict the emergence of precession and subse- quent transformations between different types of PS. We show that PS in devices with in-plane spin polarization of the current can exist at zero magnetic field without the unusual properties (i),(ii) of the spin-transfer torque. A conventional spin-transfer device with a fixed polar- izerandafreelayer(Fig. 1)isconsidered. Themacrospin magnetization of the free layer M=Mnhas a constant absolute value Mand a direction given by a unit vector n(t). The LLG equation [2, 5] reads: ˙n=γ M/bracketleftbigg −δE δn×n/bracketrightbigg +u(n)[n×[s×n]]+α[n×˙n].(1) Hereγis the gyromagnetic ratio, E(n) is the magnetic energy,αis the Gilbert damping constant, sis the spin- polarizer unit vector. The spin transfer strength u(n) is proportional to the electric current I[5, 20]. In general, it is a function of the angle between the polarizer and the free layer u(n) =f[(n·s)]I, with the function f[(n·s)] being material and device specific [22, 23, 24]. The LLG equation can be written in terms of the polar angles(θ,φ) ofvector n. Planardevicesarecharacterized by the energy form E= (K⊥/2)cos2θ+Er(θ,φ) with K⊥≫ |Er|. The first term provides the dominating easy plane anisotropy and ensures that the low energy motion happens close to the θ=π/2 plane. The residual energy Erhas an arbitrary form. The smallness of δθ=θ(t)− π/2 allows to derive a single effective equation on the in-plane angle φ(t) by performing the expansion in small parameter |Er|/K⊥[21]. For time-independent current and polarizer direction sone obtains: 1 ω⊥¨φ+αeff˙φ=−γ M∂Eeff ∂φ, (2) whereω⊥=γK⊥/M. General expressions for αeff(φ)2Eeff 0 π −π−φ m φm(1)(2)(3)(4) 0 π −π(1)(2)(4) FIG.2: (Color online)Evolutionofeffectiveenergyprofilea nd stablesolutions withspin-transferstrength(graphsares hifted up asubecomes more negative) for a device with collinear polarizer. Left: low-field 0 < h <˜ω||regime. Right: high- fieldh >˜ω||regime. Evolutionstage(3)ismissinginthehigh- fieldregime duetotheabsenceofthesecondenergyminimum. The red parts of the energy graphs mark the αeff<0 regions. Filled and empty circle gives represent the effective partic le. andEeff(φ) for arbitrary function Er(θ,φ) and polar- izer direction sare given in Ref. 21. In a special case frequently found in practice the polarizer sis directed in the easy plane at the angle φs, and the residual en- ergy satisfies ( ∂Er/∂θ)θ=π/2= 0, i.e. does not shift the energy minima away from the plane. We will also use the simplest form f[(n·s)] = const for the spin transfer strength. A more realistic function can be employed if needed. With these assumptions [21]: αeff=α+2ucos(φs−φ) ω⊥, (3) Eeff=Er(π/2,φ)−Mu2 2γω⊥cos2(φs−φ). Equation (2) has the form of Newton’s equation of mo- tion for a particle in external potential Eeff(φ) with a variable viscous friction coefficient αeff(φ). The advan- tage of such a description is that the motion of the effec- tive particle can be qualitatively understood by applying the usualenergyconservationand dissipation arguments. In the absence of current, the effective friction is a posi- tive constant, so after an initial transient motion the sys- tem always ends up in one of the minima of Er(π/2,φ). When current is present, effective friction and energy are modified. Such a modification reflects the physical pos- sibility of extracting energy from the current source, and leads to the emergence of the qualitatively new dynamic regime of persistent oscillations. These oscillations of φ correspond to the motion of nalong the highly elongated (δθ≪1) closed orbits (see examples in Fig. 3, inset), i.e. constitute the limiting form of the precession states[2, 5, 7, 25] in spin-transfer systems. To illustrate the advantages of the effective parti- cle description, consider a specific example of PS in the nanopillar experiment [7] where Eris an easy axis anisotropy energy with magnetic field Hdirected along that axis, Er(φ) = (K||/2)sin2φ−HMcosφ. The po- larizersis directed along the same axis with φs= 0 (collinear polarizer). With the definitions ω||=γKa/M, h=γH, the effective energy becomes [21] γ MEeff=˜ω||(u) 2sin2φ−hcosφ , (4) with ˜ω||=ω||+u2/ω⊥. Effective energy profiles are shown in Fig. 2. For low fields, |h|<˜ω||(u), the minima atφ= 0,πare separated by maxima at ±φm(h). According to Eq. (3), the effective friction can become negative at φ= 0 orφ=πat the critical value of spin- transfer strength |u|=u1=αω⊥/2. If this value is exceeded, the position of the system in the energy min- imum becomes unstable. Indeed, the stability of any equilibrium in one dimension depends on whether it is a minimum or a maximum of Eeffand on the sign of αeffat the equilibrium point. Out of four possible com- binations, only an energy minimum with αeff>0 is stable. A little above the threshold, αeffis negative in a small vicinity of the minimum where the system in now characterized by negative dissipation. In this situation any small fluctuation away from the equilibrium initiates growing oscillations. As the oscillations amplitude ex- ceeds the size of the αeff<0 region, part of the cycle starts to happen with positive dissipation. Eventually the amplitude reaches a value at which the energy gain during the motion in the αeff<0 region is exactly com- pensated by the energy loss in the αeff>0 region: thus a stable cycle solution emerges (Fig. 2, profile (2)). The requirement of zero total dissipation means that an integral over the oscillation period satisfies/integraltext αeff(φ)(˙φ)2dt= 0. In typical collinear systems [25] Gilbert damping satisfies α≈0.01≪/radicalbig ω||/ω⊥≈0.1≪ 1, hence the oscillator (2),(4) operates in the lightly damped regime. In zeroth order approximation the fric- tion term in (2) can be neglected, and a first integral ˙φ2/(2ω⊥)+Eeff=E0exists. Zero dissipation condition can be then approximated by /integraldisplayφ2 φ1αeff(φ)/radicalBig E0−Eeff(φ)dφ= 0,(5) withφ1,2(u) being the turning points of the effective par- ticle trajectory, and E0=Eeff(φ1) =Eeff(φ2). Since the integrand of (5) is a known function, the formula provides an expression for the precession amplitude. Consider now the low positive field regime 0 < h <˜ω||. Atu=−u1the parallel configuration becomes unstable and a cycle emerges near the φ= 0 minimum. As u is made more negative, the oscillation amplitude grows3 until eventually it reaches the point of energy maximum atu=−u2. Equivalently, the effective particle starting at the energy maximum −φmis able to reach the other maximum at + φm(Fig. 2, left, (3)). Above this thresh- old the particleinevitably goes overthe potential hill and falls into the φ=πminimum which remains stable since αeff(π)>0 holds for negative u. In other words, the cy- cle solutionwith oscillationsaroung φ= 0 ceasestoexist. At even more negative uthe third threshold is reached when the effective particle can complete the full rotation starting from the energy maximum (Fig. 2, profile (4)). Belowu=−u3a new PS with full rotation emerges. In the high-field regime h >˜ω||the evolution of the pre- cession cycle is similar (Fig. 2, right), but stage (3) is missing since there is no second minimum. The thresh- oldu=−u2separates the finite oscillations regime and the full-rotation regime. Thresholds uican be obtained analytically from (5) by substituting the critical turning points φ1,2listed above: u2=αω⊥hφm+ω||sinφm ω||φm+hsinφm(h < ω||), (6) u2=αω⊥h ω||(h > ω||), (7) u3=αω⊥h(φm−π/2)+ω||sinφm ω||(φm−π/2)+hsinφm(h < ω||).(8) The corresponding switching diagram is shown in Fig. 3 (cf. numerically obtained Fig. 12 in Ref. 25). It shows that different hysteresis patterns are possible depending on the trajectory in the parameter space. PS in the low field regime was discussed analytically in an unpublished work [26]. However, since a conventional description with two polar angles was used, the calcula- tions were much less transparent. Numeric studies of the PS were performed in Refs. [7, 25] after the experimental observation [7] of the current induced transition between two PS in the high field regime. They had shown that indeed the low-current precession state PS 1has a finite amplitude of φ-oscillations, while the high-current state PS2exhibits full rotations of φ(Fig. 3, inset). Next, we consider the cycle solutions in a device called a spin-flip transistor [18, 27]. It differs form the setup studied above in the polarizer direction, which is now perpendicular to the easy axis with φs=π/2. No exter- nal magnetic field is applied. In this case [21] αeff=α+2usinφ ω⊥, (9) γ MEeff=¯ω||(u) 2sin2φ , (10) with ¯ω||=ω||−u2/ω⊥. As the spin-transfer strength grows, the behavior of the system changes qualitatively when ¯ω||orαeff|±π/2changesignsatthethresholds ¯ u1= ±√ω||ω⊥and ¯u2=±αω⊥/2. In accord with previous in-uPS1s u1 u2 u3PS2 PS1PS2h ω -ωω (u)∼ || z nφ xPS1PS2 FIG. 3: (Color online) Switching diagram of a device with collinear polarizer. The u-axis direction is reversed for the purpose of comparison with Refs. 7, 25. The parts of the diagram not shown can be recovered by a 180-degree rotation of the picture. Stable directions in each region are given by small arrows, the precession states are marked as PS 1,2. The large arrow shows the polarizer direction. Inset: schemati c trajectories of the PS 1,2states on the unit sphere. vestigations [16, 28] at |u|>¯u1theφ= 0,πenergy min- ima are destabilized and the parallel state φ= sgn[u]φs becomes stable. Surprisingly, for α > α ∗= 2/radicalbig ω||/ω⊥a window ¯u1< u <¯u2of stability of antiparallel configu- ration,φ=−sgn[u]φs, opens (Fig. 4). As discussed in Ref. [21], the stabilization of the antiparallel state hap- pens as the spin-transfer torque is increased in spite of the fact that this torque repels the system from that di- rection. At u= ¯u2theantiparallelstateturnsintoacycle (Fig. 4, low right panel) which we will study here. Above the ¯u2threshold the amplitude of oscillations grows until they reach the energy maximum at u= ¯u3and the cy- cle solution disappears. Although αis not small, ¯ u3can still be determined from Eq. (5) because αeffis small whenuis close to u2. Calculating the integral in (5) withφ1,2=−π,0 we get ¯u3=2 παω⊥≈1.27¯u2. (11) The usage of approximations (5),(11) is legitimate for α>∼2α∗whereαeff(¯u3)≪/radicalbig¯ω||(¯u3)/ω⊥holds. For smaller values of αnumeric calculations are required. They show the existence of a stable cycle down to α= 0.8α∗where the stabilization of the antiparallel state is impossible. For α≪α∗andu>∼¯u1the strong negative dissipation regime is realized, |αeff| ≫/radicalbig¯ω||/ω⊥. Nu- meric results show that the amplitude of the oscillations4 α*u αEeff 0+π/2 π −π −π/2 0+π/2 π −π −π/2PS1s u1u2u3 FIG. 4: (Color online) Switching diagram of a spin-flip tran- sistor. The u <0 part of the diagram can be obtained by reflection with respect to the horizontal axis. In each regio n stable directions are given by small arrows, precession sta te is marked by PS 1. The large arrow shows the polarizer di- rection. Threshold ¯ u3(α) is sketched as a dashed line where approximation (11) is not valid. Lower panels: the evolutio n of effective energy and trajectories (graphs are shifted up w ith growing u) atα << α ∗(left) and α > α ∗(right). The red part of the energy graph marks the αeff<0 region. Effective particle is shown by filled and empty circles. induced bynegativedissipationissobigthat theeffective particle always reaches the energy maximum and drops into the stable parallel state (Fig 4, low left panel). We conclude that the line ¯ u3(α) crosses the u= ¯u1line at some point and terminates there. As for the full-rotation PS, one can show analytically that it does not exists in the small dissipation limit at α>∼2α∗. Numerical simulations do not find it in the α < α∗,u >¯u1regime either. In conclusion, we have shown that the planar effective description can be very useful for studying precession so- lutions in the spin transfer systems. It was already used to describe the “magnetic fan” device with current spin polarization perpendicular to the easy plane [20]. Here the switching diagrams were obtained for the spin polar- izers directed collinearly and perpendicular to the easy direction within the plane. In collinear case we found an- alytic formulas for the earlier numeric results, while the study of precession solutions in the perpendicular case (spin-flip transistor) at large damping is new. The lat- ter shows the possibility of generating microwave oscil-lations in the absence of external magnetic field with- out the need to engineer special angle dependence of the spin-transfer torque or “field-like” terms. The inequality α >2/radicalbig ω||/ω⊥required for the existence of such oscil- lations can be satisfied by either reducing the in-plane anisotropy, or increasing αdue to spin-pumping effect [29]. Most importantly, the effective planar description allows for qualitative understanding of the precession cy- cles and makes it easy to predict their emergence, sub- sequent evolution, and transitions between different pre- cession cycle types. E.g., in the systems with one re- gion of negative effective dissipation, such as considered here, it shows that no more then two precession states, one with finite oscillations and another with full rota- tions, can exist. Numerical approaches, if needed, are then based on a firm qualitative foundation. In addition, numerical calculations in one dimension are easier then in the conventional description with two polar angles. The authorthanks G. E.W. Bauerand M.D. Stiles for discussion. Research at Leiden University was supported by the Dutch Science Foundation NWO/FOM. Part of this work was performed at Aspen Center for Physics. [1] L. Berger, J. Appl. Phys., 49, 2160 (1978); Phys. Rev. B 33, 1572 (1986); J. Appl. Phys. 63, 1663 (1988). [2] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [3] J. A. Katine et al., Phys. Rev. Lett., 84, 3149 (2000). [4] J. Z. Sun, Phys. Rev. B 62, 570 (2000). [5] Ya. B. Bazaliy et al., arXiv:cond-mat/0009034 (2000); Phys. Rev. B, 69, 094421 (2004). [6] M. Tsoi et al., Nature, 406, 46 (2000). [7] S. I. Kiselev et al., Nature, 425, 380 (2003). [8] S. Kaka et al., Nature 437, 389 (2005); [9] M. R. Pufall et al., Phys.Rev. Lett. 97, 087206 (2006); [10] J. Manschot et al., Appl. Phys. Lett. 85, 3250 (2004). [11] M. Gmitra et al, Phys. Rev. Lett., 96, 207205 (2006). [12] O. Boulle et al., Nature Physics, May 2007. [13] T. Devolder et al., J. Appl. Phys., 101, 063916 (2007) [14] A. D. Kent et al., Appl. Phys. Lett., 84, 3897 (2004) [15] K. J. Lee et al., Appl. Phys. Lett., 86, 022505 (2005). [16] X. Wang et al., Phys. Rev. B, 73, 054436 (2006). [17] D. Houssameddine et al., Nature Materials, April 2007. [18] A. Brataas et al., Phys. Rep., 427, 157 (2006). [19] C. Garcia-Cervera et al., J. Appl. Phys., 90, 370 (2001). [20] Ya. B. Bazaliy et al., arXiv:0705.0406v1 (2007), to be published in J. Nanoscience and Nanotechnology. [21] Ya. B. Bazaliy, arXiv:0705.0508 (2007). [22] J. C. Slonczewski, JMMM, 247, 324 (2002). [23] A. A. Kovalev et al., Phys. Rev. B, 66, 224424 (2002) [24] J. Xiao et al., Phys. Rev. B, 70, 172405 (2004). [25] J. Xiao et al., Phys. Rev. B, 72, 014446 (2005) [26] T. Valet, unpublished preprint (2004). [27] A. Brataas et al., Phys. Rev. Lett. 84, 2481 (2000); [28] H. Morise et al., Phys. Rev. B, 71, 014439 (2005). [29] Ya. Tserkovnyak et al., Rev. Mod. Phys., 77, 1375 (2005).

2007-06-04

Current induced precession states in spin-transfer devices are studied in the case of large easy plane anisotropy (present in most experimental setups). It is shown that the effective one-dimensional planar description provides a simple qualitative understanding of the emergence and evolution of such states. Switching boundaries are found analytically for the collinear device and the spin-flip transistor. The latter can generate microwave oscillations at zero external magnetic field without either special functional form of spin-transfer torque, or ``field-like'' terms, if Gilbert constant corresponds to the overdamped planar regime.

Generation of microwave radiation in planar spin-transfer devices

0706.0529v1

arXiv:0706.3160v3 [cond-mat.mes-hall] 9 Jan 2008Spin pumping by a field-driven domain wall R.A. Duine Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands (Dated: October 26, 2018) We present the theory of spin pumping by a field-driven domain wall for the situation that spin is not fully conserved. We calculate the pumped current in a m etallic ferromagnet to first order in the time derivative of the magnetization direction. Irre spective of the microscopic details, the result can be expressed in terms of the conductivities of the majority and minority electrons and the dissipative spin transfer torque parameter β. The general expression is evaluated for the specific case of a field-driven domain wall and for that case depends st rongly on the ratio of βand the Gilbert damping constant. These results may provide an expe rimental method to determine this ratio, which plays a crucial role for current-driven domain -wall motion. PACS numbers: 72.25.Pn, 72.15.Gd I. INTRODUCTION Adiabatic quantum pumping of electrons in quantum dots1,2has recently been demonstrated experimentally forboth charge3and spin4. Currently, the activityin this fieldismostlyconcentratedontheeffectsofinteractions5, dissipation6, and non-adiabaticity7. Complementary to these developments, the emission of spin current by a precessing ferromagnet — called spin pumping — has been studied theoretically and experimentally in single- domain magnetic nanostructures8,9,10. One of the differ- ences between spin pumping in single-domain ferromag- nets and quantum pumping in quantum dots is that in thelatterthehamiltonianoftheelectronicquasi-particles is manipulated directly, usually by varying the gate volt- age of the dot. In the case of ferromagnets, however, it is the order parameter — the magnetization direction — that is driven by an external (magnetic) field. The coupling between the order parameter and the current- carrying electrons in turn pumps the spin current11. The opposite effect, i.e., the manipulation of magnetization with spin current, is called spin transfer12,13,14,15. Recently, the possibility of manipulating with cur- rent the position of a magnetic domain wall via spin transfer torques has attracted a great deal of theoretical16,17,18,19,20,21,22,23,24,25,26,27,28,29,30and experimental31,32,33,34,35,36,37,38interest. Although the subject is still controversial18,21, it is by now established that in the long-wavelength limit the equation of motion for the magnetization direction Ω, which in the absence of current describes damped precession around the effective field −δEMM[Ω]/(/planckover2pi1δΩ), is given by /parenleftbigg∂ ∂t+vs·∇/parenrightbigg Ω−Ω×/parenleftbigg −δEMM[Ω] /planckover2pi1δΩ/parenrightbigg =−Ω×/parenleftbigg αG∂ ∂t+βvs·∇/parenrightbigg Ω, (1) and contains, to lowest order in spatial derivatives of the magnetization direction, two contributions due the pres- ence of electric current.The first is the reactivespin transfertorque16,17, which corresponds to the term proportional to ∇Ωon the left- hand side of the above equation. It is characterized by the velocity vsthat is linear in the curent and related to the external electric field Eby vs=(σ↓−σ↑)E |e|ρs, (2) whereσ↑andσ↓denote the conductivities of the major- ity and minority electrons, respectively, and ρsis their density difference. (The elementary charge is denoted by |e|.) The second term in Eq. (1) due to the current is the dissipative spin transfer torque39that is proportional to β19,20,21. Both this parameter, and the Gilbert damping parameter αG, have their microscopic origin in processes in the hamiltonian that break conservation of spin, such as spin-orbit interactions. It turns out that the phenomenology of current-driven domain-wallmotion depends crucially on the value of the ratioβ/αG. For example, for β= 0 the domain wall is intrinsically pinned18, meaning that there is a criti- cal current even in the absence of inhomogeneities. For β/αG= 1ontheotherhand, thedomainwallmoveswith velocity vs. Although theoretical studies indicate that generically β/ne}ationslash=αG26,27,28,30, it is not well-understood what the relative importance of spin-dependent disorder and spin-orbit effects in the bandstructure is, and a pre- cise theoretical prediction of β/αGfor a specific mate- rial has not been attempted yet. Moreover, the determi- nation of the ratio β/αGfrom experiments on current- driven domain wall motion has turned out to be hard because of extrinsic pinning of the domain and nonzero- temperature29,38effects. In this paper we present the theory of the current pumped by a field-driven domain wall for the situation that spin is not conserved. In particular, we show that a field-driven domain wall in a metallic ferromagnet gen- erates a charge current that depends strongly on the ra- tioβ/αG. This charge current arises from the fact that a time-dependent magnetization generates a spin cur- rent, similar to the spin-pumping mechanism proposed2 by Tserkovnyak et al.8for nanostructures containing fer- romagnetic elements. Since the symmetry between ma- jority and minority electrons is by definition broken in a ferromagnet, this spin current necessarily implies a charge current. In view of this, we prefer to use the term “spin pumping” also for the case that spin is not fully conserved, and defining the spin current as a conserved current is no longer possible. The generation of spin and charge currents by a mov- ing domain wall via electromotiveforces is discussed very recently by Barnes and Maekawa40. We note here also the work by Ohe et al.41, who consider the case of the Rashba model, and the very recent work by Saslow42, Yanget al.43, and Tserkovnyak and Mecklenburg44. In addition to these recent papers, we mention the much earlier work by Berger, which discusses the current in- duced by a domain wall in terms of an analogue of the Josephson effect45. Barnes and Maekawa40consider the case that spin is fully conserved. In this situation it is convenient to perform a time and position dependent rotation in spin space, such that the spin quantization axis is locally par- allel to the magnetization direction. As a result of spin conservation, the hamiltonian in this rotated frame con- tains nowonly time-independent scalarand exchangepo- tential terms. The kinetic-energy term of the hamilto- nian, however, will acquire additional contributions that have the form of a covariant derivative. Perturbation theory in these terms then amounts to performing a gra- dient expansion in the magnetization direction17. Hence, the fact that Barnes and Maekawa consider the case that spin is fully conserved is demonstrated mathemat- ically by noting that in Eq. (5) of Ref. [40] there are no time-dependent potential-energy terms. Generaliz- ing this approach to the case of spin-dependent disorder or spin-orbit coupling turns out to be difficult. Never- theless, Kohno and Shibata were able to determine the Gilbert damping and dissipative spin transfer torques us- ing the above-mentioned method46. Since Barnes and Maekawa40consider the situation that spin is fully con- served, they effectively are dealing with the case that αG=β= 0. This is because both the Gilbert damping parameter αGandthedissipativespintransfertorquepa- rameterβarise from processes in the microscopic hamil- tonian that do not conserve spin26,27,28,30. Hence, for the case that αG=β= 0 our results agree with the results of Barnes and Maekawa40. The remainder of this paper is organized as follows. In Sec. II we derive a general expression for the electric cur- rent induced by a time-dependent magnetizationtexture. This general expression is then evaluated in Sec. III for a simple model of field-driven domain wall motion. We end in Sec. IV with a short discussion, and present our conclusions and outlook.II. ELECTRIC CURRENT Quite generally, the expectation value of the charge current density, defined by j=−cδH/δAwithcthe speed oflight, Hthe hamiltonian, and Athe electromag- netic vector potential, is given as a functional derivative of the effective action /an}b∇acketle{tj(x,τ)/an}b∇acket∇i}ht=cδSeff δA(x,τ), (3) withτthe imaginary-time variable that runs from 0 to /planckover2pi1/(kBT). (Planck’s constant is denoted by /planckover2pi1andkBTis the thermal energy.) First, we assume that spin is con- served meaning that the hamiltonian is invariant under rotations in spin space. The part of the effective action for the magnetization direction that depends on the elec- tromagnetic vector potential is then given by17 Seff=/integraldisplay dτ/integraldisplay dx/angbracketleftbig jz s,α(x,τ)/angbracketrightbig˜Aα′(Ω(x,τ))∇αΩβ(x,τ), (4) where a summation over Cartesian indices α,α′,α′′∈ {x,y,z}is implied throughout this paper. In this ex- pression, jα s,α′(x,τ) =/planckover2pi12 4mi/bracketleftbig φ†(x,τ)τα∇α′φ(x,τ) −/parenleftbig ∇α′φ†(x,τ)/parenrightbig ταφ(x,τ)/bracketrightbig +|e|/planckover2pi1 2mcAα′φ†(x,τ)ταφ(x,τ),(5) is the spin current, given here in terms of the Grassman coherent state spinor φ†= (φ∗ ↑,φ∗ ↓). Furthermore, ταare the Pauli matrices, and mis the electron mass. (Note that since we are, for the moment, considering the situa- tion that spin is conserved there are no problems regard- ing the definition of the spin current.) The expectation value/an}b∇acketle{t···/an}b∇acket∇i}htis taken with respect to the current-carrying collinear state of the ferromagnet. Finally, ˜Aα(Ω) is the vector potential of a magnetic monopole in spin space [not to be confused with the electromagnetic vector po- tentialA(x,τ)] that obeys ǫα,α′,α′′∂˜Aα′/∂Ωα′′= Ωαand is well-known from the path-integral formulation for spin systems47. Eq. (4) is most easily understood as arising from the Berry phase picked up by the spin of the elec- trons as they drift adiabatically through a non-collinear magnetization texture16,17. Variation of this term with respect to the magnetization direction gives the reactive spin transfer torque in Eq. (1). The expectation value of the spin current is given by /angbracketleftbig jz s,α(x,τ)/angbracketrightbig =/integraldisplay dτ′/integraldisplay dx′Πz α,α′(x−x′;τ−τ′)Aα′(x′,τ′) /planckover2pi1c. (6) The zero-momentum low-frequency part of the response function Πz α,α′(x−x′;τ−τ′)≡/angbracketleftbig jz s,α(x,τ)jα′(x′,τ′)/angbracketrightbig 0, with/an}b∇acketle{t···/an}b∇acket∇i}ht0the equilibrium expectation value, is deter- mined by noting that for the vector potential A(x,τ) =3 −cEe−iωτ/ωthe above equation [Eq. (6)] should in the zero-frequency limit reduce to Ohm’s law /an}b∇acketle{tjz s/an}b∇acket∇i}ht0= −/planckover2pi1(σ↑−σ↓)E/(2|e|). Using this result together with Eqs. (3-6), we find, after a Wick rotation τ→itto real time, that /an}b∇acketle{tjα/an}b∇acket∇i}ht=−/planckover2pi1 2|e|V(σ↑−σ↓)∂ ∂t/integraldisplay dx˜Aα′(Ω(x,t))∇αΩα′(x,t), (7) withVthe volume of the system. We note that the time- derivative of the Berry phase term is also encountered by Barnes and Maekawa in discussing the electromotive force in a ferromagnet40. Such Berry phase terms are known to occur in adiabatic quantum pumping48. We now generalize this result to the situation where spin is no longer conserved, for example due to spin- orbit interactions or spin-dependent impurity scatter- ing. Linearizing around the collinear state by means of Ω≃(δΩx,δΩy,1−δΩ2 x/2−δΩ2 y/2) we find that the part of the effective action that contains the electromagnetic vector potential reads30 Seff=/integraldisplay dτ/integraldisplay dx/integraldisplay dτ′/integraldisplay dx′/integraldisplay dτ′′/integraldisplay dx′′[δΩa(x,τ) ×Kab(x,x′,x′′;τ,τ′,τ′′)·A(x′′,τ′′)δΩb(x′,τ′)],(8) where a summation over transverse indices a,b∈ {x,y} is implied. The spin-wave photon interaction vertex Kab(x,x′,x′′;τ,τ′,τ′′) = ∆2 8/planckover2pi1c/an}b∇acketle{tφ†(x,τ)τaφ(x,τ)φ†(x′,τ′)τbφ(x′,τ′)j(x′′,τ′′)/an}b∇acket∇i}ht0, (9) given in terms of the exchange splitting ∆, is also en- countered in a microscopic treatment of spin transfer torques30. The reactive part of this interaction vertex determines the reactive spin transfer torque and, via Eqs. (3) and (8), reproduces Eq. (7). The zero-frequency long-wavelength limit of the dissipative part of the spin- wave photon interaction vertex determines the dissipa- tive spin transfer torque. (Note that in this approach the definition of the spin current does not enter in deter- mining the spin transfer torques.) Although Eq. (9) may be evaluated for a given microscopic model within some approximation scheme30, we need here only that varia- tion of the action in Eq. (8) reproduces both the reactive and dissipative spin torques in Eq. (1). The final result for the electric current density is then given by /an}b∇acketle{tjα/an}b∇acket∇i}ht=−/planckover2pi1 2|e|V(σ↑−σ↓)/bracketleftbigg β/integraldisplay dx∂Ω(x,t) ∂t·∇αΩ(x,t) +∂ ∂t/integraldisplay dx˜Aα′(Ω(x,t))∇αΩα′(x,t)/bracketrightbigg .(10) The above equation is essentially the result of a linear- response calculation in ∂Ω/∂t, and is the central result of this paper. We emphasize that the way in which thetransport coefficients σ↑andσ↓and theβ-parameter en- ter does not rely on the specific details of the underlying microscopic model. Note that the above result reduces to that of Barnes and Maekawa (Eq.(9) of Ref. [40]) if we takeβ= 0. III. FIELD-DRIVEN DOMAIN WALL MOTION To bring out the qualitative physics, we evaluate the result in Eq. (10) using a simple model for field- driven domain wall motion in a magnetic wire of length L. In polar coordinates θandφ, defined by Ω= (sinθcosφ,sinθsinφ,cosθ), we choose the micromag- netic energy functional EMM[θ,φ] =ρs/integraldisplay dx/braceleftbiggJ 2/bracketleftBig (∇θ)2+sin2θ(∇φ)2/bracketrightBig +K⊥ 2sin2θsin2φ−Kz 2cos2θ+gBcosθ/bracerightbigg ,(11) whereJis the spin stiffness, and K⊥andKzare anisotropy constants larger than zero. The external field in the negative z-direction leads to an energy splitting 2gB >0. We solve the equation of motion in Eq. (1) within the variational ansatz18,49 θ(x,t) =θ0(x,t)≡2tan−1/bracketleftBig e−(rdw(t)−x)/λ/bracketrightBig ,(12) together with φ(x,t) =φ0(t), that describes a rigid do- main wall with width λ=/radicalbig J/Kzat position rdw(t). The chirality of the domain wall is determined by the angleφ0(t) and the magnetization direction is assumed to depend only on xwhich is taken in the long direction of the wire. The equations ofmotion for the variationalparameters are given by18,29,49 ˙φ0(t)+αG/parenleftbigg˙rdw(t) λ/parenrightbigg =gB /planckover2pi1; /parenleftbigg˙rdw(t) λ/parenrightbigg −αG˙φ0(t) =K⊥ 2/planckover2pi1sin2φ0(t).(13) Note that the velocity vsis absent from these equations since we consider the generation of electric current by a field-driven domain wall. The above equations pro- vide a description of the field-driven domain wall and, in particular, of Walker breakdown49. That is, for an external field smaller than the Walker breakdown field Bw≡αGK⊥/(2g) the domain wall moves with a con- stant velocity. For fields B > B wthe domain wall under- goesoscillatorymotion, whichinitially makestheaverage velocity smaller. Solving the equations of motion results in ˙φ0=1 (1+α2 G)Re /radicalBigg/parenleftbigggB /planckover2pi1/parenrightbigg2 −/parenleftbiggαGK⊥ 2/planckover2pi1/parenrightbigg2 ; ˙rdw λ=gB αG/planckover2pi1−˙φ0 αG, (14)4 0 0.5 1 1.5 2 0 0.5 1 1.5 2 |j/j0| B/Bwβ=0.015 β=0.01 β=0.005 β=0 FIG. 1: Current generated by a field-driven domain wall in units of j0= 2L/[|e|(σ↑−σ↓)αGK⊥], forαG= 0.01 and various values of β. The result is plotted as a function of magnetic field in units of the Walker breakdown field Bw≡ αGK⊥/(2g). where the ···indicates taking the time-averaged value. Insertingthe variational ansatzintoEq. (10) leads in first instance to /an}b∇acketle{tjx/an}b∇acket∇i}ht=−/planckover2pi1 |e|L(σ↑−σ↓)/bracketleftbiggβ˙rdw(t) λ+˙φ0(t)/bracketrightbigg ,(15) which, using Eq. (14), becomes /an}b∇acketle{tjx/an}b∇acket∇i}ht=−/planckover2pi1 |e|L(σ↑−σ↓) βgB αG/planckover2pi1 +/parenleftBigg 1−β αG 1+α2 G/parenrightBigg Re /radicalBigg/parenleftbigggB /planckover2pi1/parenrightbigg2 −/parenleftbiggαGK⊥ 2/planckover2pi1/parenrightbigg2 .(16) As shown in Fig. 1, this result depends strongly on the ratioβ/αG. In particular, for β > α Ga local maximum appears in the current as a function of magnetic field. SinceαGis determined independently from ferromag- netic resonanceexperiments, measurementofthe slopeof thecurrentforsmallmagneticfieldsenablesexperimental determination of β. We note that within the present ap- proximation the current does not depend on the domain wall width λ. Furthermore, in the limit of zero Gilbert dampingand β, thedissipationlesslimit, wehavethatthe current density is equal to /an}b∇acketle{tjx/an}b∇acket∇i}ht= (σ↓−σ↑)gB/(|e|L). This is the result of Barnes and Maekawa40that corre- sponds to the situation that αG=β= 0, as discussed in the Introduction. We point out that, within our ap- proximation for the description of domain-wall motion, puttingβ=αGin Eq. (16) gives the same result as us- ing Eqs. (13) and (15) with αG=β= 0. That the situation discussed by Barnes and Maekawa40is indeed that ofαG=β= 0 is seen by comparing their result [Eqs. (8) and (9) of Ref. [40], and the paragraph follow- ing Eq. (9)] with our results in Eqs. (10) and (13).IV. DISCUSSION AND CONCLUSIONS Our result in Eq. (16) is a simple expression for the pumped current as a function of magnetic field for a field-driven domain wall. A possible disadvantage in us- ing Eq. (16), however, is that in deriving this result we assumed a specific model to describe the motion of the domain wall. This model does in first instance not in- clude extrinsic pinning and nonzero temperature. Both extrinsic pinning18and nonzerotemperature29can be in- cluded in the rigid-domain wall description. However, it is in some circumstances perhaps more convenient to directly use the result in Eq. (15) together with the ex- perimental determination of ˙ rdw(t). Since the only way in which the parameter βenters this equation is as a prefactor of ˙ rdw(t), this should be sufficient to determine its value from experiment. We note, however, that the precision with which the ratio β/αGcan be determined depends on how accurately the magnetization dynamics, and, in particular, the motion of the domain wall, is im- aged experimentally. With respect to this, we note that the various curves in Fig. 1 are qualitatively different for different values of β/αG. In particular, the results for β/αG>1 andβ/αG<1 differ substantially, and could most likely be experimentally distinguished. In view of this discussion, future research will in part be directed towards evaluating Eq. (10) for more complicated mod- els of field-driven domain-wall motion, which will benefit the experimental determination of β/αG. A typical current density is estimated as follows. For the experiments of Beach et al.50we have that L∼20 µm, andλ∼20 nm. The domain velocities measured in this experiment are ˙ rdw∼40−100 m/s. Taking as a typical conductivity σ↑∼106Ω−1m−1we find, using equation Eq. (15) with β∼0.01, typical electric current densities of the order of /an}b∇acketle{tjx/an}b∇acket∇i}ht ∼103−104A m−2. This re- sult depends somewhat on the polarization of the electric current in the ferromagnetic metal, which we have taken equal to 50% −100% in this rough estimate. Although much smaller than typical current densities required to move the domain wall via spin transfer torques, electri- cal current densities of this order appear to be detectable experimentally. In conclusion, we have presented a theory of spin pumping without spin conservation, and, in particular, proposed a way to gain experimental access to the pa- rameterβ/αGthat is of great importance for the physics of current-driven domain wall motion. We note that the mechanism for current generation discussed in this pa- per is quite distinct from the generation of eddy cur- rents by a moving magnetic domain51. In addition to improving upon the model used for describing domain- wall motion, we intend to investigate in future work whetherthe dampingtermsinEq.(1), orpossiblehigher- order terms in frequency and momentum52, have a nat- ural interpretation in terms of spin pumping, similar to the spin-pumping-enhanced Gilbert damping in single- domain ferromagnets8.5 It is a great pleasure to thank Gerrit Bauer, Maxim Mostovoy, and Henk Stoof for useful comments and dis-cussions. 1C. Bruder and H. Schoeller, Phys. Rev. Lett. 72, 1076 (1994). 2P.W. Brouwer, Phys. Rev. B 58, R10135 (1998). 3M. Switkes, C. M. Marcus, K. Campman, A. C. Gossard, Science283, 1905 (1999). 4Susan K. Watson, R. M. Potok, C. M. Marcus, and V. Phys. Rev. Lett. 91, 258301 (2003). 5P. Sharma and C. Chamon, Phys. Rev. Lett. 87, 096401 (2001). 6D. Cohen, Phys. Rev. B 68, 201303 (2003). 7Michael Strass, Peter H¨ anggi, and Sigmund Kohler, Phys. Rev. Lett. 95, 130601 (2005). 8Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. 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Zapperi, arXiv:0706.2122v1. 52See for example Eq. (27) of Ref. [26] for possible higher- order terms in the presence of current.

2007-06-21

We calculate the charge current in a metallic ferromagnet to first order in the time derivative of the magnetization direction. Irrespective of the microscopic details, the result can be expressed in terms of the conductivities of the majority and minority electrons and the non-adiabatic spin transfer torque parameter $\beta$. The general expression is evaluated for the specific case of a field-driven domain wall and for that case depends strongly on the ratio of $\beta$ and the Gilbert damping constant. These results may provide an experimental method to determine this ratio, which plays a crucial role for current-driven domain-wall motion.

Spin pumping by a field-driven domain wall

0706.3160v3

arXiv:0708.3827v3 [physics.class-ph] 27 Mar 2008Linear frictional forces cause orbits to neither circularize nor precess P.M. Hamilton1,2and M. Crescimanno1 1Dept. of Physics and Astronomy, Youngstown State University 2Dept. of Physics, University of Maryland E-mail:pmham@umd.edu, mcrescim@cc.ysu.edu PACS numbers: 46.40Ff, 45.20.Jj, 45.50.Pk, 34.60.+z Submitted to: J. Phys. A: Math. Gen.Linear frictional forces cause orbits to neither circulari ze nor precess 2 Abstract: For the undamped Kepler potential the lack of precession has histo rically beenunderstoodintermsoftheRunge-Lenzsymmetry. Forthed ampedKeplerproblem this result may be understood in terms of the generalization of Poiss on structure to dampedsystems suggestedrecentlybyTarasov[1]. Inthisgenera lizedalgebraicstructure the orbit-averaged Runge-Lenz vector remains a constant in the linearly damped Kepler problem to leading order in the damping coefficient. Beyond Kepler, we prove that, for any potential proportional to a power of the radius, the orbit shape and precession angle remain constant to leading order in the linear friction coefficient . 1. Introduction What happens to orbits subject to linear frictional drag? In typica l physical settings, such as Rydberg atoms or stellar binaries, the effective frictional f orces are nonlinear and, typically, lead to the circularization of the orbit. Orbital evolut ion under linear friction is special in that, as we show below, the eccentricity and the apsides do not change to leading order in the damping. The purpose of this note is to understand this elementary result from the underlying dynamical symmetry of the K epler problem, thus demonstrating theutility ofa Hamiltoniannotionin itsnon-Hamiltoniang eneralization. In many astrophysical situations, the secular evolution due to fric tion of orbits in a two body system is towards circular orbits. In an orbit in a centra l field the angular momentum scales with the momentum while the energy genera lly scales with the momentum-squared. Friction, assumed to be spatially isotropic and homogeneous but time odd, typically scales the momentum. This means generally tha t the resultant secular evolution in central force systems is that in which the energ y is minimized at fixed angular momentum. This is clearly the circular orbit. The velocity dependence of the frictional force is quite relevant, in particular as reference d against the velocity dispersion of the (undamped) motionin that central potential. Clea rly, under the action of such dissipative forces, a consequence of symmetry is that the flow in orbital shape (not size!) has two fixed points, circular orbits and strictly radial (in fall) orbits. Few physical problems have received more scrutiny than bounded o rbits in the two- and few- body system. Among these, the two-body Kepler problem is arguably the most experimentally relevant and best studied example, having been illumina ted by intense theoretical inquiry spanning hundreds of years leading to importan t insights even in relatively recent times[2,3,4,5,6,7,8,9,10,11,12,13,14,15, 16,17,18,19,20,21,22]. We do not present a systematic review or histiography of this celebr ated problem (though we thankfully acknowledge also [23, 24, 25, 28, 26, 27, 29, 30] which we have found quite useful for our study). We do not aim to contribute to t he vast literature on astrophysically and microphysically relevant models of friction in or bital problems (though the interested reader may find references [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43] a useful launching point for such review). Instead, theourpurposehereistoaccomodatefromthedynamic al symmetry group point-of-view the result that linear frictional damping (to leading or der) preserves theLinear frictional forces cause orbits to neither circulari ze nor precess 3 orbit’s shape. Although Hamiltonian systems may lose dynamical symm etry completely when dissipative forces are included, it can be shown that some stru cture may remain under a modified symplectic form. After a brief introduction to the m ethod by which Tarasov extends symplectic structure of Hamiltonian mechanics to dissipative systems, we apply it to the determination of the time averages of dynamical qu antities. Damping invariably introduces new dynamical timescales and the time averagin g we implement is overtimesshortcomparedwiththesetimescales(butstilllongcomp aredwiththeorbital timescales in the undamped problem). Tarasov’s construction reve als the relevance of the dynamical symmetry algebra to the damped Kepler problem. We then compare this aproach to the classic “variations of constan ts” method of orbit parameter evolution by describing an improvement that follows from our study. The elementary method can be generalized to non-Kepler homogene ous potentials and alsodetermines orbitalshapeevolutionforlinearlydampedKepler or bitsbeyondleading order. 2. Dynamical Symmetry and Tarasov’s Construction In the undamped Kepler problem the lack of precession is generally un derstood as a consequence of a dynamical symmetry, the celebrated so(4) symmetry formed from the two commuting so(3), one from the angular momentum /vectorL=/vector r×/vector pthe other from the Runge-Lenz vector, /vectorS=/vectorL×/vector p+k/vector r |/vector r|([6, 7, 8]) being the maximal set of local, algebraicallyindependent operatorsthatcommutewiththeHamilton ian,H=/vector p2 2+V(r), withV(r) =krαfork <0 andα=−1. (though see [22] for a more precise and general statement of the connection between algebra and orbits in a centr al field). {Li,Lj}= 2ǫijkLk{Li,Sj}= 2ǫijkSk{Si,Sj}=−2HǫijkSk (2.1) The length of /vectorSis proportional to the eccentricity (and points along the semi-majo r axis of the orbit, in the direction to the periastron from the focus) . Defining /vectorLand /vectorShas utility beyond their being constants in the 2-body Kepler problem , for example, parameterizing the secular evolution of orbits under various Hamilto nian perturbations [29, 44]. This so(4) is one of the maximal compact factor groups of the so(4,2) (the conformal group) extended symmetry formed by /vectorL,/vectorA,H, the generalization of the scaling operator R=/vector r·/vector pand the Virial operator V=/vector p2 2−r 2∂rV(r) ([17, 20]) The other central potential posessing an easily recognizable dyna mical symmetry is the multi-dimensional harmonic oscillator ( Vas given with k >0,α= 2). As is well known, the isotropic D-dimensional harmonic oscillator’s naive O(D) symmetry is part of a larger U(D) dynamical symmetry. For D= 2 harmonic oscillator, note that the U(2) symmetry does enlarge further to a so(3,2) when including R,Vand their generalization (the virial subalgebra Equation (4.16) through Equation (4.19) of each oscillator alone and closes to a sl(2,R) subgroup of the so(3,2)). Note further that it is this later algebra that is isomorphic to the dimensionally reduc edso(4,2) of the 3-d Kepler problem, by which we mean the reduction of that algeb ra to generatorsLinear frictional forces cause orbits to neither circulari ze nor precess 4 associatedwiththeorbitalplaneonly. These considerationscanals obeunderstoodfrom the KS construction[26, 27] of the Kepler problem, in which a four-d imensional isotropic harmonic oscillator is the starting point. In that construction the u(4) =su(4)×u(1) is, of itself, not preserved by the KS construction. Instead, it is t heu(2,2) subgroup of the four identical, independent oscillator’s sp(8,R) symmetry in which the overall u(1) can be isolated as the angular momentum constraint of the KS const ruction[28]. The residual symmetry su(2,2)∼so(4,2) is that of the 3-d Kepler problem. The analytical connection between the Kepler problem and the isotropic harmonic o scillator has deep historical roots, going back to Newton and Hooke (see [45] and ref erences therein). Finally, the geometric construction of the undamped Kepler problem as geodesic flow on (spatial) a 3-manifolds of constant curvature relates the so(4) dynamical symmetry to the isometry group generated by Killing vector fields on the spatia l slice[19, 21, 23]. These various connections between the Kepler problem and the isot ropic harmonic oscillator do not lead to a simple structural connection between the associated damped problems. To leading order in the damping, Kepler orbits subject to linear frictio nal force do not change shape or precess as they decay. It would be satisfying to understand this elementary result as a consequence of the preservation of the dy namical algebra under linear friction. Although this is reminiscent of the damped N-dimension al harmonic oscillator, there is no simple way to relate the damped problems. Since the subgroup associated with the shape and precession (through the /vectorS) is rank one it is suggestive that the entire group structure is preserved to leading order in th e linear friction. A recent paper by Tarasov[1] suggests a straightforward gener alization of the Poisson structure to systems with dissipative forces. There are m any other approaches to addressing structural questions of dissipative systems (for o ne example, see [46, 47]). We find the approach of [1] to be most useful for addressing quest ions of the dynamical symmetries that survive including dissipation. For completeness we n ow briefly review Tarasov’s construction, and apply it to dissipation in the central fie ld problem in the following section. To preserve as much of the algebraic structure as possible, Taras ov constructs a one-parameter family of two forms (that define a generalized Poiss on structure) that -in a sense- interpolate between different dampings. In the zero damp ing limit it smoothly matches onto the canonical symplectic form. Dimensionally, any dam ping parameter introduces a new time scale into the problem, thus this new interpolat ing two form must also be explicitely time-dependent. Tarasov requires this family of tw o forms to have the following useful properties (1) Non-degeneracy: The two form ω=ωij(t)dxi∧dxjis antisymmetric and non- degenerate along the entire flow. The xiare the 2 N(local) phase space co-ordinates. In positive terms, the inverse ωijωjk=δi jexists almost globally ‡. ‡Since we do not formulate this entirely in the exterior calculus, we mus t allow for higher codimension singularites that may not be resolvable in the dissipative system.Linear frictional forces cause orbits to neither circulari ze nor precess 5 (2) Jacobi Identity: the two form is used to define a new Poisson br acket{A,B}T= ωij∂iA∂jBthat forms an associative algebra. Explicitely it satisfies. {A,{B,C}T}T+{B,{C,A}T}T+{C,{A,B}T}T= 0 (2.2) Here we use the subscript ’ T’ to distinguish this bracket from the Poisson bracket of the undamped problem. (3) Derivation property of time translation: with respect to this ne w bracket the time derivative of the new Poisson bracket satisfies the derivation p roperty (also called the Liebnitz rule) d dt{A,B}T={dA dt,B}T+{A,dB dt}T (2.3) These requirements are remarkable for several reasons. First, property (1) indicates that (2) and (3) are possible. The deeper relevance of property ( 1) is that we can regard the two-form as (essentially a) global metric on the phase s pace. Property (2) indicates local mechanical observables in this ’dissipation deformed’ algebra form a lie algebra. Property (3) is key to the utility of Tarasov’s constructio n for understanding constants of motion in dissipative systems. It stipulates that time d evelopment in the dissipative system, while no longer just {,H}(or even {,H}T), must be compatible with the structure of the symplectic algebra in the new bracket and thus the (new) bracket of time independent quantities in the dissipative system are themselves time independent. Thus, just as in the Hamiltonian case, time independen t quantities form a closed subalgebra. Note that for a Hamiltonian system property ( 3) is automatic since in that case time translation is an inner automorphism of the sym plectic algebra. In a dissipative system by contrast the Hamiltonian is no longer the op erator of time translation, but, if Tarasov’s construction can be implemented, tim e translation is still anautomorphismofthealgebra,andassuchmayberegardedasan outerautomorphism. Finally, fromproperty (3)it follows after a brief calculation that the two-form ωmust be time idependent in the full dissipative system,dω dt= 0. In terms of symplectic geometry, this is metric compatibility of the dissipative flow. To proceed with the construction, consider the general flow ˙ xi=χi(/vector x,t). Again, these are not assumed to be Hamiltonian flows. Assuming property ( 1) and using ωij to form a bracket {A,B}T=ωij∂iA∂jB, property (2) leads to the condition ωim∂mωjk+ωjm∂mωki+ωkm∂mωij= 0. (2.4) Total time derivatives and derivatives along phase space directions do not commute in the flow, [d dt,∂i]A=−∂jA∂iχj. (2.5) Using this and the jacobi identity (2.2), one sees that property (3 ) implies a condition relating the form ωand the flow χi, ∂ωij(t) ∂t=∂iχj−∂jχiwhere χj=ωjk(t)χk(2.6)Linear frictional forces cause orbits to neither circulari ze nor precess 6 Givenχi, we proceed by solving (2.4) for an ωijthat satisfies (2.6). This completes Tarasov’s construction. Webreiflyoffer afewfurther remarkshelpful toorientthereader . First, inthemore familiar context of Hamiltonian flows, there ˙ xi=χi={xi,H}for a local function Hon the phase space. For this case we can compute in the Darboux fram e and learn that the usual symplectic form (automatically satisfying (2.4)) is a solution als o to (2.6) since the RHS in that case is zero. We recognize the RHS of (2.6) as exactly the obstruction to the flow, χibeing Hamiltonian. Conformal transformation of the two-form, ˜ ω= Ωω, where Ω is a a scalar function, can only relate two solutions of (2.4) and (2.6) IFF the Ω is a constant of the motion dΩ dt= 0. For in that case (2.6) indicates that ∂Ω ∂tωij=χj∂iΩ−χi∂jΩ (2.7) whereas (2.4) yields ωjk∂jΩ+ωkl∂jΩ+ωlj∂kΩ = 0 (2.8) so, contracting by χkand comparing with (2.7), we learn that Ω must be a constant of the motion. Thus, each solution is conformally unique. Wedonotknowwhatconditionson χileadtotheexistence ofevenonenon-singular simultaneous solutionωof (2.4) and (2.6). Tarasov[1] provides an explicit solution for a general Hamiltonian system ammended by a general linear frictional force. The general question of the existence of ω(t) for a more general χiis at this point unclear, but beyond the scope of this present effort. 3. Dynamical Symmetry in a Damped System Consider damped orbital motion in a central field; ˙/vector x=/vector p (3.1) ˙/vector p=−∂rV/vector x r−β(p)/vector p (3.2) withr=|/vector x|andV(r) the interparticle potential (throughout we take the reduced ma ss to be normalized to 1). The function β(p) is some general function parameterizing the speed dependence of the damping, and this form of the damping fun ction is the most general consistent with isotropy and homogeniety of the damping f orces. Note that we can understand this set as descending from a limit in which the centra l mass is very much larger than the orbital mass though, as in general, damping do es inextricably mix the center of mass motion and the relative motion. We call linear damp ing the choice ofβconstant. The Equation (2.6) takes the form, ∂ωxp(t) ∂t=∂x(ωpxχx)−∂p(ωxpχp) =∂pωxp′(β(p)p′) (3.3)Linear frictional forces cause orbits to neither circulari ze nor precess 7 Again, we do not know if solutions to Equation (3.3) exist and satisfy J acobi for every choice of β(p). However, for β(p) =const.there is a simple solution to Equation (3.3) that satisfies Jacobi[1], ωij(t) =eβtˆωij (3.4) where ˆωis the usual symplectic form of the undamped Kepler problem. Physic ally this corresponds to the uniform shrinkage of phase space volumes und er linear damping. Clearly, in going from {,}(Poisson bracket) to the new bracket {,}Tthe relations in Equation (2.1) gain a factor of e−βt. The algebra in the new bracket resulting from this simple rescaling is still so(4). The utility of this simple change to the algebra of Equation (2.1) (which was for the undamped system) is that it is now c ompatible with the evolution under Equation (3.1) and Equation (3.2) of the damped system. To see this in an example, take the first relation in Equation (2.1) and take th e (total) time derivative of both sides. Then noted{Li,Lj} dt=−2β(2ǫijkLk)/ne}ationslash= 2ǫijk˙Lk;i.e.the usual Poisson bracket is no longer compatible with time evolution. Duplicating the previous linefor{Li,Lj}T= 2e−βtǫijkLkonelearnsthat thisiscompatiblewiththeflowEquation (3.1)andEquation(3.2). Similarly, onemaycheckthatallthebracke tsinEquation(2.1) (after replacing {,}with{,}T) are as well. Also note that {Li,H}T= 0 ={Si,H}T, though since brackets with Hno longer delineate time evolution, these equations do not imply that /vectorLand/vectorSare constants of the motion in the dissipative system (also clear from Equation (3.10) below). The critique here is familiar to any attempt to reconcile symplectic str ucture and dissipation; fundamentally, Equation (3.1) and Equation (3.2) st ill treat xand pdifferently so that time evolution is no longer an element in the dynamica l algebra of {,}or{,}T. Torelaxthecategoryof’constantsofthemotion’sufficientlyford issipative systems, consider to what extent dynamical quantities averaged over some number of orbits change on a longer time scale, i.e.on a timescale relevant to the dissipation (note 1/β(p) is essentially that timescale). Let <>denote time averages over many orbits, O a classical observable, and suppose that ωis a solution to Equation (2.6) and the Jacobi identity for the system as in Equation (3.1) and Equation (3.2). In ge neral, <{O,H}T>=< ωxp(t)(˙x∂xO −(−˙p−β(p)p)∂pO)> (3.5) =< ω(t)/bracketleftBigdO dt−∂O ∂t/bracketrightBig +ωxp(t)β(p)p∂pO> (3.6) Note that sums are implied in the x,pindices of the ωxp(t), the new symplectic form. Above we have used isotropy to rewrite the sum in the first term in te rms of the (normalized) symplectic trace of ωxp(t) which we denote simply as ω(t). To show one intermediate step, integrating by parts and using Equation (3.3) we arrive at <{O,H}T>=1 T∆(ωTO)+< ωxαωpβ(∂αχβ−∂βχα+χl∂lωαβ)O +ωxpβ(p)p∂pO−ω∂O ∂t> (3.7)Linear frictional forces cause orbits to neither circulari ze nor precess 8 Where ∆( G) refers simply to the overall change of the quantity Gover time T. Finally, using the Equation (2.5) and the fact that ωxpsatisfies the Jacobi identity we reduce the above to <{O,H}T>=1 T∆(ωO)+<(ωxp′∂p′χp−ωpx′∂x′χx)O+ωxpβ(p)p∂pO−ω∂O ∂t>(3.8) We now specialize to vector fields of the general form Equation (3.1) and Equation (3.2) to find, 1 T∆(ωO) =<{O,H}T+ω∂O ∂t−ωxp(∂p(β(p))pO−β(p)p∂pO)> (3.9) and so making the RHS zero indicates conserved quantities in the non -Hamiltonian system. Again, this last result was derived for general β(p), which assumes only that the friction is isotropic and homogeneous. In the linear friction case β(p) =β=const. For that case, using O=L/ω2in the above equation implies that L/ωare constants of the motion in this system. Similarly, taking O=S/ωindicates that ∆ Sis proportional to (2β/vectorL/ω)×< ω/vector p >which, again, is zero to first order in β. This result then applied to the case of bounded Kepler orbits with linear damping indicates tha t the (orbit- averaged) Runge-Lenz vector, and thus the dynamical algebra o f the Kepler problem, is conserved to leading order in the linear friction coefficient. In elementary terms, although angular momentum /vectorLand/vectorSare constants in the Hamiltonian system for V(r)∼1 rthey evolve under linear damping of (3.1), (3.2) as, ˙/vectorL=−β/vectorL˙/vectorS=−2β/vectorL×/vector p. (3.10) Note that in the weak damping limit, since /vectorLis conserved to O(β0), the second equation time averages to −2< β/vector p >×</vectorL >. Thus, again we learn that if the damping were strictly linear ( βconstant) then since < /vector p >= 0, the time average of˙/vectorSis 0, again indicating that the eccentricity vector would be conserved to leadin g order. Note also that it is straightforward to integrate the /vectorLequation explicitely, finding /vectorL=/vectorL0e−βtthe initial condition /vectorL0being identified now a conserved quantity of the dissipative system. We use these results in the next section of this paper to ammend the ’textbook’ orbital secular evolution equations. 4. The Damped Kepler Problem The previous section suggests that (linear-) damped bounded Kep ler orbits shrink but retain their aspect ratio and do not precess to leading order in the d amping. It is well known that superlinear damping does lead to circularization whereas sublinear damping leads to infall orbits in the Kepler case. So far this begs the question s of whether this generalizes to other central field problems, and, if so, then at wha t order in the linear damping coefficient do orbits undergo shape and precessional chan ge. In this section we address both questions, first describing a problem that arises u sing a time-honored pertubative method for treating general perturbing forces in th e Kepler problem, and second, generalize the result of the preceeding section to a broad class of central fieldLinear frictional forces cause orbits to neither circulari ze nor precess 9 potentials. We then establish in precise terms the fate of Kepler orb its under linear damping. Consider the usual secular orbital evolution method (called “the va riations of constants”) most common in literature on cellestial mechanics, for example, in [48] (Chapter 11 Section 5, pg. 323, though see also the treatments o f non-linear friction in [49, 50, 51]). In the “variations of constants’ method, orbital r esponse to an applied force/vectorF=R/vector x+N/vectorL+B/vectorL×/vector x, in the orbit’s tilt Ω, the orbital plane’s axis, i, the eccentricity ǫ, the angle of the ascending node ωthe semi-major axis aand the period T= 2π/n(in their notation) evolve following[48], dΩ dt=nar√ 1−ǫ2Nsinu sini(4.1) di dt=nar√ 1−ǫ2Ncosu (4.2) dω dt=na2√ 1−ǫ2 ǫ[−Rcosθ+B(1+r P)sinθ]−cosidΩ dt(4.3) dǫ dt=na2√ 1−ǫ2[Rsinθ+B(cosθ+cosE)] (4.4) da dt= 2na2[Raǫ√ 1−ǫ2sinθ+Ba2 r√ 1−ǫ2] (4.5) and where dn dt=−3n 2ada dt(4.6) withu=θ+ωand for the unperturbed Kepler orbit,P r= 1 +ǫcosθ,Pis the latus rectum, and Eis the anomaly, i.e.r=P(1−ǫcosE). The central angle θis found via the usual definition of angular momentum. When we specialize thes e Kepler orbit evolution equations to the case of isotropic and homogeneous frict ion we learn that (see [48], Chapter 11, section 7 but using β(p)pforTin that reference) da dt= 2pa2β(p)p (4.7) dω dt=2sinθ ǫβ(p) (4.8) and dǫ dt= 2(cosθ+ǫ)β(p) (4.9) We can now specialize further to the marginal case, linear friction β(p) =β=const. To integratetheseequations, note r2dθ dt=L=L0e−βtand, intermsoftheforcecomponents, N= 0, and R=β(p)pcosυandB=β(p)psinυwhereυis the angle between the radius vector and the tangent to the orbit. That angle can be written usin g the parametericLinear frictional forces cause orbits to neither circulari ze nor precess 10 20 40 60 80Time /Minus0.02/Minus0.010.010.020.030.04Eccentricity Figure 1. The eccentricity in the actual damped Kepler problem (solid curve) compared with the eccentricity from (4.10) and (4.11) (dashed cur ve) versus time. Note that for the later the eccentricity can oscillates through zer o and can even, as in this case, asymptote to a negative value. form ofrin terms of the constants of the orbit and the angle θ, (sinυ=L/rpand cosυ=ǫsinθ/Lp) resulting in a self-contained pair of ODE’s in ǫ,θandt, dθ dt=e+3βt L3 0(1+ǫcosθ)2(4.10) dǫ dt=−2β(cosθ+ǫ) (4.11) If we integrate these to leading order in βonly (by, for example, using the first equation to eliminate the time derivative to leading order in β) we do indeed find that the eccentricity is an orbit-averaged constant of the motion. But diffic ulty arises when we try to understand these equations beyond leading order in the dam ping, as a direct numerical integration of the equation set reveals (Figure 1). For a broad set of initial angles and small initial eccentricities, the ǫpasses through zero and goes negative. For comparison, the eccentricity (i.e. the square root of the lengt h of the /vectorSvector) computed by numerical integration of the original equations of mot ion for precisely the same mechanical parameters and initial conditions is included on that figure. Even if one only wanted to assign importance to the asymptotic chan ge in the eccentricity, that asymptotic change from integrating the equat ion pair (4.10) , (4.11) does not scale correctly with the damping coefficient, as may be chec ked numerically (see [48] for further admonisions against using the “variations of c onstants” methodLinear frictional forces cause orbits to neither circulari ze nor precess 11 over long timescales). Clearly the “variations of constants” metho d at higher orders in the evolution leads to unphysical results at short and long timesca les. The fault is traceabletothefactthatinhigherorderthereare β-(thedampingcoefficient)dependent terms in the orbit shape whose contributions are ignored substitut ing forrusing the undamped Kepler shape of the ellipse. This substitution is however ine ctricably part of the“variationofconstants” method. Tofurtherclarifythisprob lemwiththe“variations of constants” method, it is not due to some ambiguity in the eccentr icity of a non- closed orbit, since eccentricity itself, rendered as the length of th e/vectorSvector, has a local definition. Algebraically, withthisdefinitionoftheeccentricity, note thatǫ2−2L2U=k2 in the 1/rpotential even under arbitrary damping . A more useful algebraically identical form isǫ2= 4V2r2−2R2H, from which, since His negative for any damping function on a bounded orbit, we see immediately that ǫ2is bounded away from zero. We now, in two parts, describe an approach emphasising the secular evolution of the dynamical symmetry, that addresses this mismatch with the us ual “variation of constants” method. For simplicity we focus in the main on potentials w ith fixed scaling wieghtα, deined through V(r) =krα. Orbits in any central potential are characterized by a fixed orbital plane and a single dimensionless parameter, the rat iod/cof the perihelion distance dto the aphelion distance c. LetLdenote the angular momentum so thatVeff(r) =L2 2r2+V(r) is the effective potential. Then from Veff(c) =U=Veff(d) whereUis the total energy for a V(r) of a fixed α, we have, U k=cα+2−dα+2 c2−d2L2 2k=cα−dα c2−d2c2d2(4.12) that then can be reduced to a ’dispersion relation’ between UandL, Uα+2 k2L2α=f(d/c) (4.13) wherefin this case is a monotonic function on [0 ,1]. Note also that f(x) =f(1/x). We calld/cthe aspect ratio of the orbit (related to the eccentricity in the α=−1 case). Thus in leading order (only) in the damping we think of the RHS as a func tion of the orbital eccentricity only. In applications, the differential form of ( 4.13) is particularly useful, /bracketleftBig (α+2)δU δL−2αU L/bracketrightBigδL U=f′ fδ(d/c) (4.14) Equations of this sort are often written down when referring to th e secular evolution of orbital system (see for example [52] and references therein). As before consider further onlydamping forcesthatareisotropicandhomogeneous; theycan bewrittenintheform from the previous section, /vectorFdrag=−β(p)/vector p. (to simplify notation we henceforth drop the vector symbol over the pdenoting by pboth|p|and/vector p, unambigious by context). In the limit of weak damping we expect Lto be approximately constant so that, time averaging, we arrive at < δL >=−< β(p)> Lto leading order in β(p). Note also that δU=−β(p)p2to leading order in β. Notethatthetimederivativeof RisthesumofaPoissonbracket withHplusaterm probprtional to β(see Equation (4.16)). This is,dR dt= 2V+O(β) which, averaged overLinear frictional forces cause orbits to neither circulari ze nor precess 12 bounded orbits, indicates (the virial theorem) that <V>=O(β). Thus for V(r) =krα this implies that < p2>=α < V > +O(β) so that < U >=α+2 2< V >+O(β), which to leading order in βin Equation (4.14) indicates −2α < p2>/parenleftBig < β(p)p2>−< β(p)>< p2>/parenrightBig =f′ fδ(d/c) (4.15) Thus restricted to linear damping ( βconstant), but for any α, the averages in (4.15) factorize trivially and the aspect ratio is unchanged to leading order under linear damping. The Equation (4.15) also indicates that this will, in general, no t be the case for a velocity dependent damping coefficient. Although for pot entials with a fixed scaling exponent αthere is but one dimensionless parameter (See LHS Equation (4.13)) , the introduction of the damping coefficient βintroduces new length and time scales, indicating that the orbital aspect ratio d/cmay be a function of βand time. The fact thatβis time odd does apparently not preclude its inclusion to linear order in t he orbital aspect ratio in general. Thus, we repeat, the conclusion th at for any monomial potentialslineardampingpreserves theorbitalshapeisnotaconse quence ofdimensional analysis and discrete symmetries. As a final check, note that relation Equation (4.15) and Equation (3 .9) are both consistent with the attractors of the secular flow in the orbital sh ape. For circular orbits p2isaconstant ofthemotion(againtoleading order in β)andthustheLHSofEquation (4.15) is zero, as expected by symmetry. Note that in contrast to (4.15) in Equation (3.9) the change in the eccentricity is proportional to the eccentr icity for any β(p), and since the eccentricity vanishes in this limit its orbit-averaged change by (3.9) does as well. Also the strictly radial infall orbit limit is one in which the inner radius ,d→0, and so the LHS of Equation (4.15) being non-zero in this limit looks incon clusive. But, by the definition of fvia Equation (4.13) we see that in this limit f→0 orf→ ∞ depending on the sign of α. Thus, by (4.12), L= 0 and remain zero for any β(p). In Tarasov’s formulation, since (3.9) isfully vector covariant forisotr opic andhomogeneous (but otherwise arbitrary β(p)) the change in the /vectorSmust be along the vector itself for the radial infall case. Furthermore, as indicated in the discussion follow ing (3.9), the change ∆/vectorSis linear in /vectorL(for any β(p)) which vanishes in the radial infall case. In summary, both prescriptions indicate that circular orbits and radial infall mus t satisfy <˙/vectorS >= 0 for any damping function as expected on the grounds by symmetry. Furthermore, it is straightforward to go further and perturbat ively show using (4.15) andthe equations of motionthat β=const.istheonlyshape-preserving damping functionfortheKeplerpotential( α=−1). Theresultisclearlycommontoallmonomial central potentials only, as it is straightforward to demonstrate a counterexample in a more complicated potential. This is due to the fact that there are no additional length scales in the potential and is not the case with other potentials, suc h as the effective potential in General Relativity (where the Schwarzschild radius aris es asa second length scale in the potential). Returning to the rather general statement (3.9), in the Tarasov formulation, the explicit time dependence of a candidate constant of motion Ogives a second term whichLinear frictional forces cause orbits to neither circulari ze nor precess 13 cancels the last two terms. If the operator has a fixed momentum s caling weight (for example, Lis weight 1 and Sis essentially wieght 2), the last two terms will be of that same scaling weight only for the case of linear friction, β(p) =const.Note that this argument does not rule out the existence of additional consta nts of the motion in the dissipative system that scale to zero as one goes to the Hamilto nian limit. The argument does, however, certify that in the case of linear fric tion the original Hamiltonian symmetries do survive to leading order in that friction. Havingshownthatlinearfrictionpreserves theeccentricity tolead ingorderbegsthe question of what happens in higher order in the damping. In the spirit of the discussion after (4.13) where the Virial played a key role, consider the time evo lution of that part of the dynamical algebra ˙R={R,H}−β(p)R= 2V −β(p)R (4.16) ˙V={V,H}−2β(p)(2V −H) =−1 r(∂rV+1 2∂r(r∂rV))R−2β(p)(2V −H) (4.17) ˙H=−2β(p)(2V −H) (4.18) and for completeness, we have {R,V}=H+V −V+r 2∂rV+r 2∂r(r∂rV) (4.19) To orient the reader to the content of these, first note the Hamilt onian limit ( i.e. β(p)→0 limit) for the Kepler case ( α=−1), both <V>→0 and<R/r3>→0 as expected. We thus expect both of these time averages to be at least proportional to some positive power of β. Now, in the abscence of damping Vis time even and Ris time odd. Formally, taking βto be time odd preserves this discrete symmetry of the above evolution equations. Since we expect the <V>and<R>to be analytic functions of β, it must thus be that <V>vanishes quadratically as β→0. An elementary argument now certifies that the <V>must be nonpositive in the damped system. Take β(p) =βa constant. Consider the radial component of the velocity, R/r. It must average to zero in the β→0 limit. Since the damped orbit must shrink, we thus expect <R/r >∼<−Cβ >for some positive quantity C(a function of the other orbital parameters, etc.). But now take the evolution e quation (4.16) divide byrand time average. Clearly, integrating by parts, <˙R/r >=−<R/r2˙r >= −<R2/r3>implying that the time average <2V/r−βR/r >must also be strictly negative. But since <R/r >must already be negative, the <V>must also be strictly negative in the damped system. Specializing to Kepler ( α=−1), differentiating ǫ2=|/vectorS|2in time and applying the equations of motion of the system with friction, we learn thatd dtǫ2=−8βL2V. Using the fact that <V>is negative and order β2and integrating both sides, we learn that the asymptotic change in the eccentricity to leading order is positive and also of order β2(note the integral itself scales as 1 /β). Furthermore, since these are exact evolution equations, we have shown that the integration is well behaved thro ughout. Thus linear friction causes Kepler orbits to become more eccentric by a fixed am ount that scales with the square of the linear damping coefficient.Linear frictional forces cause orbits to neither circulari ze nor precess 14 5. Conclusion TypicallyHamiltoniansymmetrieslosethierrelevancetothegeometry ofthetrajectories when damping forces are added to the Hamiltonian system. If the da mping is weak, homogeneous and isotropic, then for linear damping in monomial pote ntials, we have shown that orbit-averaged shape is stationary. This can underst ood most easily through Tarasov’s generalization of conserved quantities from the Hamilton ian context to the non-Hamiltonian setting. This approach also quantifies in precise ana lytic terms the fate and subsequent utility of the dynamical symmetry algebra in th e associated non- Hamiltonian system. There are three main frameworks for understanding orbital motio n in a perturbed central field. The first is directly from the equations of motion; this admits straightforward generalization to the non-Hamiltonian case but so mewhat obscures the structure and fate of the dynamical symmetry group. The secon d, namely the KS construction, embedstheKeplerorbitprobleminthehigherdimensio nal setofharmonic oscillators with constraints; this illuminates the dynamical symmetry group but does not seem to readily admit a generalization to the non-Hamiltonian syst em. Lastly, the geometrical approach, namely that which associates the Keple r Hamilton equations to geodesic flow on manifolds of constant curvature, also illuminates the dynamical symmetry groupwhile making thegeneralizationtothenon-Hamiltonia ncasesomewhat unclear. In light of these difficulties, we used Tarasov’s framework (and applie d to the damped central field problem here) for extending Poisson symmetr ies to dissipative systems, emphasising its utility in making crisp connections between d ynamics, algebra and the geometric character of the solutions. Finally, the dynamica l algebra remains whole in first order in the linear dissipative system, but flow at higher o rder is not trivial. The secular perturbative method “variation of constants” is not adequate to explain this, however an elementary method based on the Virial suba lgebra explains the change in the shape of kepler orbits in higher order in linear damping. Acknowledgments This work was supported in part by the National Science Foundation through a grant to the Institute of Theoretical Atomic and Molecular Physics at Har vard University and the Smithsonian Astrophysical Observatory, where this work was begun and by a fellowship from the Radcliffe Institute for Advanced Studies where this work was concluded. It is a pleasure to acknowledge useful discussions with H arvard-Smithsonian Center for Astrophysics personell Mike Lecar, Hosein Sadgehpou r, Thomas Pohl and Matt Holman.Linear frictional forces cause orbits to neither circulari ze nor precess 15 Bibliography [1] Tarasov, V E 2005 J. Phys. A 38#10/11 2145 [2] Ermanno, G J 1710 G. Lett. Ital. 2447 [3] Hermann, J 1710 Hist. Acad. R. Sci: M´ em. Math. Phys. 519 [4] Bernoulli, J 1710 Hist. Acad. R. Sci: M´ em. Math. Phys. 521 [5] Laplace, P S 1827 A Treatise on Celestial Mechanics (Dublin) [6] Runge, C 1919 Vektoranalysis 1(Hirzel, Leipzig) [7] Lenz, W 1924 Z. Phys. 24197 [8] Pauli, W 1926 Z. Phys. 36336 [9] Hulth´ en, L 1933 Z. Phys. 8621 [10] Fock V 1935 Z. Phys. 98145 [11] Bargmann V 1936 Z. Phys99576 [12] Moser J 1970 Commun. Pure Appl. Math. 23609 [13] Belbruno E A 1977 Celest. Mech. 15467 [14] Ligon T and Schaaf M 1976 Rep. Math. Phys. 9281 [15] Cushman R H and J. J. Duistermaat J J 1997 Commun. Pure Appl. Math. 50773 [16] Stehle P and Han M Y 1967 Phys. Rev. 159#5 1077 [17] Iosifescu M and Scutaru H 1980 J. Math. Phys. 21(8) 2033 [18] Gy¨ orgyi G 1968 Nuovo Cimento ,A 53717 and 1969 62449 ; also 1969 Magyar. Fiz. Foly. 17, 39 and 1972 20163 [19] Guillemin V and Steinberg S 1990 Am. Math. Soc. Colloquium Publications 42 [20] Cordani B 2003 The Kepler Problem (Birkh¨ auser verlag (Basel/Boston)) [21] Keane A J Barrett R. K. and Simmons J F L 2000 J. Math Phys 41#12 8108 [22] Fradkin D M 1967 Prog. of Theor. Phys. 37798 [23] Keane A J 2002 J. Phys. A: Math. Gen. 358083 [24] Goldstein, H. 1980 Classical Mechanics (Reading, MA. 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C. 1985 Orbit Circularization Time in Bina ry Stellar Systems, 1985 IAUS..105..411M [34] Mardling R A 1993 The Role of Chaos in the Circularization of Tidal Ca pture Binaries II Long Time Evolution Preprint astro-ph/9312054 [35] Kozai Y 1962 The Astronomical Journal 67#9 591 [36] Innanen K A Zheng J Q Mikkola S and Valtonen M J 1997 The Astronomical Journal 113#5 1915 [37] Apostolatos T A 1994 Topics in General Relativity: Naked Singular ities, and Theoretical Aspects of Gravitational Waves from Merging Compact Binaries ThesisDissertation Abstracts International 56-01 [38] Dotti M Colpi M and Haardt F 2006 Mon. Not. R. Astron. Soc. 367103Linear frictional forces cause orbits to neither circulari ze nor precess 16 [39] Glampedakis K Hughes S A and Kennefick D 2002 Phys. Rev. D 66064005 [40] Apostolakis T Kennefick D Ori A and Poisson E 1993 Phys. Rev. D 47#12 5376 [41] PressWHPriceRHTeukolskySALightmanAP1975 Problem Book in Relativity and Gravitation (Princeton, Princeton University Press) ISBN-13 978-06910816 25 [42] Schutz B F 1985 A First Course In General Relativity (Cambridge, Cambridge University Press) ISBN-13 978-0521277037 [43] Bellomo P and Stroud C R , Jr. Classical evolution of quantum elliptic statesPreprint physics/9812022 [44] Morehead,J 2005 Am. J. Phys. ,73(3) 234 [45] Grant A K Rosner J L 1994 Am. J. Phys. 62(4) 310 [46] Sergi A Ferrario M 2001 Phys. Rev. E64056125 [47] Sergi A 2003 Phys. Rev. E67021101, 2004 Phys. Rev. E69021109,Phys. Rev. E72066125 (2005). [48] Danby J. M. A. 1988 Fundamentals of Celestial Mechanics 2nd Ed. (Richmond, Va. , Willman-Bell Inc.) [49] Watson J. C. 1964 Theoretical Astronomy (NY, Dover Press) [50] Murray, C. D. and Dermott S. F. 2000 Solar System Dynamics (Cambridge, Cambridge Univ. 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2007-08-28

For the undamped Kepler potential the lack of precession has historically been understood in terms of the Runge-Lenz symmetry. For the damped Kepler problem this result may be understood in terms of the generalization of Poisson structure to damped systems suggested recently by Tarasov[1]. In this generalized algebraic structure the orbit-averaged Runge-Lenz vector remains a constant in the linearly damped Kepler problem to leading order in the damping coe

Linear frictional forces cause orbits to neither circularize nor precess

0708.3827v3

arXiv:0708.4164v1 [cs.IT] 30 Aug 2007Asymptotic improvement of the Gilbert-Varshamov bound for linear codes Philippe Gaborit∗Gilles Z´ emor† August 29, 2007 Abstract The Gilbert-Varshamov bound states that the maximum size A2(n,d) of a binary code of length nand minimum distance dsatisfiesA2(n,d)≥ 2n/V(n,d−1) where V(n,d) =/summationtextd i=0/parenleftbign i/parenrightbig stands for the volume of a Ham- ming ball of radius d. Recently Jiang and Vardy showed that for binary non-linear codes this bound can be improved to A2(n,d)≥cn2n V(n,d−1) forca constant and d/n≤0.499. In this paper we show that certain asymptotic families of linearbinary [ n,n/2]randomdouble circulantcodes satisfy the same improved Gilbert-Varshamov bound. These result s were partially presented at ISIT 2006 [3]. Index terms: Double circulant codes, Gilbert-Varshamov bound, linear codes, random coding. 1 Introduction The Gilbert-Varshamov bound asserts that the maximum size Aq(n,d) of a q-ary code of length nand minimum Hamming distance dsatisfies Aq(n,d)≥qn /summationtextd−1 i=0/parenleftbign i/parenrightbig (q−1)i. (1) This result is certainly one of the most well-known in coding theory, it was originally stated in 1952 by Gilbert [5] and improved by Vars hamov in [15]. In 1982 Tsfasman, Vladuts and Zink [14] improved the GV bound on the number of codewords by an exponential factor in the block length, bu t this spectacular result only holds for some classes of non-binary codes. Rece ntly Jiang and ∗XLIM, Universit´ e de Limoges, 123, Av. Albert Thomas, 87000 Limoges, France. gaborit@unilim.fr †Universit´ e de Bordeaux 1, Institut de Math´ ematiques de Bo rdeaux, 351 cours de la Lib´ eration, 33405 Talence. zemor@math.u-bordeaux1.fr 1Vardy [6] improved the GV bound for non-linear binary codes b y a linear factor in the block length nto A2(n,d)≥cn2n V(n,d−1), (2) ford/n≤0.499, for a constant cthat depends only on the ratio d/nand whereV(n,d) =/summationtextd i=0/parenleftbign i/parenrightbig stands for the volume of a Hamming ball of radius d. This new bound asymptotically surpasses previous improvem ents of the binary Gilbert-Varshamov bound which only managed to multiply the right hand side in (1) by a constant (see [6] for references). The method used by Jiang and Vardy relies on a graph-theoretic framework and more specifi cally on locally sparse graphs which are used to yield families of non-linear codes (their result was later slighlty improved in [16]). In this paper we also im prove on the the Gilbert-Varshamov bound by a linear factor in the block leng th but for linear codes, thereby solving one of the open problems of [6]. The me thod we use is not related to graph theory and relies on double circulant ra ndom codes. Double circulant codes are [2 n,n] codes which are stable under the action of permutations composed of two circular permutations of or dernacting si- multaneously on two differents halves of the coordinate set. T hese codes can also be seen as quasi-cyclic codes, a natural generalizatio n of cyclic codes [13]. Their study started in 1969 in [8] and since they gave some ver y good codes it was natural to wonder whether they could be made to satisfy the Gilbert- Varshamov bound. A first step in that direction was made by Che n, Peterson and Weldon in [1] who prove that when 2 is a primitive root of th e ringZ/pZfor pa prime, double circulant [2 p,p] random codes satisfy the Gilbert-Varshamov bound; unfortunately it is still unknown (this is Artin’s ce lebrated conjecture, 1927) whether an infinity of such pexists. Later Kasami [9], building on this idea, extended the result of [1] to the case of powers of such p, and obtained a bound which is worse than the Gilbert-Varshamov bound by an exponential factor in the block length (though a very small one). Later Ka sami’s work was generalized to other cases in [7, 11, 12], and, in particu lar in [2], bounds were proven for certain classes of quasi-cyclic codes that a re worse than the Gilbert-Varshamov bound only by a subexponential factor in the block length. In this paper, building anew on Kasami’s idea we prove, by usi ng a proba- bilistic approach, that randomly chosen double circulant c odes not only satisfy the Gilbert-Varshamov bound with high probability, but als o the same linear improvement as that of Jiang and Vardy (2). The paper is organized as follows: in Section 2 we cover the ma in ideas involved. We start by recalling the probabilistic method fo r deriving lower boundsontheminimumdistanceoflinearcodes(section 2.0) , thenweintroduce double circulant codes in section 2.1 and derive (5) an upper bound on the probability that a random double circulant code contains a n on-zero vector of weight not more than a given w. In section 2.2 we study the probability that a given vector belongs to a randomly chosen double circulant code. Finally in section 2.3 we derive our improved lower bound on the minimum distance in the simple case when the codelength is 2 pand 2 is a primitive root of Z/pZ: the result is given in Theorem 4. 2in Section 3, we develop our method in the more complicated ca se of block- lengths 2 pm,pa “Kasami” prime, in order to obtain an infinite family of doub le circulant codes with an improved minimum distance. Section 3.1 starts by giv- ingan informal sketch of the content of section 3, which is in tended to give some guidance to the reader and discuss the technical issues invo lved. Section 3.2 shows how to derive our main result, which is Theorem 8, from a proposition on the weight distribution of a certain class of cyclic codes . Finally section 3.3 is devoted to a proof of this last proposition. Section 4 concludes by some comments and side results. 2 Overview of the method, the simple cases 2.0 The Gilbert Varshamov bound for linear codes and its im- provement To put the rest of the paper into perspective and introduce no tation, let us recall how the probabilistic method derives the Gilbert Var shamov bound for linear codes. Rather than bounding the code size from below b y a function of the minimum distance, as in (2), we fix a lower bound on the code rate and find a lower bound on the minimum distance. We limit ourselves to the rate 1/2 case because it will be our main object of study. LetCrandbethe randomcode of length 2 nand dimension k≥nobtained by choosing randomly and uniformly a n×2nparity-check matrix in {0,1}n×2n. The probability that a given nonzero vector x= (x1...x2n) is a codeword is clearly 1 /2n. Letwbe a positive number, not necessarily an integer. We are interested in the random variable X(w) equal to the number of nonzero codewords of Crandof weight not more than w. In other words X(w) =/summationdisplay x∈B2n(w)Xx (3) whereB2n(w) denotes the set of nonzero vectors xofV2n={0,1}2nof weight at most w, andXxis the Bernoulli random variable equal to 1 if x∈Crand and equal to zero otherwise. Now whenever we prove that the pr obability P(X(w)>0) is less than 1, we prove the existence of a [2 n,k,d] code with k≥nandd > w. Since the variable X(w) is integer valued we have P(X(w)>0)≤E[X(w)] =/summationdisplay x∈B2n(w)E[Xx] =|B2n(w)|P(x∈Crand) =|B2n(w)|1 2n. Hence, for every positive integers nandwsatisfying |B2n(w)|<2nthere exists a linear code of parameters [2 n,n,d > w ]. Reworded, we have the following lower bound on d, essentially equivalent to (1). Theorem 1 (GV bound) For every positive integer nthere exists a linear code of parameters [2n,n,d]satisfying |B2n(d)| ≥2n. 3In the present paper we shall prove : Theorem 2 There exists a positive constant band an infinite sequence of in- tegersnand[2n,n,d]linear codes satisfying |B2n(d)| ≥bn2n. This result, equivalent to (2) for rate 1 /2, will be obtained by again choosing random matrices, but from a restricted class, namely the set of parity-check matrices of double circulant codes. 2.1 Double circulant codes A binary double circulant code is a [2n,n] linear code Cwith a parity-check matrix of the form H= [In|A] whereInis then×nidentity matrix and A= a0an−1... a 1 a1a0... a 2 a2a1... a 3 .................... an−1an−2... a 0 . There is a natural action of the group Z/nZon the space V2n={0,1}2nof vectorsx= (x1...xn,xn+1...x2n) namely, Z/nZ×V2n→V2n (j,x)/ma√sto→j·x where 1·x= (xn,x1...xn−1,x2n,xn+1,...x2n−1) andj·x= (j−1)·(1·x). The double circulant code Cis clearly invariant under this group action. Consider now Cto be the random code Crandobtained by choosing the vector a= (a0...an−1) randomly and uniformly in {0,1}n. As before, we are interested in the random variable X(w) defined by (3) and equal to the number of nonzero codewords of Crandof weight not more than w. We are interested in the maximum value of wfor which we can claim that P(X(w)>0)<1, for this will prove the existence of codes of parameters [2n,n,d > w ]. The core remark is now that, if y=j·x, then Xy=Xx whereXx(Xx) is the Bernoulli random variable equal to 1 if x∈Crand(y∈ Crand) and equal to zero otherwise. Let now B′ 2n(w) be a set of representatives of the orbits of the elements of B2n(w), i.e. for any x∈B2n(w),|{j·x,j∈ Z/nZ} ∩B′ 2n(w)|= 1. We clearly have X(w)>0 if and only if X′(w)>0 where X′(w) =/summationdisplay x∈B′ 2n(w)Xx. 4Denote by ℓ(x) the length (size) of the orbit of x, i.e.ℓ(x) = #{j·x,j∈Z/nZ}. We have X′(w) =/summationdisplay x∈B2n(w)Xx ℓ(x)(4) By writing P( X(w)>0) = P(X′(w)>0)≤E[X′(w)], together with (4) we obtain P(X(w)>0)≤/summationdisplay d|n/summationdisplay wt(x)≤w ℓ(x)=dE[Xx] d. (5) Suppose in particular that nis a prime, in that case orbits are of size 1 or n, and ifw < nthen clearly the orbit of xhas sizenfor anyx∈B2n(w), so that (5) becomes P(X(w)>0)≤E[X(w)]/n. If we can manage to prove that E[X(w)]≤ |B2n(w)|c 2n(6) for constant c, then we will have proved the existence of double circulant c odes of parameters [2 n,n,d > w ], for any wsuch that |B2n(w)|<1 cn2n. 2.2 The behaviour of P(x∈Crand) To prove equality (6) we need to study carefully the quantiti es E[Xx], for x∈B2n(w), since E[X(w)] =/summationdisplay x∈B2n(w)E[Xx]. Forx∈V2n, let us write x= (xL,xR) withxL,xR∈ {0,1}n. Consider the syndrome function σ σ:V2n→Vn x/ma√sto→σ(x) =xtH=σL(x)+σR(x) whereσL(x) =xLandσR(x) =xRtA. Foranybinaryvectoroflength n,u= (u0,...,u n−1), denoteby u(Z) =u0+ u1Z+···+un−1Zn−1its polynomial representation in the ring F2[Z]/(Zn+1). For any u∈Vn, letC(u) denote the cyclic code of length ngenerated by the polynomial representation u(Z) ofu. SinceσR(x) has polynomial representa- tion equal to xR(Z)a(Z), we obtain easily Lemma 3 The right syndrome σR(x)of any given x∈V2nis uniformly dis- tributed in the cyclic code C(xR). Therefore, the probability P(x∈Crand)that xis a codeword of the random code Crandis •P(x∈Crand) = 1/|C(xR)|ifxL∈C(xR), •P(x∈Crand) = 0 ifxL/ne}ationslash∈C(xR). 52.3 The case nprime and 2primitive modulo n Ifnis prime and 2 is primitive modulo nthen, over F2[Z], the factorization of Zn+1 into irreducible polynomials is Zn+1 = (1+ Z)(1+Z+Z2+···+Zn−1) and there is only one non-trivial cyclic code of length n, namely the [ n,n− 1,2] even-weight code. Therefore P( X(w)>0) = P(X′(w)>0)≤E[X′(w)] together with (4) and Lemma 3 give P(X(w)>0)≤/summationdisplay wt(xL)+wt(xR)≤w wt(xR) odd1 n2n+/summationdisplay wt(xL)+wt(xR)≤w wt(xR) even wt(xL) even1 n2n−1(7) P(X(w)>0)≤2|B2n(w)|1 n2n. We therefore have the following result: Theorem 4 Ifpis prime and 2is primitive modulo p, then there exist double circulant codes of parameters [2p,p,d > w ]for any positive number wsuch that 2|B2p(w)|< p2p. Unfortunately, it is not known (though it is conjectured) wh ether there existsaninfinitefamilyofprimes pforwhich2isprimitivemodulo p. Therefore, to obtain Theorem 2 we will envisage cases when nis non-prime. This will involve two technical difficulties, namely dealing with non- trivial divisors dof nin (5), and non-trivial cyclic codes C(xR) of length nin Lemma 3. 3 An infinite family of double circulant codes 3.1 Preview In this section we will study the behaviour of the minimum dis tance of random double circulant codes for the infinite sequences of blockle ngths 2nintroduced by Kasami : we will have n=pmfor suitably chosen p. We will first specialise inequality (5) to this case, for which all the possible orbit sizesℓare powers ofp,ps,s≤m. Applying Lemma 3 will lead us to an upper bound (13) on P(X(w)>0) that involves the weight distributions of the cyclic code s of length n. This upper bound can be essentially thought of as the same as (7), plus a number of parasite terms involving all vectors x= (xL,xR) ofB2n(w) for which bothxLandxRare codewords of some cyclic code of length nthat is neither the whole space {0,1}nnor the [ n,n−1,2] even-weight subcode. The problem at hand is to control the parasite terms so that they do not pol lute too much the main term i.e. the right hand side of (7). To do this, the cr ucial part will be to bound from above with enough precision terms of the form /summationdisplay i+j≤wAi(C)Aj(C)1 |C|(8) 6whereCis a cyclic code of length nandAi(C) is the number of codewords of weighti. In section 3.2 we shall state such an upper bound, namely Pro posi- tion 5, and show how it leads to the desired result which will b e embodied by Theorem 8. Section 3.3 will then be devoted to proving Proposition 5. It is not easy in general to estimate the weight distribution of cyclic cod es that don’t have extra properties, but it turns out that for these particular code lengths of the formn=pm, all cyclic codes Chave a special degenerate structure. Either C consists of a collection of vectors of the form ( x,x,...,x ) wherexis a subvector oflength n/pandisrepeated ptimes, or Cisthedualofsuchacode. Section 3.2 will have reduced the problem to the latter class of cyclic co des only. Ideally, we would like to claim that the cyclic codes Chave a binomial distribution of weights, i.e. Ai(C)≈|C| 2n/parenleftbign i/parenrightbig , however this is not true, the cyclic codes C have many more low-weight codewords than would be dictated b y the binomial distribution. The problem of the unbalanced couples ( i,j), (ismall and jlarge or vice versa) in the sum (8) is therefore dealt with by the tri vial upper bound Ai(C)≤/parenleftbign i/parenrightbig : Lemma 11 will show that these terms account for a sufficiently small fraction of |B2n(w)|/2n. Lemma 10 is the central result of section 3.3 which gives a more refined upper bound on Ai(C) foriwell enough separated from 0, i.e. i≥κnfor constant positive κ. Fortunately, we do not need Ai(C) to be too close to the binomial distribution, and the cruder u pper bound of Lemma 11 will suffice to derive Proposition 5. 3.2 Reducing the problem to the study of the weight distribu- tion of certain cyclic codes Following Kasami [9], let us consider nof the form n=pmwhere 2 is primi- tive modulo pand 2p−1/ne}ationslash= 1 mod p2. It will be implicit that all the primes p considered in the remainder of section 3 will satisfy this pr operty. Let us also suppose m≥2, since the case m= 1 is covered by Theorem 4. It is known [9] that the irreducible factors of Zn+ 1 inF2[Z] are 1 + Z together with all the polynomials of the form 1+Q(Z)+Q(Z)2+···Q(Z)p−1(9) forQ(Z) =Z,=Zp,Zp2,...,Zpm−1. Sincenis a prime power, (5) gets rewritten through Lemma 3 as: P(X(w)>0)≤m/summationdisplay s=1/summationdisplay wt(x)≤w ℓ(x)=ps C(xL)⊂C(xR)1 ps|C(xR)|(10) Note that x∈V2nhas orbit length ℓ(x)< nif and only if both xLandxR are made up of psuccessive identical subvectors of length n/p. Equivalently xL andxReach belong to the cyclic code generated by the polynomial Pn(Z) = 1+Zn/p+Z2n/p+···+Z(p−1)n/p. (11) 7LetCndenote the set of those cyclic codes of length nwhose generator poly- nomial is nota multiple of Pn(Z). All the other cyclic codes of length nare obtained by duplicating ptimes some cyclic code of length n/p. Therefore, for s=m, the inner sum in (10) can be bounded from above by: /summationdisplay C∈Cn/summationdisplay i+j≤wAi(C)Aj(C)1 n|C|(12) whereAi(C) denotes the number of codewords of Cof weight i. Applying (12) recursively, we obtain from (10) P(X(w)>0)≤m−1/summationdisplay s=0/summationdisplay C∈Cn/ps/summationdisplay i+j≤w/psAi(C)Aj(C)1 |C|n/ps.(13) We now proceed to evaluate the righthandside of (13). The mos t technical part of our proof of Theorem 2 is contained in the following Pr oposition. Proposition 5 There exist positive constants q,K,c1andγ <1such that, for anyn=pmwithp≥q, we have |B2n(2Kn)| ≤2nand for any positive real numberw,K≤w/2n≤1/4, and for any cyclic code CofCn, we have /summationdisplay i+j≤wAi(C)Aj(C)1 |C|≤c1|B2n(w)| 2nγn−dimC. Suitable numerical values of the constants are q= 143,K= 0.1,γ= 1/21/5, c1= 26/5. Before proving Proposition 5, let us derive the consequence s on the proba- bility P(X(w)>0). That will lead us to our main result, namely Theorem 8, the consequence of which is Theorem 2. We have: Lemma 6 There exists a constant c2such that, for any n=pm,p > q, and for anyK≤w/2n≤1/4, /summationdisplay C∈Cn/summationdisplay i+j≤wAi(C)Aj(C)1 |C|≤c2|B2n(w)| 2n. A suitable numerical value for c2isc2= 4.3. Proof:From Proposition 5 it is enough to show that the sum/summationtext C∈Cnγn−dimC is upperbounded by a constant for any γ <1. Choosing a code CinCn is equivalent to choosing its generator polynomial, and fro m the list (9) of irreducible factors of Zn+ 1, we see that if we order all possible generator polynomials by increasing degrees, we have 1 and 1+ Z, then 2 polynomials of degree at least p−1, then 4 polynomials of degree at least p(p−1), ... then 2i 8polynomials of degree at least p(p−1)i−1and so on. Therefore, since n−dimC equals the degree of the generator polynomial, we obtain /summationdisplay C∈Cnγn−dimC≤1+γ+2γp−1+/summationdisplay i≥22iγp(p−1)i−1 ≤1+γ+2γp−1+/parenleftbigg2 p−1/parenrightbigg2/summationdisplay i≥2(p−1)iγ(p−1)i ≤1+γ+2γp−1+/parenleftbigg2 p−1/parenrightbigg2/summationdisplay j≥1jγj ≤1+γ+2γp−1+/parenleftbigg2 p−1/parenrightbigg2γ (1−γ)2. With the values γ= 21/5,c1= 26/5andp≥143given in Proposition 5 we obtain that c2= 4.3 is suitable. From (13) and Lemma 6 we obtain that P(X(w)>0)≤c21 n|B2n(w)| 2n+c2m−1/summationdisplay s=1ps n|B2n/ps(w/ps)| 2n/ps (14) to deal with this last sum we invoke: Lemma 7 For any prime p >143and for any positive number wsuch that |B2n(w)| ≤n2n, we have m−1/summationdisplay s=1ps n|B2n/ps(w/ps)| 2n/ps≤2 p Proof:Chooseptimes a vector of length 2 n/pand weight not more than w/p: concatenate the resulting vectors and one obtains a vector o f length 2 nand weight not more than w. Therefore |B2n/p(w/p)|p≤ |B2n(w)|and we have m−1/summationdisplay s=1ps n|B2n/ps(w/ps)| 2n/ps≤m−1/summationdisplay s=1ps n/parenleftbigg|B2n(w)| 2n/parenrightbigg1/ps ≤m−1/summationdisplay s=1ps nn1/ps. The result follows from routine computations. We see therefore from (14) and Lemma 7 that, if we choose wsuch that |B2n(w)| ≤bn2n, forb <1, then, provided the conditions of Proposition 5 are satisfied, we have P( X(w)>0)≤bc2+2c2/p. Forc2= 4.3 and any p >143this quantity is less than 1 when b≤0.23. The largest wfor which |B2n(w)| ≤bn2n is readily seen to satisfy K≤w 2n≤1 4which means that all conditions of Proposition 5 are satisfied, so that we have proved: Theorem 8 There exist positive constants b≤0.23andq, such that for any primep≥qsuch that 2is primitive modulo pand2p−1/ne}ationslash= 1 mod p2, and for any power n=pmofp, there exist double circulant codes of parameters [2n,n,d > w ]for anywsuch that |B2n(w)| ≤bn2n. A suitable value of qis q= 143and the first suitable prime pisp= 2789. 93.3 Proof of Proposition 5 Our remaining task is now to prove Proposition 5. We start by n oting that Proposition 5 is stated with a positive real number w, because the discussion starting from (13) involves balls of non-integer radius. Ho wever, it clearly is enough to prove it only for integer values of w. The crucial part of the proof will be to bound from above the we ight distri- bution of C, forC∈Cn. Let us note that, since the polynomial Pn(Z) defined in (11) is an irreducible factor of Zn+1, the code Cbelongs to Cnif and only ifPn(Z) divides the generator polynomial of the dual code C⊥. This means that any codeword of C⊥must be obtained by repeating ptimes a subvector of lengthn/p. Equivalently, a generating matrix of C⊥, i.e. a parity-check matrix ofCis of the form HC= [A|A|···|A] meaning that it equals the concatenation of pidentical copies of an r×n/p matrixA. We shall need the following lemma. Lemma 9 LetHtr= [Ir|Ir|···|Ir]be ther×trmatrix obtained by concate- natingtcopies of the r×ridentity matrix. Let σtrbe the associated syndrome function: σtr:{0,1}tr→ {0,1}r x/ma√sto→σtr(x) =xtHtr. Letw≤trbe an integer. Then, for any s∈ {0,1}r, the number of vectors of lengthtrand of weight wthat map to sbyσtris not more than: √ 2rt/parenleftbigg1+|1−2ω|t 2/parenrightbiggr/parenleftbiggtr w/parenrightbigg wherew=ωtr. Proof: LetXbe a random vector of length trobtained by choosing indepen- dently each of its coordinates to equal 1 with probability ω. The probabilities that any given coordinate of σtr(X) equals 0 or 1 are those of a sum of tinde- pendent Bernoulli random variables of parameter ω, namely: 1+(1−2ω)t 2and1−(1−2ω)t 2. Since all the coordinates of σtr(X) are clearly independent, max s∈{0,1}rP(σtr(X) =s) =/parenleftbigg1+|1−2ω|t 2/parenrightbiggr . (15) Now letW= wt(X) be the weight of X. We have P(W=w) =/parenleftbiggtr w/parenrightbigg ωw(1−ω)tr−w=/parenleftbiggtr ωtr/parenrightbigg 2−trh(ω) 10wherehdenotes the binary entropy function, h(x) =−xlog2x−(1−x)log2(1− x). By a variant of Stirling’s formula [13][Ch. 10, §11,Lemma 7] /parenleftbiggn w/parenrightbigg ≥2nh(ω)//radicalbig 8nω(1−ω), (16) therefore: P(W=w)≥1/radicalbig 8trω(1−ω)≥1√ 2tr. For given s, letNwdenote the number of vectors of length trand weight wthat have syndrome s. Since P( σtr(X) =s|W=w) =Nw//parenleftbigtr w/parenrightbig we have P(σtr(X) =s)≥P(σtr(X) =s|W=w)P(W=w)≥Nw/parenleftbigtr w/parenrightbig1√ 2tr. Hence, by (15), Nw≤√ 2tr/parenleftbigg1+|1−2ω|t 2/parenrightbiggr/parenleftbiggtr w/parenrightbigg which is the claimed result. Lemma 10 Let0< κ <1/4. There exist q, such that for any p > q,n=pm, and for any code C∈Cn, the following holds: •eitherC={0,1}norCequals the even-weight code, •or, the weight distribution of Csatisfies, for any i,κn≤i≤n/2, Ai(C)≤1 23r/5/parenleftbiggn i/parenrightbigg wherer=n−dimC. Forκ= 0.07a suitable value of qisq= 143. Proof: Ifr= 0 orr= 1, i.e. Cequals the whole space {0,1}nor the even- weight code, there is nothing to prove. Suppose therefore r >1. From the factorization (9) of Zn+ 1 into irreducible factors we see that we must have r≥p−1. From the discussion preceding Lemma 9 we must have r≤n−pm−1(p−1) =n/p (17) and a parity-check matrix of Cis made up of pidentical copies of some r×n/p matrixA. Therefore, after permuting coordinates, there exists a pa rity-check matrix of Cof the form HC= [B|Ir|Ir|···|Ir] whereBissomer×(n−rt)matrixandisfollowed by tcopies ofthe r×ridentity matrix. The integer tcan be chosen to take any value such that 1 ≤t≤p: we shall impose the restriction t≤p1/3. (18) 11For anyx∈ {0,1}n, writex= (x1,x2) wherex1is the vector made up of the firstn−trcoordinates of xandx2consists of the remainding trcoordinates Now the syndrome function σassociated to HCtakes the vector x∈ {0,1}n toσ(x) =x1tB+σtr(x2) whereσtris the function defined in Lemma 9. The codeCis the set of vectors xsuch that σ(x) = 0, therefore by partitioning the set of vectors of weight iinto all possible values of x1we have, from Lemma 9: Ai(C)≤√ 2trtr/summationdisplay j=0/parenleftBigg 1+|1−2j tr|t 2/parenrightBiggr/parenleftbiggtr j/parenrightbigg/parenleftbiggn−tr i−j/parenrightbigg (19) for anyisuch that i≥tr. (20) Notice that:/parenleftbiggtr j/parenrightbigg/parenleftbiggn−tr i−j/parenrightbigg =/parenleftbigi j/parenrightbig/parenleftbign−i tr−j/parenrightbig /parenleftbign tr/parenrightbig/parenleftbiggn i/parenrightbigg so that (19) becomes Ai(C)≤√ 2trtr/summationdisplay j=0/parenleftBigg 1+|1−2j tr|t 2/parenrightBiggr/parenleftbigi j/parenrightbig/parenleftbign−i tr−j/parenrightbig /parenleftbign tr/parenrightbig/parenleftbiggn i/parenrightbigg ≤√ 2tr(tr+1)/parenleftbiggn i/parenrightbigg max 0≤j≤tr/parenleftBigg 1+|1−2j tr|t 2/parenrightBiggr/parenleftbigi j/parenrightbig/parenleftbign−i tr−j/parenrightbig /parenleftbign tr/parenrightbig.(21) Seti=ιnandj=αtr, we have: /parenleftbigi j/parenrightbig/parenleftbign−i tr−j/parenrightbig /parenleftbign tr/parenrightbig≤ij(n−i)tr−j /parenleftbign tr/parenrightbig j!(tr−j)! ≤ιj(1−ι)tr−jntr /parenleftbign tr/parenrightbig j!(tr−j)! ≤ιj(1−ι)tr−jntr (n−tr)tr/parenleftbigtr j/parenrightbig−1since/parenleftbign tr/parenrightbig ≥(n−tr)tr/(tr)! ≤ιj(1−ι)tr−j/parenleftbigtr j/parenrightbig (1−tr n)tr. We have seen (17) that r≤n/pandt≤p1/3(condition (18)), therefore tr/n≤ 1/p2/3≤1/2: by usingthe inequality 1 −x≥2−2x, valid whenever 0 ≤x≤1/2, we therefore have /parenleftbigi j/parenrightbig/parenleftbign−i tr−j/parenrightbig /parenleftbign tr/parenrightbig≤22t2r2/nιj(1−ι)tr−j/parenleftbiggtr j/parenrightbigg and by using/parenleftbigtr j/parenrightbig ≤2trh(α)we finally get /parenleftbigi j/parenrightbig/parenleftbign−i tr−j/parenrightbig /parenleftbign tr/parenrightbig≤2tr(2tr n−D(α||ι)) 12whereD(x||y) =xlog2x y+(1−x)log21−x 1−y. Together with (21) we get: Ai(C)≤2r(β+f(ι))1 2r/parenleftbiggn i/parenrightbigg with f(ι) = max 0≤α≤1g(α,ι) (22) where g(α,ι) = log2(1+|1−2α|t)−tD(α||ι) (23) andβ=1 rlog2√ 2tr+1 rlog2(tr+1)+2t2r/n. Write log2(tr+1)≤1+log2tr to getβ≤(3 2+3 2log2(tr))/r+2t2r/n. By using t < p1/3andp−1≤r≤n/p, we get 3 2log2tr r<3 2log2(r+1)4/3 r= 2log2(r+1) r≤2log2p p−1 and β≤3 2(p−1)+2log2p p−1+2 p1/3. We see that βcan be made arbitrarily small by increasing the value of p. A numerical computation gives us β <0.152 for all p >143. Since we have supposed i≤n/2, we have ι≤1/2 so that the definition (22) and (23) of fcan be replaced by the equivalent f(ι) = max 0≤α≤ιg(α,ι) g(α,ι) = log2(1+(1−2α)t)−tD(α||ι) from which we easily see that gandfare decreasing functions of ι. We see that f(κ) can be made arbitrarily small, for all κ >0, by choosing tbig enough. Numerically, by choosing t= 14,κ= 0.07 andp >143, we see that (20) is satisfied and we get, for all 0 .07≤ι,f(ι)≤f(κ)≤0.24. We obtain therefore that, for all κn≤i≤n/2, Ai(C)≤2−0.608r/parenleftbiggn i/parenrightbigg which proves the lemma. To prove Proposition 5, we need a final technical lemma, of a pu rely enu- merative nature. Lemma 11 Let0< κ < K < 1/4. There exist an integer n0andε >0such that, for any n≥n0,w= 2ωnwithK≤ω <1/4, 2/summationdisplay i+j≤w i<κn/parenleftbiggn i/parenrightbigg/parenleftbiggn j/parenrightbigg ≤1 2εn|B2n(w)|. Forκ= 0.07,K= 0.1,n0= 143, a suitable value of εisε= 0.004. 13Proof: Clearly we have: 2/summationdisplay i+j≤w i<κn/parenleftbiggn i/parenrightbigg/parenleftbiggn j/parenrightbigg ≤κn2/parenleftbiggn κn/parenrightbigg/parenleftbiggn w−κn/parenrightbigg ≤κn22n(h(κ)+h(2ω−κ))=κn222nh(ω) 2n(2h(ω)−h(κ)−h(2ω−κ)) ≤κn2 22nh(ω) 2n(2h(K)−h(κ)−h(2K−κ)) since 2h(ω)−h(κ)−h(2ω−κ) is an increasing function of ω. By (16) we have 22nh(ω)≤√ 16n|B2n(w)|, so that we obtain, since κ≤1/4, 2/summationdisplay i+j≤w i<κn/parenleftbiggn i/parenrightbigg/parenleftbiggn j/parenrightbigg ≤n5/2|B2n(w)| 2n(2h(K)−h(κ)−h(2K−κ))≤|B2n(w)| 2εn for anyn≥n0withε≤2h(K)−h(κ)−h(2K−κ)−5 2log2n0 n0. Proof of Proposition 5: IfC={0,1}nor ifCis the even-weight subcode, thenAi(C)≤/parenleftbign i/parenrightbig , and/summationtext i+j≤wAi(C)Aj(C)≤/summationtext i+j≤w/parenleftbign i/parenrightbig/parenleftbign j/parenrightbig =|B2n(w)|. The result clearly holds for any c1≥2/γ. LetC∈Cnwithr=n−dimC >1. Let us write: 1 |C|/summationdisplay i+j≤wAi(C)Aj(C) =S1+S2 with S1=1 |C|/summationdisplay i+j≤w κn≤i,jAi(C)Aj(C) and S2=2 |C|/summationdisplay i+j≤w i<κnAi(C)Aj(C). By Lemma 10 we have S1≤1 |C|/summationdisplay i+j≤w/parenleftbiggn i/parenrightbigg/parenleftbiggn j/parenrightbigg1 26r/5≤|B2n(w)| 2n1 2r/5. To upperbound S2we simply write Ai(C)≤/parenleftbign i/parenrightbig . By Lemma 11, we have S2≤|B2n(w)| 2n2r 2εn=|B2n(w)| 2n2r (2εp)n/p≤|B2n(w)| 2n2r (2εp)r since we have seen (17) that r≤n/p. By choosing p≥6 5εwe obtain S2≤|B2n(w)| 2n1 2r/5. This proves the result with γ= 1/21/5andc1= 26/5. 144 Comments The probabilistic method we used easily shows that almost al l double circulant codes of the asymptotic family presented here satisfy an imp roved bound of the form (2). Actually we suspect that this is also the case fo r most choices ofn: this is suggested by computer experiments with randomly ch osen double circulant codes of small blocklengths. We have tried to strike a balance between giving readable pro ofs and deriv- ing a non-astronomical lower bound on the prime pin Theorem 8. In principle, the numerical values could be refined. In particular, the con stantbof Theorem 8 could bemade to approach 1 /2 (as in Theorem 4) but at the cost of a larger p. If we convert the formulation of Theorem 8 in the form (2) (whi ch just involves switching from |B2n(d)|in Theorem 2 to |B2n(d−1)|in (2)) we obtain a con- stantcwhich is of the same order of magnitude, but somewhat worse, t han the improved constant c≈0.102 of [16] for Jiang and Vardy’s method. In this paper we only consider the binary case with codes of ra te 1/2 but the method can be straightforwardly generalized to the case of d ifferent alphabets and to quasi-cyclic codes of any rational rate (though at the cost of a wors- ening of the constant b) by considering for parity check matrices vertical and horizontal concatenations of random circulant matrices. Finally, a natural question is to wonder whether the ideas de veloped in this paper can be extended to Euclidean lattices in a way similar t o the generaliza- tion of Jiang and Vardy’s method to sphere-packings of Eucli dean spaces [10]. A positive answer to this question is given in the paper [4]. References [1] C. L. Chen, W. W. Peterson and E. J. Weldon, “Some results o n quasi- cyclic codes,” Inform. Control , Vol. 15, no. 5, pp. 407–423, 1969. [2] V.V.Chepyzhov, “Newlower boundsforminimumdistanceo flinearquasi- cyclicandalmost linearcycliccodes,” Problemy Peredachi Informatsii , Vol. 28, no 1, pp. 39–51, 1992. [3] P. Gaborit and G. Z´ emor, “Asymptotic improvement of the Gilbert- Varshamov bound for linear codes”, ISIT 2006, Seattle, p.28 7-291. [4] P. Gaborit and G. Z´ emor, “On the construction of dense la ttices with a given automorphism group,” Annales de l’Institut Fourier , vol. 57 No. 4 (2007), pp. 1051–1062. [5] E. N. Gilbert, ” A comparison of signalling alphabets”, Bell. Sys. Tech. J. , 31, pp. 504-522, 1952. [6] T. Jiang and A. Vardy, “Asymptotic improvement of the Gil bert- Varshamov bound on the size of binary codes,” IEEE Trans. Inf. Theory , Vol. 50, no. 8, pp. 1655–1664, 2004. 15[7] G. A. Kabatiyanskii, ”On the existence of good cyclic alm ost linear codes over non prime fields”, Problemy Peredachi Informatsii , Vol. 13, no 3, pp. 18–21, 1977. [8] M. Karlin, ” New binarycoding results by circulant”, IEEE Trans. Inform. Theory15, pp. 81–92, 1969. [9] T. Kasami, “A Gilbert-Varshamov bound for quasi-cyclic codes of rate 1/2,”IEEE Trans. Inf. Theory , Vol. 20, no. 5, pp. 679–679, 1974. [10] M. Krivelevich, S. Litsyn, A. Vardy, ”A lower bound on th e density of sphere packings via graph theory”, Int. Math. Res. Not , no. 43, 2271–2279, 2004. [11] E. Krouk, “On codes with prescribed group of symmetry,” Voprosy Kiber- netiki, Vol. 34, pp. 105–112, 1977. [12] E. Krouk and S. Semenov, “On the existence of good quasi- cyclic codes,” proc. of 7th joint Swedish-Russian International Workshop on Information Theory, St-Petersburg, Russia, june 1995, pp. 164–166. [13] F.J. MacWilliams and N.J.A. Sloane, “The Theory of Erro r-Correcting Codes,” North-Holland, Amsterdam 1977. [14] M. A. Tsfasman, S.G. Vladuts and Zink, ”Modular curves, Shimura curves and Goppa codes better than Varshamov-Gilbert bound”, Math. Nach. , 104, pp. 13–28, 1982. [15] R.R. Varshamov, ”Estimate of the number of signals in er ror-correcting codes”,Dokl. Acad. Nauk ,117, pp. 739–741, 1957 (in Russian). [16] V. Vu and L. Wu, ”Improving the Gilbert-Varshamov bound for q-ary codes”,IEEE Trans. Inf. Theo. ,51(9), pp. 3200–3208, 2005 16

2007-08-30

The Gilbert-Varshamov bound states that the maximum size A_2(n,d) of a binary code of length n and minimum distance d satisfies A_2(n,d) >= 2^n/V(n,d-1) where V(n,d) stands for the volume of a Hamming ball of radius d. Recently Jiang and Vardy showed that for binary non-linear codes this bound can be improved to A_2(n,d) >= cn2^n/V(n,d-1) for c a constant and d/n <= 0.499. In this paper we show that certain asymptotic families of linear binary [n,n/2] random double circulant codes satisfy the same improved Gilbert-Varshamov bound.

Asymptotic improvement of the Gilbert-Varshamov bound for linear codes

0708.4164v1

arXiv:0709.2937v2 [cond-mat.mes-hall] 23 Jan 2008Theory of current-driven magnetization dynamics in inhomogeneous ferromagnets Yaroslav Tserkovnyak,1Arne Brataas,2and Gerrit E. W. Bauer3 1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, U SA 2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway 3Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands (Dated: November 8, 2018) Abstract We give a brief account of recent developments in the theoret ical understanding of the interac- tion between electric currents and inhomogeneous ferromag netic order parameters. We start by discussingthephysicaloriginofthespintorquesresponsi bleforthisinteraction andconstructaphe- nomenological description. We then consider the electric c urrent-induced ferromagnetic instability and domain-wall motion. Finally, we present a microscopic j ustification of the phenomenological description of current-driven magnetization dynamics, wi th particular emphasis on the dissipative terms, theso-called Gilbertdamping αandtheβcomponent oftheadiabatic current-driventorque. 1I. INTRODUCTION Ferromagnetism is a correlated state in which, at sufficiently low temp eratures, the elec- trons align their spins in order to reduce the exchange energy. Belo w the Curie temperature Tc, the free energy F[M] then attains its minimum at a finite magnetization M∝negationslash= 0 with an arbitrary direction, thus spontaneously breaking the spin-rot ational symmetry. Crystal anisotropies that are caused by the spin-orbit interaction and sha pe anisotropies governed by the magnetostatic dipolar interaction pin the equilibrium magnetiza tion direction to a certain plane or axis. At low temperatures, T≪Tc, fluctuations of the magnitude of the magnetization around the saturation value Ms(the so-called Stoner excitations) become energetically unfavorable. The remaining low-energy long-waveleng th excitations are spin waves (or magnons, which, in technical terms, are viewed as Goldst one modes that restore the broken symmetry). These are slowly varying modulations of the magnetization direction in space and time. A phenomenological description of the slow collective magnetization d ynamics without dissipation proceeds from the free energy F[M(r)] as a functional of the inhomogeneous (and instantaneous) magnetic configuration M(r) [1]. The equation of motion ∂M(r,t) ∂t=γM(r,t)×δF[M] δM(1) preserves the total free energy of the system, since the rate o f change of the magneti- zation is perpendicular to the “gradient” of the free energy. The f unctional derivative of the free energy with respect to the local magnetization is called t heeffective field: Heff(r,t) =−δF[M]/δM. [In the following, we use the abbreviations ∂t=∂/∂tfor the partial derivative in time and ∂M=δ/δMfor the functional derivative with respect to M.] In the presence of only an externally applied magnetic field H,F[M] =−/integraltext d3rM(r)·H(r), soγis identified as the effective gyromagnetic ratio. In general, Heffincludes the crys- tal anisotropy due to spin-orbit interactions, modulation of the ex change energy due to magnetization gradients, and demagnetization fields due to dipole-d ipole interactions. The Landau-Lifshitz equation (1) qualitatively describes many ferroma gnetic resonance (FMR) [2, 3] and Brillouin light scattering [4] experiments. With M=Msm, wheremis the magnetic direction unit vector, we may rewrite Eq. (1) as ∂tm(r,t) =−γm(r,t)×Heff(r,t). (2) 2By definition, the effective field on the right-hand side of Eq. (2) is de termined by the instantaneous magnetic configuration. This can be true only if the m otion is so slow that all relevant microscopic degrees of freedom manage to immediately r eadjust themselves to the varying magnetization. If this is not the case, the effective field acquires a finite time lag that to lowest order in frequency can be schematically expanded as ˜Heff→ −∂MF[M(r,t−τ)]≈Heff−τ(∂tM·∂M)Heff, (3) whereτisa characteristic delay time. This dynamic correction tothe instant aneous effective field,δHeff, leads to a new term in the equation of motion ∝m×δHeff. Although in general nonlocal and anisotropic [5], it makes sense to identify first the simple st, i.e., local and isotropic contribution. We can then construct two new terms out o f the vectors mand ∂tm. The first one, ∝m×∂tm, is dissipative, meaning that it is odd under time reversal (i.e., under the transformation t→ −t,H→ −H, andm→ −m), and thus violates the time-reversal symmetry of the Landau-Lifshitz equation (2). Th is argument leads to the Landau-Lifshitz-Gilbert (LLG) equation [6, 7]: ∂tm(r,t) =−γm(r,t)×Heff(r,t)+αm(r,t)×∂tm(r,t), (4) introducing the phenomenological Gilbert damping constant α. The second local and isotropic term linear in ∂tmand perpendicular to m, that can be composed out of m and∂tmis proportional to ∂tmand can be combined with the left-hand side of the LLG equation. In principle, any physical process that contributes to t he Gilbert damping can thus also renormalize the gyromagnetic ratio γ. The latter should therefore be interpreted as an effective parameter in the equation of motion (4). The second law of thermodynamics requires that αγ≥0, which guarantees that the dissipation of energy P∝Heff·∂tm≥0 (assuming the magnetization dynamics are slow and isolated from any external sinks of entropy). Since the implications of small modifications of the gyroma gnetic ratio are mi- nor, we will be mainly concerned with the Gilbert damping constant α. Furthermore, we will focus on dissipative effects due to spin dephasing by magnetic or s pin-orbit impurities [8, 9, 10, 11, 12, 13], noting that many other Gilbert damping mechan isms have been pro- posed in the past [14, 15, 16, 17, 18, 19, 20, 21]. According to the fl uctuation-dissipation theorem, the dissipation, whatever its microscopic origin is, must be accompanied by a stochastic contribution h(r,t) to the effective field. Assuming Gaussian statistics with a 3white noise correlator [22], which is valid in the classical limit with charac teristic frequen- cies that are sufficiently small compared to thermal energies: ∝angb∇acketlefthi(r,t)hj(r′,t′)∝angb∇acket∇ight= 2kBTα γMsδijδ(r−r′)δ(t−t′). (5) Eqs. (1)-(5) form the standard phenomenological basis for unde rstanding dynamics of fer- romagnets [2, 3, 4], in the absence of an applied current or voltage b ias. II. CURRENT-DRIVEN MAGNETIZATION DYNAMICS In order to understand recent experiments on current-biased m agnetic multilayers [23, 24, 25, 26, 27, 28, 29, 30, 31] and nanowires [32, 33, 34, 35, 36, 3 7, 38, 39], Eq. (4) has to be modified [40, 41, 42]. The leading correction has to take into account the finite divergence of the spin-current density in conducting ferromagnets, with mag netization texture that has to be brought into compliance with the conservation of angular mome ntum. We have to introduce a new term s0∂tmi|torque=∇·ji (6) in the presence of a current density jifor spin-icomponent, where s0is the total equilibrium spin density along −m(the minus sign takes into account that electron spin and magnetic moment point in opposite directions). By adding this term to the right -hand side of Eq. (4) as a contribution to ∂tmi, we assume that the angular momentum lost in the spin current is fully added to the magnetization. This is called the spin-transfer to rque [42, 43, 44, 45]. Thesimplest approximationforthespin-current density inthebulko fisotropicferromagnets [43, 45, 46] is ji=Pjmi, wherePis a material-dependent constant that converts charge- current density jinto spin-current density. The underlying assumption here is that s pins are carried by the electric current such that the spin-polarization axis adiabatically follows the local magnetization direction. This is the case for a large exchange fi eld that varies slowly in space. This condition is fulfilled very well in transition-metal ferrom agnets. The spin conversion factor P=/planckover2pi1 2eσ↑−σ↓ σ↑+σ↓(7) characterizes the polarization of the spin-dependent conductivit yσs(s=↑ors=↓) with↑ chosen along −m. Hence ∂tm=−γm×Heff+P s0(j·∇)m, (8) 4where we took into account local charge neutrality by ∇·j= 0. The phenomenological Eq. (8) is “derived” without taking into accou nt spin relaxation processes. Its inclusion requires some care since both Gilbert damp ing and spin-transfer torque are nontrivially affected [47, 48, 49, 50, 51, 52]. Spin relaxat ion is generated by impurities with potentials that do not commute with the spin density op erator, such as a quenched random magnetic field or spin-orbit interaction associat ed with randomly dis- tributed non-magnetic impurities [13, 47, 49, 50, 51]. In the absenc e of an applied current j, imperfections with potentials that mix the spin channels contribute to the Gilbert con- stantα[8, 9, 10, 13, 50]. It is instructive to interpret the right-hand side o f the equation of motion (8) as an analytic expansion of driving and damping torques in∇and∂t. The LLG equation (4) corresponds to the most general (local and isot ropic) expression for the damping to the zeroth order in ∇and first order in ∂t. We will not be concerned with higher order terms in ∂t, since the characteristic frequencies of magnetization dynamics a re typically small on the scale of the relevant microscopic energies, at le ast in metallic systems. The contribution to the effective field due to a finite magnetic stiffnes s [1] is proportional to∇2. In the presence of inversion symmetry, the terms proportional to∇cannot appear without applied electric currents. In the following, we focus on the c urrent-driven terms lin- ear in∇, assuming that the spatial variations in the magnetization direction are sufficiently smooth to rule out higher-order contributions. The dynamics of iso tropic spin-rotationally invariant ferromagnets can then in general be described by the ph enomenological equation of motion [47, 49, 53] ∂tm=−γm×Heff+αm×∂tm+P s0(1−βm×)(j·∇)m, (9) in which αandβcharacterize those terms that break time-reversal symmetry. Both arise naturally in the presence of spin-dependent impurities [47, 49, 50]. E ven though in practice β≪1,itgivesanimportantcorrectiontothecurrent-driven spin-tra nsfertorque[47, 49, 53], asdiscussedinmoredetailbelow. Forthespecialcaseof α=β: Eq.(9)canthenberewritten (after multiplying it by 1+ αm×on the left) as ∂tm=−γ∗m×Heff+αγ∗m×Heff×m+P s0(j·∇)m, (10) whereγ∗=γ/(1 +α2). The dissipative term proportional to m×Heff×mis called the Landau-Lifshitz damping. Eq. (9) cannot be transformed into equ ation (10) if α∝negationslash=β[in 5which case Eq. (10) necessarily retains a βterm]. A special feature of Eq. (10) appears when Heff(m) is time independent and translationally invariant. A general solution m(r,t) in the absence of an electric current j= 0 (such as a static domain wall or a spin wave) can be used to construct a solution ˜m(r,t) =m(r+Pjt/s0,t) (11) of Eq. (10) for an arbitrary uniform j. This unique feature of the solutions of Eq. (9) only arises for α=β. Interestingly, the argument above has been turned around by Ba rnes and Maekawa [54], who find that Galilean invariance of a system would dictate α=β. Galilean invariance requires the existence of solutions of the form ˜m(r−vt), where ˜m(r) is an arbitrary static solution (say, a domain wall) and vis an arbitrary velocity. As explained above, this is only possible when α=β. However, the general validity of the Galilean invariance assump- tion for the current-carrying state needs to be discussed in more detail from a microscopic point of view. The Galilean invariance argument [54] implies that the bias -induced electron drift exactly corresponds to the domain-wall velocity, since other wise electron motion would persist in the frame that moves with the domain wall. Referring to Eq. (11), we must, therefore, identify v=−Pj/s0with the average electron drift velocity in the presence of the currentj. We argue in the following that this is indeed true in certain special limits , but is not generic, however. In the itinerant Stoner model for ferrom agnets, the spin-dependent Drude conductivity reads σs∝nsτs, wherensandτsare the densities and scattering times of spins, respectively. When there is no asymmetry between the scatterin g times,τ↑=τ↓, vindeed equals the electron drift velocity and Galilean invariance is effec tively fulfilled. In general, however, since the spin dependence of wave functions an d densities of states for the electrons at the Fermi energy lead to different scattering cro ss sections in conducting ferromagnets, the equality between −Pj/s0and the average drift velocity disappears. In the simplest model of perturbative white-noise impurity potentials, for example, 1 /τs∝νs, whereνsis the spin-dependent density of states. Assuming parabolic free- electron bands and weak ferromagnets, in which the ferromagnetic exchange split ting is much less than the Fermi energy, the domain-wall velocity v=−Pj/s0actually becomes 2 /3 of the average drift velocity ¯v, v=−Pj s0=n↑τ↑−n↓τ↓ n↑τ↑+n↓τ↓n↑v↑+n↓v↓ n↑−n↓≈n↑/ν↑−n↓/ν↓ (n↑−n↓)(ν↑+ν↓)/2¯v≈2 3¯v.(12) 6Here,vsis the spin- selectron drift velocity, and we used the relation νs∝n1/3 s, which is valid in three dimensions. Clearly, the potential disorder breaks Ga lilean invariance. An identity of αandβcan therefore not be deduced from general symmetry principles. Furthermore, spin-orbit interaction or magnetic disorder that st rongly affect the values of α andβ(see below) also break Galilean invariance at the level of the microsco pic Hamiltonian. Nevertheless, for itinerant ferromagnets we show below that α∼β(where by ∼we mean “of the order”), with α≈βin the simplest model of weak and isotropic spin-dephasing impurities [49], which implies that deviations from translational invarian ce are not very important in metallic ferromagnets, such as transition metals and th eir alloys, in which the Stoner model is applicable. Very recently, two independent gro ups measured α≈βin permalloy nanowires [36, 37]. Let us also consider the s−dmodel of ferromagnetism. When, as is usually done, the d- orbital lattice is assumed spatially locked, Galilean invariance is broken even in the absence of disorder. In this case, the ratio α/βdeviates strongly from unity, although it remains to be relatively insensitive to the strength of spin-dependent impuritie s. In other words, αand βscale similarly with the strength of spin-dephasing processes, and t heir ratio appears to be determined mainly by band-structure effects and the nature (r ather than the strength) of the disorder [49, 50]. A predictive material-dependent theory of magnetization damping and current-induced domain wall motion that transcends the toy m odels mentioned above is beyond the scope of our paper. The form of the equation of motion (10) for the special case α=βhas also triggered the suggestion [55] that the Landau-Lifshitz form of damping, ∝m×Heff×m, is more natural than the Gilbert form, ∝m×∂tm. In our opinion, however, such a distinction is purely semantic. Both forms are odd under time reversal, and one can easily imagine simple models in which either form arises more naturally than the other : For example, a Bloch-like T2relaxation added to the Stoner model naturally leads to a Landau-L ifshitz form of damping [49], whereas the dynamic interface spin pumping [56, 57] very generally obeys the Gilbert damping form. Moreover, mathematically, both eq uations are identical (in the absence of any additional torques), since we have shown ab ove that the Landau- Lifshitz form of damping follows from the Gilbert one simply by multiplying both sides of the LLG form by 1+ αm×from the left (and vice versa by 1 −αm×). Only at the special pointα=β, the Landau-Lifshitz form (10) does not involve the “ βterm.” In that limit, 7it may be a more transparent expression for the equation of motion . On the other hand, we noted above that in general α∝negationslash=βand the ratio α/βdepends on material and sample. The current-driven dynamics of domain walls and other spatially nonu niform magnetization distributions turn out to be very sensitive to small deviations of α/βfrom unity, which strongly reduces any advantage a Landau-Lifshitz damping formu lation might have over the Gilbert phenomenology. In general, we therefore prefer to use th e Gilbert phenomenology. Under time reversal, the electric current as well as the magnetizat ion vector change sign and the adiabatic current-induced torque is symmetric, thus nond issipative. The Ohmic dissipation generated by this current does not depend on the magn etization texture in this limit and is intentionally disregarded. Saslow [58] prefers to discuss a t orque driven by voltage rather current, which, after inserting Ohm’s law for the cu rrent, becomes odd under time reversal and thus appears dissipative (see Ref. [59] for anot her discussion of this point). Obviously, the βcorrection torque is odd for the current-biased and even for the voltage- biased configurations. The current-bias picture appears to be mo re natural, since it reflects the absence of additional dissipation by the magnetization texture in the adiabatic limit as well as the close relation between the βcorrection and the Gilbert dissipation. III. CURRENT-DRIVEN INSTABILITY OF FERROMAGNETISM Let us now pursue some special aspects of the solutions of the phe nomenological Eq. (9), highlighting the role of various parameters, before we discuss the m icroscopic derivation of the magnetization dynamics in Sec. V. It is interesting, for examp le, to investigate the possibility to destabilize a single-domain ferromagnet by sufficient ly large spin torques [43, 46]. We consider a homogeneous ferromagnet with an easy-axis anisotropy along the x axis, characterized by the anisotropy constant K, and an easy-plane anisotropy in the xy plane, with the anisotropy constant K⊥, see Fig. 1. Typically, the anisotropies originate from the demagnetization fields: For a ferromagnetic wire, for exa mple, the magnetostatic energy is lowest when the magnetization is in the wire direction, so tha t there are no stray field lines outside the ferromagnet. The effective field governing mag netization dynamics is then given by Heff= (H+Kmx)x−K⊥mzz+A∇2m, (13) 8where we also included an applied field Halong the xaxis and the exchange coupling parametrized by the stiffness constant A.A,H,K,K ⊥≥0. We then look for spin-wave solutions of the form m=x+uei(q·r−ωt), (14) plugging it into Eq. (9) with the effective field (13) in the presence of a constant current densityj, and linearizing it with respect to small deviations u. Whenα= 0 and j= 0, we recover the usual spin-wave dispersion (Kittel formula): ω0(q) =γ/radicalbig (H+K+Aq2)(H+K+K⊥+Aq2). (15) Afiniteαγ >0results ina negative Im ω(q), asrequired by thestability oftheferromagnetic state. Asufficientlylargeelectriccurrentmay, however, reverse thesignofIm ω(q)forcertain wave vectors q, signaling the onset of an instability. The critical value of the curren t for the instability corresponds to the condition Im ω(q) = 0. Straightforward manipulations based on Eqs. (9) and (13) show that this condition is satisfied when P s0/parenleftbigg 1−β α/parenrightbigg (q·jc) =±ω0(q). (16) which leads to a critical current density jc=jc0 |1−β/α|. (17) jc0is the lowest current satisfying equation ( P/s0)(q·jc0) =ω0(q) for some q, where the left-hand side can be loosely interpreted as the current-induced D oppler shift to the nat- ural frequency given by the right-hand side [46]. According to Eq. ( 17), a current-driven instability is absent when α=β. This conclusion is in line with the arguments leading to Eq. (11): For the special case of α=β, a spin-wave solution in the presence of a finite current density jwould acquire a frequency boost proportional to q·j, but with a stable amplitude. Note that in general the onset of the current-driven f erromagnetic instability is significantly modified by the existence of βeven with β≪1, provided that the ratio β/αis appreciable. In fact, αis typically measured to be ∼0.001−0.01, and the existing microscopic theories [49, 50] predict βto be not too different from α. 9FIG. 1: Transverse head-to-head (N´ eel) domain wall parall el to the yaxis in the easy xyplane. The uniform magnetization has two stable solutions m=±xalong the easy axis x, which is characterized by the anisotropy constant K. These are approached far away from the domain wall: m→ ±xatx=∓∞, respectively. In equilibrium, the magnetization directi onmis forced into the xyplane by the easy-plane anisotropy parametrized by K⊥. A weak magnetic field Hor electric currentjapplied along the xaxis can induce a slow domain-wall drift along the xaxis, during which the magnetization close to the domain wall is tilted sl ightly out of the xyplane. At larger H orj(above the so-called Walker threshold), the magnetization is significantly pushed out of the xy plane and undergoes precessional motion during the drift. I n the moving frame, the magnetization profile may remain still close to the equilibrium one. IV. CURRENT-DRIVEN DOMAIN-WALL MOTION Even more interesting phenomena are associated with the effect of the applied electric current on a stationary domain-wall. In particular, we wish to discus s how the spin torques move and distort a domain wall. These questions date back about thr ee decades [40], although only relatively recently they sparked an intense activity by several groups [49, 50, 51, 53, 54, 60]. This is motivated by the growing number of intrigu ing experiments [32, 33, 34, 35, 36, 38, 39] as well as the promise of practical pote ntial, such as in the so- called racetrack memory [61] or magnetic logics [62]. Current-induce d domain-wall motion is a central topic of the present review. For not too strong driving currents and in the absence of any signifi cant transverse dy- namics, one can make progress analytically by using the one-dimensio nal Walker ansatz, which was first employed in studies of magnetic-field driven domain-wa ll dynamics [63]. 10This approach has proven useful in the present context as well [6 0, 64, 65]. The key idea is to approximately capture the potentially complex domain-wall motio n by few parameters describing the displacement of its center and a net distortion of the domain-wall structure. In a quasi-one-dimensional set-up, such as a narrow magnetic wire , the domain wall is con- strained to move along a certain axis, whereas the transverse dyn amics are suppressed. This regime is relevant for a number of existing experiments, although it s hould be pointed out that the common vortex-type domain walls do not necessarily fall int o this category. Let us consider an idealized situation with an effective field (13) and an equilibr ium domain wall magnetization in the xyplane. The magnetization prefers to be collinear with the xaxis due to the easy-axis anisotropy K. A transverse head-to-head domain wall parallel to the y axis corresponds to a magnetization direction that smoothly rotat es in the xyplane between xatx→ −∞and−xatx→ ∞, as sketched in Fig. 1. The collective domain-wall dynamics can be described by the center p ositionX(t) and an out-of-plane tilting angle Φ( t). [For a more technically-interested reader, we note that in the effective treatment of Ref. [60], these variables are canonica lly conjugate.] There is also a width distortion, but that is usually considered less important. WhenH < K, the two uniform stable states are m=±x. When H= 0, a static transverse head-to-head domain-wall solution centered at x= 0 is given by ϕ(x)≡0,lntanθ(x) 2=x W, (18) where position-dependent angles ϕandθparametrize the magnetic configuration: m= (mx,my,mz) = (cosθ,sinθcosϕ,sinθsinϕ). (19) W=/radicalbig A/Kis the wall width, which is governed by the interplay between the stiffn essA that tends to smooth the wall extent and the easy-axis anisotrop yKthat tends to sharpen the wall. The external magnetic field Hor the current density jalong the xaxis disturb the static solution (18), distorting the domain-wall structure and displacing it s position. At weak field and current biases, magnetic dynamics can be captured by the Walk er ansatz [63, 64]: ϕ(r,t)≡Φ(t),lntanθ(r,t) 2≡x−X(t) ˜W(t). (20) Here, it is assumed that the driving perturbations ( Handj) are not too strong, such that the wall preserves its shape, except for a small change of its width ˜W(t) and a uniform 11out-of-plane tilt angle Φ( t).X(t) parametrizes the net displacement of the wall along the xaxis. Note that although ϕis assumed to be spatially uniform, it has an effect on the magnetization direction only when m∝negationslash=±x, i.e., only near the wall center. A more detailed discussion concerning the range of validity of this approximation can be found in Ref. [63]. Inserting the ansatz (20) into the equation of motion (9) with j=jx(since the other current directions do not couple to the wall), and using Eq. (13) for the effec tive field, one finds [53, 64] ˙Φ+α˙X ˜W=γH−βPj s0˜W, ˙X ˜W−α˙Φ =γK⊥sin2Φ 2−Pj s0˜W, ˜W=/radicalBigg A K+K⊥sin2Φ. (21) It iseasyto verify thatthestaticsolution(18)is consistent withth ese equations when H= 0 andj= 0. Two different dynamic regimes can be distinguished based on Eqs. (21): When the driving forces are weak, a slightly distorted wall moves at a cons tant speed, ˙X= const, and constant tilt angle, ˙Φ = 0 (assuming constant Handj). The corresponding Walker ansatz (20) then actually provides the exact solution, which is appr oached at long times after the constant driving field and/or current are switched on [6 3, 64]. Beyond certain critical values of Horj, called Walker thresholds, however, no solution with constant angle Φ and constant velocity ˙Xexist. Both undergo periodic oscillations in time, albeit with a finite average drift velocity ∝angb∇acketleft˙X∝angb∇acket∇ight ∝negationslash= 0. In the spacial case of α=β, Eqs. (21) are exact at arbitrary dc currents when H= 0: According to Eq. (11), the static domain-wall solution (of an arbitrary domain-wall shape) then simply moves with velocity −Pj/s0without any distortions. When β∝negationslash=α, the Walker threshold current diverges when βapproaches α, reminiscent of the critical current (17) discussed in the previous s ection. For subthreshold fields and currents with Φ( t)→const as t→ ∞, the steady state terminal velocity is given by [47] v=˙X(t→ ∞) =γH˜W−βPj/s0 α. (22) In particular, when j= 0, the wall depicted in Fig. 1 moves along the direction of the applied magnetic field Hin order to decrease the free energy [63]. Let us in the following 12focus on the current-driven dynamics with H= 0. At a finite but small j, the wall is slightly compressed according to 1−˜W W≈(Pj/s0)2 2γ2AK⊥/parenleftbigg 1−β α/parenrightbigg2 , (23) whereW=/radicalbig A/Kis the equilibrium width. When α=β, the domain-wall velocity v→ −Pj/s0. In this case, if we consider the electron spins following the magnetiz ation direction from ±mto∓mon traversing the domain wall with current density j, the entire angular momentum change is transferred to the domain-wall displac ement. In this sense, the ratio β/αcan be loosely interpreted as a spin-transfer efficiency from the cu rrent density to the domain-wall motion. Only when α=β, the rigidly moving domain-wall solution is exact at arbitrary current densities, leading to an infinite Walker threshold current. The latter becomes finite and decreases with β < α, approaching a finite value jt0atβ= 0 [60], see Fig. 2. In the absence of a strong disorder pinning centers, as assumed so far, jt0∝K⊥(which is also the case with the Walker threshold fieldin the absence of an applied current [63]), with an average velocity that slightly above the threshold reads ∝angb∇acketleft˙X∝angb∇acket∇ight ∝/radicalBig j2−j2 t0. (24) See theβ= 0 curve in Fig.2. At finite β, thedepinning current is determined by the pinning fields, which should be included into the effective field (13). The domain -wall velocity at currents slightly above the depinning current is predicted in Ref. [5 4] to grow linearly with j. So far in our discussion, we have completely disregarded the random noise contribution to the magnetization dynamics. As noted above, see Eq. (5), ther mal fluctuations are ubiquitous in dissipative systems. Below the (zero-temperature) d epinning currents, applied currentscandrivethedomainwallwithfiniteaveragevelocity ∝angb∇acketleftv∝angb∇acket∇ightonlybythermalactivation. The question how ln ∝angb∇acketleftv∝angb∇acket∇ightscales with the current at low temperatures and weak currents is of fundamental interest beyond the field of magnetism. Experimen ts on thermally-activated domain-wall motion in magnetic semiconductors [33, 39] reveal a “cr eep” regime [66], in which the effective thermal-activation barrier diverges at low curre nt density j, so that ln∝angb∇acketleftv∝angb∇acket∇ightscales as const −j−µ, with an exponent µ∼1/3. This is inconsistent with the theory based on the Walker ansatz for rigid domain-wall motion [67], w hich yields a simple linear scaling of the effective activation barrier and ln ∝angb∇acketleftv∝angb∇acket∇ight ∝const +j. A refinement of the 13FIG. 2: Average current-driven domain-wall velocity vnumerically calculated using the Walker ansatz [Eqs. (21)] in Ref. [53]. Here, the domain-wall width has been approximated by its equilib- rium value, ˜W≈W, assuming K⊥≪K. The curves are very similar to the full micromagnetic simulations [53]. u=−Pj/s0has the units of velocity (proportional to electron drift ve locity) and vw=γK⊥ζ/2 is its value for j=jt0. The length ζ≈W, if we assume K⊥≪K(as was done in this calculation), while ζ≈/radicalbig 2A/K⊥in the opposite limit, K⊥≫K, which is relevant for a thin-film with large demagnetization anisotropy K⊥= 4πMs(in which case ζis called exchange length) [64]. α= 0.02, and we refer to Ref. [53] for the remaining details. Walker-ansatz treatment [68] cannot explain the experiments eith er. A scaling theory of creep motion close to the critical temperature [39] does offer a qua litative agreement with measurements by Yamanouchi et al.[39]. However, the intrinsic spin-orbit coupling in p- doped (Ga,Mn)As leads to current-driven effects beyond the stan dard spin-transfer theories, see, e.g., Refs. [69, 70], which needs to be understood better in the present context. Even at zero temperature, there are stochastic spin-torque so urces in the presence of an applied current, which stem from the discreteness of the angula r momentum carried by electron spins, in analogy with the telegraph-like shot noise of electr ic current carried by discrete particles. A theoretical study of the combined thermal a nd shot-noise contributions 14to the stochastic torques for inhomogeneous magnetic configura tions [71] did not yet explore consequences forthedomain-wall dynamics. Forexample, itisnotk nown whether shot noise assists thecurrent-driven domain-wall depinning atlowtemperatu res. Questions alongthese lines pose challenging problems for future research. Effects beyond the theory discussed above are generated by non adiabatic spin torques, which lead to higher-order in ∂tand∇terms in the equation of motion (9). It is in principle possible to extend linear-response diagrammatic Green’s function c alculation [13, 49, 50] by systematically calculating higher-order terms as an expansion in t he small parameters, i.e., spin-wave frequency and momentum [72]. A dynamic correction to the spin torque in Eq. (9) has been found in Refs. [49, 72], which comes down to replacin gβ→β+n(/planckover2pi1/∆xc)∂t, where ∆ xcis the ferromagnetic exchange splitting and n= 1(2) for the Stoner ( s−d) model [49, 72]. Since this term scales like ∂t∇, it is symmetric under time reversal and therefore nondissipative. Although this dynamic correction is rath er small at the typical FMR frequencies /planckover2pi1ω≪∆xc, it can cause significant effects at large currents [72]. Starting fromaninhomogeneousequilibriumconfiguration[72, 73,74,75], suc hasamagneticspiralor a domain wall, one can capture nonadiabatic terms in the equation of m otion that vanish in linear response withrespect totheuniformmagnetizationconsider ed inRefs. [13, 49, 50, 51]. For strongly-inhomogeneous magnetic structures, perturbativ e expansions around a uni- form magnetic state fail. For example, for sharp domain walls, the eff ective equations (21) describing wall dynamics and displacement acquire a new term, which c an be understood as a force transferred by electrons reflected at the potential b arrier caused by the domain wall [60, 76]. Electron reflection at a domain wall increases the resist ance. Adiabaticity implies a vanishing intrinsic domain-wall resistance (see, however, Re f. [69] for a model with strong intrinsic spin-orbit coupling). The force term, becomes impo rtant only for abrupt walls with width W∼λxc≡/planckover2pi1vF/∆xc. Such nonadiabatic effects are not expected to be strong in metallic ferromagnets, where typically W≫λxc∼λF(the Fermi wavelength). Dilute magnetic semiconductors [such as (Ga,Mn)As] are a different c lass of materials with longerλxcand a strong spin-orbit coupling [70]. In metallic systems, effects of the spin-torque in the most relevant regime of slow dy- namics,/planckover2pi1ω≪∆xc, with smooth walls, W≫λxc, and at moderate applied currents is in our opinion captured by the adiabatic terms linear in ∂tand∇. We will now discuss the microscopic basis for Eq. (9) containing such terms. 15V. MICROSCOPIC THEORY OF MAGNETIZATION DYNAMICS Once the phenomenological equation for current-driven magnetiz ation dynamics is re- duced to the form (9), which requires smooth magnetization variat ion, slow dynamics, and isotropic ferromagnetism, the remaining key questions concern th e magnitude and relation between the two dimensionless parameters αandβ. The size of the Gilbert damping con- stantαis a long-standing open question in solid-state physics, and even a br ief review of the relevant ideas and literature is beyond the scope of this paper. A re cent model calculation highlighting the multitude of relevant energy scales that control ma gnetic damping can be found in Ref. [12]. Here, we discuss only the ratio β/α, since it is of central importance for macroscopic current-driven phenomena. As noted above, th e ratioβ/αdetermines, for example, the onset of the ferromagnetic current-driven instabilit y [see Eq. (17)] as well as the Walker threshold current (both diverging when β/α→1). The subthreshold current- driven domain-wall velocity is proportional to β/α[see Eq. (22)], while β/α= 1 is a special point, at which the effect of a uniform current density jon the magnetization dynamics is eliminated in the frame of reference that moves with velocity v=−Pj/s0[see Eq. (11)]. Although the exact ratio β/αis a system-dependent quantity, some qualitative aspects not too sensitive to the microscopic origin of these parameters have re cently been discussed [13, 49, 50]. In Ref. [49], we developed a self-consistent mean-field approach, in which itinerant elec- trons are described by a time-dependent single-particle Hamiltonian ˆH= [H0+U(r,t)]ˆ1+γ/planckover2pi1 2ˆσ·(H+Hxc)(r,t)+ˆHσ, (25) where the unit matrix ˆ1 and the vector of the Pauli matrices ˆσ= (ˆσx,ˆσy,ˆσz) form a basis for the Hamiltonian in spin space. H0is the crystal Hamiltonian including kinetic and potential energy. Uis the scalar potential consisting of disorder and applied electric-fie ld contributions. The total magnetic field consists of the applied, H, and exchange, Hxc, fields. Finally, thelasttermintheHamiltonian, ˆHσ, accountsforspin-dephasing processes, e.g, due to quenched magnetic disorder or spin-orbit scattering associate d with impurity potentials. This last term is responsible for low-frequency dissipative processe s affecting αandβin the collective equation of motion (9). In time-dependent spin-density-functional theory [44, 77, 78] o f itinerant ferromagnetism, 16the exchange field Hxcis a functional of the time-dependent spin-density matrix ραβ(r,t) =∝angb∇acketleftΨ† β(r)Ψα(r)∝angb∇acket∇ightt (26) that should be computed self-consistently from the Schr¨ odinger equation corresponding to ˆH. The spin density of conducting electrons is given by s(r) =/planckover2pi1 2Tr[ˆσˆρ(r)]. (27) Focusing on low-energy magnetic fluctuations that are long range a nd transverse, we restrict our attention to a single parabolic band. Consideration of realistic ba nd structures is pos- sible from this starting point. We adopt the adiabatic local-density ap proximation (ALDA, essentially the Stoner model) for the exchange field: γ/planckover2pi1Hxc[ˆρ](r,t)≈∆xcm(r,t), (28) with direction m=−s/slocked to the time-dependent spin density (27) (assuming γ >0). In another simple model of ferromagnetism, the so-called s-dmodel, conducting selec- trons interact with the exchange field of the delectrons which are assumed to be localized to the crystal lattice sites. The d-orbital electron spins are supposed to account for most of the magnetic moment. Because d-electron shells have large net spins and strong ferromagnetic correlations, they are usually treated classically. In a mean-field s-ddescription, therefore, conducting sorbitals are described by the same Hamiltonian (25) with an exchange field (28). The differences between the Stoner and s-dmodels for the magnetization dynamics are rather minor and subtle. In the ALDA/Stoner model, the excha nge potential is (on the scale of the magnetization dynamics) instantaneously aligned with th e total magnetization. In contrast, the direction unit vector min thes-dmodel corresponds to the dmagnetization, which is allowed to be misaligned with the smagnetization, transferring torque between the sanddmagnetic moments. Since most of the magnetization is carried by the latter, the external field Hcouples mainly to the dspins, while the sspins respond to and follow the time-dependent exchange field (28). As ∆ xcis usually much larger than the external (includ- ing demagnetization and anisotropy) fields that drive collective magn etization dynamics, the total magnetic moment will always be very close to m. A more important difference of the philosophy behind the two models is the presumed shielding of the dorbitals from exter- nal disorder. The reduced coupling with dissipative degrees of free dom would imply that 17their dynamics are much less damped. (Whether this is actually the ca se in real systems remains to be proven, however.) Consequently, the magnetization damping has to come from the disorder experienced by the itinerant selectrons. As in the case of the itinerant ferromagnets, the susceptibility has to be calculated self-consist ently with the magnetization dynamics parametrized by m. For more details on this model, we refer to Refs. [10, 49]. With the above differences in mind, the following discussion is applicable t o both models. In order to avoid confusion, we remark that the equilibrium spin density s0introduced earlier refers to the total spin density, i.e., d- pluss-electron spin density, while Eq. (27) refers only to the latter. The Stoner model is more appropriate for trans ition-metal ferromagnets because of the strong hybridization between dands,pelectrons. Magnetic semiconductors are characterized by deep magnetic impurity states for which the s-dmodel may be a better choice. The single-particle itinerant electron response to electric and magn etic fields in Hamil- tonian (25) is all that is needed to compute the magnetization dynam ics microscopically. As mentioned above, the distinction between the Stoner and s-dmodels will appear only at the end of the day, when we self-consistently relate m(r,t) to the itinerant electron spin response. Before proceeding, we observe that since the consta ntsαandβwhich parametrize the magnetic equation of motion (9) affect the linear response to a s mall transverse applied field with respect to a uniform magnetization, we can obtain them by a linear-response calculation for the single-domain bulk ferromagnet. The large-scale magnetization texture associated with a domain wall does not affect the value of these para meters, in the consid- ered limit. The linear response to a small magnetic field is complicated by the presence of an electrically-driven applied current, however. Since the Kubo for mula based on two-point equilibrium Green’s functions is insufficient to calculate the response t o simultaneous mag- netic and electric fields, we chose to pursue a nonequilibrium (Keldysh ) Green’s function formalism in Refs. [13, 49]. A technically impressive equilibrium Green’s fu nction calcula- tion has been carried out in Ref. [50], which to a large extent confirme d our results, but also contributed some important additions that will be discussed below. The central quantity in the kinetic equation approach [13, 49] is the nonequilibrium component of the 2 ×2 distribution function ˆfk(r,t). In the quasiparticle approximation, validwhen∆ xc≪EF[49], thekineticequationcanbereducedtoasemiclassical Boltzmann - like equation that accounts for electron drift in response to the ele ctric field as well as the 18spin precession in the magnetic field. The nonequilibrium component of the spin density readss′= (/planckover2pi1/2)/integraltext d3kfk/(2π)3, wherefk= Tr[ˆfkˆσ]: ∂ts′−∆xc /planckover2pi1z×s′−∆xcs /planckover2pi1z×u=−/planckover2pi1 2/integraldisplayd3k (2π)3(vk·∂r)fk−s′+su τσ. (29) shere is the equilibrium spin density of itinerant electrons, vk=∂kεks//planckover2pi1is the momentum- dependent group velocity, and the magnetization direction m=z+uis assumed to undergo a small precession urelative to the uniform equilibrium direction z. The first term on the right-hand side is the spin-current divergence and the last term is t he spin-dephasing term introduced phenomenologically in Ref. [49] and studied microscopically in Ref. [13]. As detailed in Ref. [49], the spin currents have to be calculated from the full kinetic equation and then inserted in Eq. (29). The final result (for the Stoner mod el) is given by Eq. (10) or, equivalently, Eq. (9), with α=β. The latter is proportional to the spin-dephasing rate: β=/planckover2pi1 τσ∆xc. (30) The derivation assumes ω,τ−1 σ≪∆xc//planckover2pi1, which is typically the case in real materials suffi- ciently below the Curie temperature. The s-dmodel yields the same result for β, but α=ηβ (31) is reduced by the η=s/s0ratio, i.e., the fraction of the itinerant to the total angular momentum. [Note that Eq. (31) is also valid for the Stoner model sinc e thens0=s.] For thes-dmodel, the equation of motion (9) clearly cannot be reduced to Eq. ( 10), since α∝negationslash=β. The steady-state current-driven velocity (22) for both mean-fi eld models becomes v=−βPj αs0=−Pj s, (32) wheresis the itinerant electron spin density. Interestingly, the velocity (3 2) is completely determined by properties of the conducting electrons, even for t hes−dmodel. In the Drude model, v∝Eτ m∗, (33) whereEis the applied electric field, τis the characteristic momentum scattering time, and m∗is the effective mass of the itinerant bands at the Fermi energy. We expect the velocity (33), which is essentially the conducting electron drift velocity, to b e suppressed for the 19s-dmodel if the dorbitals are coupled to their own dissipative bath, which has not been included in the above treatment. Ref. [50] refines these results by relaxing the assumption that ∆ xc≪EFand by consid- ering also anisotropic spin-dephasing impurities, which results in α∝negationslash=βfor both Stoner and s-dmodels. Ref.[51]laterofferedaKeldysh functional-integral appro achleading tothesame results. (These authors also found stochastic torques express ed in terms of thermal fluctu- ations (5) in the weak current limit; see, however, Ref. [71] for add itional current-induced stochastic terms present in the case of an inhomogeneous magnet ization.) Consider, for example, weak magnetic disorder described by the potential ˆHσ=h(r)·ˆσwith Gaus- sian white-noise correlations ∝angb∇acketleftha(r)hb(r′)∝angb∇acket∇ight ∝Uaδabδ(r−r′), where Ua=U⊥(U/bardbl) whenais perpendicular (parallel) to the equilibrium magnetization direction. (S pin-orbit interaction associated with scalar disorder gives similar results.) For isotropic dis order,U⊥=U/bardbl, and ∆xc≪EF,α/β≈ηwithη=s/s0, as was already discussed (reducing to η= 1 for the Stoner model). Even for larger exchange, the correction to this α/βratio turns out to be rather small: For parabolic bands, for example, α/β≈[1−(∆xc/EF)2/48]η. This ratio is more sensitive to anisotropies U/bardbl∝negationslash=U⊥, however, so that in general α/β∝negationslash=ηeven in the limit ∆ xc/EF→0 [50]. VI. SUMMARY AND OUTLOOK Our microscopic understanding is based on a mean-field approximatio n, in which itiner- ant electrons interact self-consistently with a space- and time-de pendent exchange field. We presented results for the local-spin-density approximation and th e mean-field s-dmodel. We identified a relation between dissipative terms parametrized by αandβand spin-dephasing scattering potentials. The central result for the collective low-fr equency long-wavelength current-driven magnetization dynamics can be formulated as a gen eralization of the phe- nomenological Landau-Lifshitz-Gilbert equation, accounting for t he current-driven torques. One should in general also include stochastic terms due to thermal fl uctuations as well as nonequilibrium shot-noise contribution in the presence of applied cur rentj[71]. Despite some recent efforts, stochastic effects remain to be relatively une xplored both theoretically and experimentally, however. The most important parameter that determines the effect of an ele ctric current on the 20collective magnetization dynamics in extended systems is the ratio β/α. We find that this ratio is not universal and in general depends on details of the ba nd structure and spin-dephasing processes. Nevertheless, simple models give α∼βwith the special limit α≈βfor the Stoner model with weak and isotropic spin-dephasing disord er. Solving the magnetization equation of motion for a domain wall is rather straight forward at low dc currents, when the wall is only slightly compressed. The domain-wall motion can then be modeled within the Walker ansatz, based on parametrizing the magne tic dynamics in terms of wall position and spin distortion. Two regimes can then be distinguis hed: At the lowest currents, the wall moves steadily in the presence of a constant un iform current, while above the so-called Walker threshold, the magnetization close to the wall c enter starts oscillating, resulting in a singular dependence of the average velocity on the app lied current. The values of the αandβparameters are not affected by the magnetization textures. Micromagnetic simulations can provide better understanding of exp erimental results in the regimeswheredomainwallsarenotwell describedbyaone-dimensiona l model. Experiments can contribute to the understanding by studying ferromagnets w ith systematic variations of impurity types and concentration, for Py and other different ma terials. Experimental investigation of creep in metallic ferromagnets at temperatures fa r below the critical ones, as compared to studies [33, 39] on magnetic semiconductors close t o the Curie transition, are highly desirable in order to advance our understanding. Besides realistic microscopic evaluations of the key parameters αandβ, the collective current-driven magnetization dynamics pose many theoretical ch allenges, in the spirit of classical nonlinear dynamical systems. Current-driven magnetism displays a rich behavior well beyond what can be achieved by applied magnetic fields only. At su percritical currents, ferromagnetism becomes unstable, possibly leading to chaotic dyna mics [79], although al- ternative scenarios have been also suggested [80]. Domain-wall dyn amics in a medium with disordered pinning potentials pose an interesting yet, at weak applie d currents, tractable problem. Spin torques and dynamics in sharp walls and the role of stro ng intrinsic spin-orbit coupling (relevant for dilute magnetic semiconductors) are not yet completely understood. Oscillatory domain-wall motion under ac currents and in curved geom etries is also starting to attract attention both experimentally and theoretically [35, 81]. Another direction of recent activities concern the backaction of a moving domain wall on t he charge degrees of freedom [82, 83, 84, 85, 86]. 21With the exciting recent and forthcoming experimental developmen ts, the questions con- cerning interactions of the collective ferromagnetic order with elec tric currents will certainly challenge theoreticians for many years to come. The prospects of using purely electric means to efficiently manipulate magnetic dynamics are also promising for prac tical applications. VII. ACKNOWLEDGMENTS Wewould like tothanktheEditors forcarefullyreading themanuscrip t andmaking many useful comments. 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2007-09-18

We give a brief account of recent developments in the theoretical understanding of the interaction between electric currents and inhomogeneous ferromagnetic order parameters. We start by discussing the physical origin of the spin torques responsible for this interaction and construct a phenomenological description. We then consider the electric current-induced ferromagnetic instability and domain-wall motion. Finally, we present a microscopic justification of the phenomenological description of current-driven magnetization dynamics, with particular emphasis on the dissipative terms, the so-called Gilbert damping $\alpha$ and the $\beta$ component of the adiabatic current-driven torque.

Theory of current-driven magnetization dynamics in inhomogeneous ferromagnets

0709.2937v2

arXiv:0802.1740v1 [cond-mat.other] 12 Feb 2008Temperature dependent magnetization dynamics of magnetic nanoparticles A. Sukhov1,2and J. Berakdar2 1Max-Planck-Institut f¨ ur Mikrostrukturphysik, Weinberg 2, D-0 6120 Halle/Saale, Germany 2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle-Wittenbe rg, Heinrich-Damerow-Str. 4, 06120 Halle, Germany Abstract. Recent experimental and theoretical studies show that the switc hing behavior of magnetic nanoparticles can be well controlled by extern al time-dependent magnetic fields. In this work, we inspect theoretically the influence o f the temperature and the magnetic anisotropy on the spin-dynamics and the switching properties of single domain magnetic nanoparticles (Stoner-particles). Our theo retical tools are the Landau-Lifshitz-Gilbert equation extended as to deal with finit e temperatures within a Langevine framework. Physical quantities of interest are t he minimum field amplitudes required for switching and the corresponding rever sal times of the nanoparticle’s magnetic moment. In particular, we contrast the ca ses of static and time-dependent external fields and analyze the influence of dampin g for a uniaxial and a cubic anisotropy. PACS numbers: 75.40.Mg, 75.50.Bb, 75.40.Gb, 75.60.Jk, 75.75.+aTemperature dependent magnetization dynamics of magnetic nanoparticles 2 1. Introduction Inrecent years, therehasbeenasurgeofresearchactivities fo cusedonthespindynamics and the switching behavior of magnetic nanoparticles [1]. These stud ies are driven by potential applications in mass-storage media and fast magneto- electronic devices. In principle, various techniques are currently available for controllin g or reversing the magnetization of a nanoparticle. To name but a few, the magnetizat ion can be reversed by a short laser pulse [2], a spin-polarized electric current [3, 4] or an alternating magnetic field [5, 6, 7, 8, 9, 10, 11, 12, 13]. Recently [6], it has been sh own for a uniaxial anisotropythattheutilizationofaweak time-dependent ma gneticfieldachieves a magnetization reversal faster than in the case of a static magne tic field. For this case [6], however, the influence of the temperature and the different ty pes of anisotropy on the various dependencies of the reversal process have not be en addressed. These issues, which are the topic of this present work, are of great impor tance since, e.g. thermal activation affects decisively the stability of the magnetizat ion, in particular when approaching the superparamagnetic limit, which restricts the density of data storage [14]. Here we study the possibility of fast switching at finite t emperature with weak external fields. We consider magnetic nanoparticles with an ap propriate size as to displayalong-rangemagneticorderandtobeinasingledomainremane ntstate(Stoner- particles). Uniaxial and cubic anisotropies are considered and show n to decisively influence the switching dynamics. Numerical results are presented and analyzed for iron-platinum nanoparticles. In principle, the inclusion of finite tempe ratures in spin- dynamics studies is well-established (cf. [19, 20, 23, 15, 16, 1] and references therein) and will be followed here by treating finite temperatures on the level of Langevine dynamics. For the analysis of switching behaviour the Stoner and Wo hlfarth model (SW) [17] is often employed. SW investigated the energetically metas table and stable position of the magnetization of a single domain particle with uniaxial an isotropy in the presence of an external magnetic field. They showed that the minimum static magnetic field (generally referred to as the Stoner-Wohlfarth (SW ) field or limit) needed to coherently reverse the magnetization is dependent on the direc tion of the applied field with respect to the easy axis. This dependence is described by t he so-called Stoner-Wohlfarth astroid. The SW findings rely, however, on a sta tic model at zero temperature. Application of a time-dependent magnetic field reduc es the required minimum switching field amplitude below the SW limit [6]. It was, however, no t yet clear how finite temperatures will affect these findings. To clarify th is point, we utilize an extension of the Landau-Lifshitz-Gilbert equation [18] including fi nite temperatures on the level of Langevine dynamics [19, 20, 23]. Our analysis shows t he reversal time to be strongly dependent on the damping, the temperature and th e type of anisotropy. These dependencies are also exhibited to a lesser extent by the crit ical reversal fields. Thepaperisorganizedasfollows: nextsection2presents detailsof thenumerical scheme and the notations whereas section 3 shows numerical results and a nalysis for Fe 50Pt50 and Fe 70Pt30nanoparticles. We then conclude with a brief summary.Temperature dependent magnetization dynamics of magnetic nanoparticles 3 2. Theoretical model In what follows we focus on systems with large spins such that their m agnetic dynamics can be described by the classical motion of a unit vector Sdirected along the particle’s magnetization µ, i.e.S=µ/µSandµSisthe particle’s magnetic moment at saturation. The energetics of the system is given by H=HA+HF. (1) whereHA(HF) stands for the anisotropy (Zeeman energy) contribution. Furt hermore, the anisotropy contribution is expressed as HA=−Df(S) withDbeing the anisotropy constant. Explicit formof f(S)isprovidedbelow. Themagnetizationdynamics, i.e. the equation of motion for S, is governed by the Landau-Lifshitz-Gilbert (LLG) equation [18] ∂S ∂t=−γ (1+α2)S×/bracketleftBig Be(t) +α(S×Be(t))/bracketrightBig . (2) Here we introduced the effective field Be(t) =−1/(µS)∂H/∂Swhich contains the external magnetic field and the maximum anisotropy field for the unia xial anisotropy BA= 2D/µS.γis the gyromagnetic ratio and αis the Gilbert damping parameter. The temperature fluctuations will be described on the level of the Lang evine dynamics [19]. This means, a time-dependent thermal noise ζ(t) adds to the effective field Be(t) [19]. ζ(t) is a Gaussian distributed white noise with zero mean and vanishing time correlator /angbracketleftζi(t′)ζj(t)/angbracketright=2αkBT µsγδi,jδ(t−t′). (3) i,jare Cartesian components, Tis the temperature and kBis the Boltzmann constant. It is convenient to express the LLG in the reduced units b=Be BA, τ=ωat, ωa=γBA. (4) The LLG equation reads then ∂S ∂τ=−1 (1+α2)S×/bracketleftBig b(τ) +α(S×b(τ))/bracketrightBig , (5) where the effective field is now given explicitly by b(τ) =−1 µSBA∂H ∂S+Θ(τ) (6) with /angbracketleftΘi(τ′)Θj(τ)/angbracketright=ǫδi,jδ(τ−τ′);ǫ=2αkBT µsBA. (7) Thereducedunitsareindependent ofthedampingparameter α. Inthefollowingsections we use extensively the parameter q=kBT D. (8) qis a measure for the thermal energy in terms of the anisotropy ene rgy. And d=D/(µSBA) expresses the anisotropy constant in units of a maximum anisotro pyTemperature dependent magnetization dynamics of magnetic nanoparticles 4 energy for the uniaxial anisotropy and is always 1 /2. The stochastic LLG equation (5) in reduced units (4) is solved numerically using the Heun method which c onverges in quadratic mean to the solution of the LLG equation when interprete d in the sense of Stratonovich [20]. For each type of anisotropy we choose the time s tep ∆τto be one thousandth part of the corresponding period of oscillations. The v alues of the time interval in not reduced units for uniaxial and cubic anisotropies are ∆tua= 4.61·10−15s and ∆tca= 64.90·10−15s, respectively, providing us thus with correlation times on the femtosecond time scale. The reason for the choice of such small tim e intervals is given in [19], where it is argued that the spectrum of thermal-agitation forc es may be considered as white up to a frequency of order kBT/hwithhbeing the Planck constant. This value corresponds to 10−13sfor room temperature. The total scale of time is limited by a thousand of such periods. Hence, we deal with around one million iter ation steps for a switching process. Details of realization of this numerical scheme c ould be found in references [21, 22, 20]. We note by passing, that attempts have b een made to obtain, under certain limitations, analytical results for finite-temperatur e spin dynamics using the Fokker-Planck equation (cf. [15, 16] and references therein ). For the general case discussed here one has however to resort to fully numerical appro aches. 3. Results and interpretations Weconsider a magneticnanoparticleina singledomainremanent state (Stoner-particle) with aneffective anisotropy whose origin can be magnetocrystalline, magnetoelastic and surface anisotropy. We assume the nanoparticle to have a spheric al form, neglecting thus the shape anisotropy contributions. In the absence of exte rnal fields, thermal fluctuations may still drive the system out of equilibrium. Hence, the stability of the system as the temperature increases becomes an important is sue. The time tat which the magnetization of the system overcomes the energy barr ier due to the thermal activation, also called the escape time , is given by the Arrhenius law t=t0eD kBT, (9) where the exponent is the ratio of the anisotropy to the thermal e nergy. The coefficient t0may be inferred when D≫kBTand for high damping [19] (see [25] for a critical discussion) t0=1+α αγπµS 2D/radicalBigg kBT D. (10) Here we focus on two different types of iron-platinum-nanoparticle s: The compound Fe50Pt50which has a uniaxial anisotropy [26, 27], whereas the system Fe 70Pt30possesses a cubic anisotropy [24]. Furthermore, the temperature dependen ce will be studied by varyingq(cf. eq.(8)). For Fe 50Pt50the important parameters for simulations are the diameter of the nanoparticles 6 .3nm, the strength of the anisotropy Ku= 6·106J/m3, the magnetic moment per particle µp= 21518 ·µBand the Curie-temperature Tc= 710K[26, 27].Temperature dependent magnetization dynamics of magnetic nanoparticles 5 The relation between KuandDuisDu=KuVu, whereVuis the volume of Fe 50Pt50 nanoparticles. In the calculations for Fe 50Pt50nanoparticles the following qvalues were chosen: q1= 0.001,q2= 0.005 orq3= 0.01 which correspond to the real temperatures 56 K, 280Kor 560K, respectively (these temperatures are below the blocking temperature). The corresponding escape times are tq1≈2·10217s,tq2≈1075s andtq3≈7·1031s, respectively. In some cases we also show the results for an additio nal temperature q01= 0.0001 with the corresponding real temperature to be equal to 5 K. Thecorrespondingescapetimeforthisis tq01≈104300s. Thesetimesshouldbecompared with the measurement period which is about tm≈5ns, endorsing thus the stability of the system during the measurements. For Fe 70Pt30the parameters are as follows: The diameter of the nanoparticles 2 .3nm, the strength of the anisotropy Kc= 8·105J/m3, the magnetic moment per particle µp= 2000·µB, the Curie-temperature is Tc= 420K[24], and Dc=KcVc(Vcis the volume.) For Fe 70Pt30nanoparticles the values of qwe choose in the simulations are q4= 0.01,q5= 0.03 orq6= 0.06 which means that the temperature is respectively 0.3K, 0.9Kor 1.9K. The escape times are tq4≈1034s,tq5≈2·105sandtq6≈2·10−2s, respectively. Here we also choose an intermediate value q04= 0.001 and the real temperature 0 .03Kwith the corresponding escape time to be equal to tq04≈10430s. The measurement period is the same, namely about 5 ns. All values of the escape times were given for α= 0.1. Central to this study are two issues: The critical magnetic field and the corresponding reversal time . The critical magnetic field we define as the minimum field amplitude needed to completely reverse the magnetization. The reversal tim e is the corresponding time for this process. In contrast, in other studies [6] the rever sal time is defined as the time needed for the magnetization to switch from the initial position t o the position Sz= 0, our reversal time is the time at which the magnetization reaches the very proximity of the antiparallel state (Fig. 1). The difference in the defi nition is in so far important as the magnetization position Sz= 0 at finite temperatures is not stabile so it may switch back to the initial state due to thermal fluctuations an d hence the target state is never reached. 3.1. Nanoparticles having uniaxial anisotropy: Fe 50Pt50 A Fe50Pt50magnetic nanoparticle has a uniaxial anisotropy whose direction defi nes the zdirection. The magnetization direction Sis specified by the azimuthal angle φand the polar angle θwith respect to z. In the presence of an external field bapplied at an arbitrarily chosen direction, the energy of the system in dimensio nless units derives from ˜H=−dcos2θ−S·b. (11) The initial state of the magnetization is chosen to be close to Sz= +1 and we aim at the target state Sz=−1.Temperature dependent magnetization dynamics of magnetic nanoparticles 6 00.511.522.533.544.5 5 Time, [ns]-1-0.500.51Magnetization SzT0=0 K T3=560 K Figure 1. (Color online) Magnetization reversal of a nanoparticle when a stat ic field is applied at zero Kelvin ( q0= 0, black) and at reduced temperature q3= 0.01≡560K (blue). The strengths of the fields in the dimensionless units (4) and (8) areb= 1.01 andb= 0.74, respectively. The damping parameter is α= 0.1. The start position of the magnetization is given by the initial angle θ0=π/360 between the easy axis and the magnetization vector. 3.1.1. Static field For an external static magnetic field applied antiparallel to the z direction ( b=−bez) eq.(11) becomes ˜H=−dcos2θ+bcosθ. (12) To determine the critical field magnitude needed for the magnetizat ion reversal we proceed as follows (cf. Fig. 1): At first, the external field is increa sed in small steps. When the magnetization reversal is achieved the corresponding va lues of the critical field versus the damping parameter αare plotted as shown in the inset of Fig. 3. The reversal times corresponding to the critical static field amplitudes of Fig. 3 are plotted versus damping in Fig. 4. In the Stoner-Wohlfarth (static) model the mechanism of magnet ization reversal is not due to damping. It is rather caused by a change of the energy profi le in the presence of the field. The curves displayed on the energy surface in Fig. 2 mark t he magnetization motion in the E(θ,φ) landscape. The magnetization initiates from φ0= 0 and θ0and ends up at θ=π. As clearly can be seen from the figure, reversal is only possible if th e initial state is energetically higher than the target state. This ”low d amping” reversal is, however, quite slow, which will be quantified more below. For the re versal at T= 0, the SW-model predicts a minimum static field strength, namely bcr=B/BA= 1 (the dashed line in Fig. 3 ). This minimum field measured with respect to the anisotropy field stren gth does not depend on the damping parameter α, provided the measuring time is infinite. For T >0 the simulations were averaged over 500 cycles with the result shown in Fig. 3. The one- cycle data are shown in the inset. Fig. 3 evidences that with increasin g temperature thermal fluctuations assist a weak magnetic field as to reverse the magnetization. Furthermore, the required critical field is increased slightly at very large and strongly at very small damping with the minimum critical field being at α≈1.0. The reason forTemperature dependent magnetization dynamics of magnetic nanoparticles 7 Figure 2. (Color online) The trajectories of the magnetization unit vector parameterized by the angles θandφat zero temperature. Other parameters are as in Fig. 1 for q0. 00.511.522.533.544.5 5 5.5 6 Damping α00.20.40.60.81Critical DC field T0=0 K (SW) T01=5 K T1=56 K T2=280 K T3=560 K0 1 2 3 4 5 6 Damping α00.20.40.60.81Critical DC field T3=560 K Figure 3. (Color online) Critical static field amplitudes vs. the damping paramet ers for different temperatures averaged over 500 times. Inset show s not averaged data for q3= 0.01≡560K. this behavior is that for low damping the second term of equation (2) is much smaller than the first one, meaning that the system exhibits a weak relaxat ion. In the absence of damping, higher fields are necessary to switch the magnetization . For high α, both terms in equation (2) become small (compared to a low-damping case ) leading to a stiff magnetization and hence higher fields are needed to drive the ma gnetization. For moderate damping, we observe a minimum of switching fields which is due to an optimal interplay between precessional and damping terms. Obviously, finit e temperatures do not influence this general trend. For the case of q0= 0, the Landau-Lifshitz-Gilbert equation of motion can be solved analytically in spherical coordinates. The details of the solution can b e found in Ref. [20] (eq. (A1)-(A8)). The final result of the solution in this refere nce differs, however, from the one given here due to to different geometries in these syst ems. In contrast to our alignment of the magnetization and the external field, the stat ic field in Ref. [20] is applied parallel to the initial position of the magnetization. For the so lution, we assume that the magnetization starts at θ=θ0=π/360 and arrives at θ=π. Note, that the expression θ/negationslash= 0 is important only for zero Kelvin since the switching is not possible ifTemperature dependent magnetization dynamics of magnetic nanoparticles 8 the magnetization starts at θ0= 0 (the vector product in equation (2) vanishes). The reversal time in the SW-limit is then given by trev=g(θ0,b)1+α2 α, (13) wheregis defined as g(θ0,b) =µS 2γD1 b2−1ln/parenleftBiggtg(θ/2)bsinθ b−cosθ/parenrightBigg/vextendsingle/vextendsingle/vextendsingleπ θ0. (14) Fromthisrelationweinferthatswitchingispossibleonlyiftheappliedfie ldislargerthan the anisotropy field and the reversal time decreases with increasin gb. This conclusion is independent of the Stoner-Wohlfarth model and follows directly fr om the solution of the LLG equation. An illustration is shown by the dashed curve in Fig. 4, wh ich was a test to compare the appropriate numerical results with the analytical o ne. As our aim is the study of the reversal-time dependence on the magnetic moment an d on the anisotropy constant, we deem the logarithmic dependence in Eq.(14) to be weak and write g(b,µS,D)≈µS γ2D B2µ2 S−4D2. (15) This relation indicates that an increase in the magnetic moment result s in a decrease of the reversal time. The magnetic moment enters in the Zeeman en ergy and therefore the increase in magnetic moment is very similar to an increase in the mag netic field. An increase of the reversal time with the increasing anisotropy orig inates from the fact that the anisotropy constant determines the height of the poten tial barrier. Hence, the higher the barrier, the longer it takes for the magnetization to ove rcome it. For the other temperatures the corresponding reversal times ( also averaged over 500 cycles) are shown in Fig. 4. In contrast to the case T= 0, where an appreciable dependence on damping is observed, the reversal times for finite t emperatures show a weaker dependence on damping. If α→0 only the precessional motion of the magnetization is possible and therefore trev→ ∞. At high damping the system relaxes on a time scale that is much shorter than the precession time, giving t hus rise to an increase in switching times. Additionally, one can clearly observe the in crease of the reversal times with increasing temperatures, even though these time remain on the nanoseconds time scale. 3.1.2. Alternating field As was shown in Ref. [6, 7, 15] theoretically and in Ref. [5] experimentally, a rotating alternating field with no static field being ap plied can also be used for the magnetization reversal. A circular polarized microwa ve field is applied perpendicularly to the anisotropy axis. Thus, the Hamiltonian might b e written in form of equation (11) and the applied field is b(t) =b0cosωtex+b0sinωtey, (16) whereb0is the alternating field amplitude and ωis its frequency. For a switching of the magnetization the appropriate frequency of the applied altern ating field should beTemperature dependent magnetization dynamics of magnetic nanoparticles 9 00.511.522.533.544.5 5 5.5 6 Damping α012345Reversal time, [ns] Theory T0=0 K T01=5 K T1=56 K T2=280 K T3=560 K 0 1 2 3 4 5 6 Damping α012345Reversal time, [ns] T3=560 K Figure 4. (Color online) Reversal times corresponding to the critical static fi elds in Fig. 3 vs. damping averaged over 500 cycles. Inset shows the as-c alculated numerical results for q3= 0.01≡560K(one cycle). 00.050.10.150.20.25 Time, [ns]-1-0.500.51Magnetization SzT0=0 K 0 1 2 3 4 5 Time, [ns]-1-0.500.51Magnetization Sz T3=560 K Figure 5. (Color online) Magnetization reversal in a nanoparticle using a time dependentfieldfor α= 0.1andatazerotemperature. Thefieldstrengthandfrequency in the units (4) are respectively b0= 0.18 andω=ωa/1.93. Inset shows for this case the magnetization reversal for the temperature q3= 0.01≡560Kwithb0= 0.17 and the same frequency. chosen. In Ref. [15] analytically and in [6] numerically a detailed analysis of the optimal frequency is given which is close to the precessional frequency of t he system. The role of temperature and different types of anisotropy have not yet be en addressed, to our knowledge. Fig. 5 shows our calculations for the reversal process at two differ ent temperatures. In contrast to the static case, the reversal proceeds through many oscillations on a time scale of approximately ten picoseconds. Increasing the tempe rature results in an increase of the reversal time. Fig. 6 shows the trajectory of the magnetization in the E( θ,φ) space related to the case of the alternating field application. Compared with the situation depicted in Fig. 2, the trajectory reveals a quite delicate motion of the magnetizat ion. It is furthermore, noteworthy that the alternating field amplitudes needed for the re versal (cf. Fig. 7) are substantially lower than their static counterpart, meaning that th e energy profile of theTemperature dependent magnetization dynamics of magnetic nanoparticles 10 Figure 6. (Color online) Trajectories followed by magnetization as specified by θand φforq0= 0. Other parametersare b0= 0.18,α= 0.1 andω=ωa/1.93. Energy-profile variations due to the oscillating external field are not visible on this sc ale. system is not completely altered by the external field. Fig. 7 inspects the dependence of the minimum switching field amplitude on damping. The critical fields are obtained upon averaging over 500 cy cles. The SW- limit lies by 1 on this scale. In contrast to the static case, the critical fields increase with increasing α. In the low damping regime the critical field is smaller than in the case of a static field. This behavior can be explained qualitatively by a r esonant energy- absorptionmechanism when thefrequencies oftheappliedfieldmatc hes thefrequency of the system. Obviously, at very low frequencies (compared to the p recessional frequency) the dynamics resembles the static case. The influence of the temperature on the minimum alternating field amp litudes is depicted in Fig. 7. With increasing temperatures, the minimum amplitud es become smaller due to an additional thermal energy pumped from the enviro nment. The curves in this figure can be approached with two linear dependencies with diffe rent slopes for approximately α <1 and for α >1; for high damping it is linearly dependent on α, more specifically it can be shown that for high damping the critical field s behave as bcr≈1+α2 α. (17) The proportionality coefficient contains the frequency of the alter nating field and the critical angle θ. The solution (17) follows from the LLG equation solved for the case when the phase of the external field follows temporally that of the m agnetization, which we checked numerically to be valid. The reversal times associated with the critical switching fields are s hown in (Fig. 8). Qualitatively, we observe the same behavior as for the case of a static field. The values of the reversal times for T= 0 are, however, significantly smaller than for the static case. For the same reason as in the static field case, an incre ased temperature results in an increase of the switching times.Temperature dependent magnetization dynamics of magnetic nanoparticles 11 00.511.522.533.544.5 5 5.5 6 Damping α00.511.522.5 Critical AC fieldT0=0 K T1=56 K T2=280 K T3=560 K0 1 2 3 4 5 6 Damping α00.511.522.5 Critical AC fieldT3=560 K Figure 7. (Color online) Critical alternating field amplitudes vs. damping for different temperatures averaged over 500 times. Inset shows no t averaged data for q3= 0.01≡560K. 00.511.522.533.544.5 5 5.5 6 Damping α012345Reversal time, [ns]T1=56 K T2=280 K T3=560 K 0 1 2 3 4 5 6 Damping α00.10.20.30.40.5Reversal time, [ns]T0=0 K Figure 8. (Coloronline) The damping dependence ofthe reversaltimes corre sponding to the critical field amplitudes of Fig. 7 for different temperatures. Inset shows the case of zero Kelvin. 3.2. Nanoparticles with cubic anisotropy: Fe 70Pt30 Now we focus on another type of the anisotropy, namely a cubic anis otropy which is supposed to be present for Fe 70Pt30nanoparticles [24]. The energetics of the system is then described by the functional form ˜H=−d(S2 xS2 y+S2 yS2 z+S2 xS2 z)−S·b, (18) or in spherical coordinates ˜H=−d(cos2φsin2φsin4θ+cos2θsin2θ)−S·b. (19) In contrast to the previous section, there are more local minima or in other words more stable states of the magnetization in the energy profile for the Fe 70Pt30nanoparticles. It can be shown that the minimum barrier that has to be overcome is d/12 which is twelve times smaller than that in the case of a uniaxial anisotropy. Th e maximal one is onlyd/3. The magnetization of these nanoparticles is first relaxed to the initia l state close toTemperature dependent magnetization dynamics of magnetic nanoparticles 12 Figure 9. (Color online) Trajectories of the magnetization in the θ(φ) space (q0= 0). In the units (4) we choose b= 0.82 andα= 0.1. φ0=π/4 andθ0= arccos(1 /√ 3), whereas in the target state it is aligned antiparallel to the initial one, i. e. φe= 3π/4 andθe=π−arccos(1/√ 3). In order to be close to the starting state for the uniaxial anisotropy case we choose φ0= 0.2499·π,θ0= 0.3042·π. 3.2.1. Static driving field A static field is applied antiparallel to the initial state of the magnetization, i.e. b=−b/√ 3(ex+ey+ez). (20) In Fig. 9 the trajectory of the magnetization in case of an applied st atic field is shown. Similar to the previous section the energy of the initial state lies highe r than that of the target state. The magnetization rolls down the energy landsca pe to eventually end up by the target state. The trajectory the magnetization fo llows is completely different from the one for the uniaxial anisotropy. Fig. 10 suppleme nts this scenario of the magnetization reversal by showing the time evolution of the Szvector. Because of the different anisotropy type, the trajectory is markedly differ ent from the case of the uniaxial anisotropy and a static field. Here we show only the Szmagnetization component even though the other components also have to be tak en into account in order to avoid a wrong target state. The procedure to determine the critical field amplitudes is similar to th at described in the previous section. In Fig. 11 the critical fields versus the dampin g parameter for different temperatures are shown. For q0, the critical field strength is smaller than 1. This is consistent insofar as the maximum effective field for a cubic anis otropy is2 3BA. In principle, the critical field turns out to be constant for all αbut for an infinitely large measuring time. Since we set this time to be about 5 nanoseconds, th e critical fields increase for small and high damping. On the other hand, at lower tem peratures smaller critical fields are sufficient for the (thermal activation-assisted) reversal process. The behaviour of the corresponding switching times presented in Fig . 12 only supplements the fact of too low measuring time, which is chosen as 5 nsfor a better comparison of these results with ones for uniaxial anisotropy. Ind eed, constant jumps in the reversal times for T= 0Kas a function of damping can be observed. The reasonTemperature dependent magnetization dynamics of magnetic nanoparticles 13 00.511.522.533.544.5 5 Time, [ns]-1-0.500.51Magnetization Sz T0=0 K T6=1.9 K Figure 10. (Color online) Magnetization reversal of a nanoparticle when a stat ic field b= 0.82 is applied and for α= 0.1 at zero temperature (black). The magnetization reversal for α= 0.1,b= 0.22 andq6= 0.06≡1.9Kis shown with blue color. 0 1 2 3 4 56 Damping α00.20.40.60.811.2 Critical DC field T0=0 K T04=0.03 K T4=0.3 K T5=0.9 K T6=1.9 K0 1 2 3 4 5 6 Damping α00.51Critical DC field T6=1.9 K Figure 11. (Color online) Critical static field amplitudes vs. the damping paramet ers for different temperatures averaged over 500 times. Inset show s not averaged data for q6= 0.06≡1.9K. 0 1 2 3 4 56 Damping α012345Reversal time, [ns] T0=0 K T04=0.03 K T4=0.3 K T5=0.9 K T6=1.9 K Figure 12. (Color online) Reversal times corresponding to the critical static fi elds of Fig. 11 vs. damping averaged over 500 times. why the reversal times for finite temperatures are lower is as follow s: The initial state forT= 0Kis chosen to be very close to equilibrium. This does not happen for finit eTemperature dependent magnetization dynamics of magnetic nanoparticles 14 Figure 13. (Color online) Trajectories of the magnetization vector specified b y the angles θandφat zero temperature. The chosen parameters are b0= 0.055 and ω= ˜ωa/1.93, where ˜ ωa= 2/3ωa. temperatures, where the system due to thermal activation jump s out of equilibrium (cf. see Fig. 10). 3.2.2. Time-dependent external field Here we consider the case of an alternating field that rotates in the plain perpendicularly to the initial state of the ma gnetization. It is possible to switch the magnetization with a field rotating in the xy−plane but the field amplitudes turn out to be larger than those when the field rotates p erpendicular to the initial state. For the energy this means that the field entering equa tion (19) reads b(t) = (b0cosω1tcosφ0+b0sinω1tsinφ0cosθ0)ex +(−b0cosω1tsinφ0+b0sinω1tcosφ0cosθ0)ey+(−b0sinθ0sinω1t)ez,(21) whereb0is the alternating field amplitude and ω1is the frequency associated with the field. This expression is derived upona rotationof thefield planeby th e anglesφ0=π/4 andθ0= arccos(1 /√ 3). The magnetization trajectories depicted in Fig. 13 reveal two inter esting features: Firstly, particularly for small damping, the energy profile changes v ery slightly (due to the smallness of b0) while energy is pumped into the system during many cycles. Secondly, thesystemswitchesmostlyinthevicinityoflocalminimatoa cquireeventually the target state. Fig. 14 hints on the complex character of the ma gnetization dynamics in this case. As in the static field case with a cubic anisotropy the critic al field amplitudes shown in Fig. 15 are smaller than those for a uniaxial anisot ropy. Obviously, the reason is that the potential barrier associated with this anisot ropy is smaller in this case, giving rise to smaller amplitudes. As before an increase in tempe rature leads to a decrease in the critical fields. The reversal times shown inFig. 16 exhibit the same feature asin the cases for uniaxial anisotropy: With increasing temperatures the corresponding rev ersal times increase. A physically convincing explanation of the (numerically stable) oscillation s for the reversal times is still outstanding.Temperature dependent magnetization dynamics of magnetic nanoparticles 15 00.511.522.533.544.5 5 Time, [ns]-1-0.500.51Magnetization Sz T0=0 K T6=1.9 K Figure 14. (Color online) Magnetization reversal in a nanoparticle using a time de pendent field for α= 0.1 andq0(black) and for q6= 0.06≡1.9K(blue). Other parameters are as in Fig. 13. 0 1 2 3 4 5 6 Damping α00.51Critical AC fieldT6=1.9 K 00.511.522.533.544.5 5 5.5 6 Damping α00.511.52Critical AC fieldT0=0 K T4=0.3 K T5=0.9 K T6=1.9 K Figure 15. (Color online) Critical alternating field amplitudes vs. damping for diffe rent temperatures averaged over 500 cycles. Inset shows the single cycle data at q6= 0.06≡1.9K. 00.511.522.533.544.5 5 5.5 6 Damping α012345Reversal time, [ns] T4=0.3 K T5=0.9 K T6=1.9 K 0 1 2 3 4 5 6 Damping α012345Reversal time, [ns]T0=0 K Figure 16. (Color online) The damping dependence of the reversal times corre sponding to the critical fields of the Fig. 15 for different temperatures averaged over 500 runs. Inset shows the T= 0 case. 4. Summary In this work we studied the critical field amplitudes required for the m agnetization switching of Stoner nanoparticles and derived the corresponding r eversal times forTemperature dependent magnetization dynamics of magnetic nanoparticles 16 static and alternating fields for two different types of anisotropies . The general trends for all examples discussed here can be summarized as follows: Firstly , increasing the temperature results in a decrease of all critical fields regardless o f the anisotropy type. Anisotropy effects decline with increasing temperatures making it ea sier to switch the magnetization. Secondly, elevating thetemperature increases th e corresponding reversal times. Thirdly, thesametrendsareobservedfordifferenttemper atures: Thecriticalfield amplitudes for a static field depend only slightly on α, whereas the critical alternating field amplitudes exhibit a pronounced dependence on damping. In the case of a uniaxial anisotropy we find the critical alternating field amplitudes to be smalle r than those for a static field, especially in the low damping regime and for finite temperat ures. Compared withastaticfield, alternating fieldsleadtosmaller switching times( T= 0K). However, this is not the case for the cubic anisotropy. The markedly different trajectories for the two kinds of anisotropies endorse the qualitatively different magnet ization dynamics. In particular, one may see that for a cubic anisotropy and for an alt ernating field the magnetization reversal takes place through the local minima lea ding to smaller amplitudesoftheappliedfield. Generally, acubicanisotropyissmallert hantheuniaxial one giving rise to smaller slope of critical fields, i.e. smaller alternating fi eld amplitudes. It is useful to contrast our results with those of Ref. [15]. Our re versal times for AC-fields increase with increasing temperatures. This is not in contr adiction with the findings of [15] insofar as we calculate the switching fields at first, an d then deduce the corresponding reversal times. If the switching fields are kept con stant while increasing the temperature [15] the corresponding reversal times decreas e. We note here that experimentally known values of the damping parameter are, to our k nowledge, not larger than 0 .2. The reason why we go beyond this value is twofold. Firstly, the valu es of damping are only well known for thin ferromagnetic films and it is not clear how to extend them to magnetic nanoparticles. For instance, in FMR exper iments damping values are obtained from the widths of the corresponding curves o f absorption. The curves for nanoparticles can be broader due to randomly oriented easy anisotropy axes and, hence, the values of damping could be larger than they actually are. Secondly, due toaverystrongdependenceofthecriticalAC-fields(Fig. 7, e.g.) t heycanevenbelarger than static field amplitudes. This makes the time-dependent field disa dvantageous for switching in an extreme high damping regime. Finally, as can be seen from all simulations, the corresponding rever sal times are much more sensitive a quantity thantheir critical fields. This follows from t he expression (13), where a slight change in the magnetic field bleads to a sizable difference in the reversal time. This circumstance is the basis for our choice to average all the reversal times and fields over many times. This is also desirable in view of an experimental r ealization, for example, in FMR experiments or using a SQUID technique quantities like critical fields and their reversal times are averaged over thousands of times. T he results presented in this paper are of relevance to the heat-assisted magnetic recor ding, e.g. using a laser source. Our calculations do not specify the source of therma l excitations but they capture the spin dynamics and switching behaviour of the syst em upon thermalTemperature dependent magnetization dynamics of magnetic nanoparticles 17 excitations. Acknowledgments This work is supported by the International Max-Planck Research School for Science and Technology of Nanostructures. References [1]Spindynamics in confined magnetic structures III B. Hillebrands, A. Thiaville (Eds.) (Springer, Berlin, 2006); Spin Dynamics in Confined Magnetic Structures II B. Hillebrands, K. Ounadjela (Eds.) (Springer, Berlin, 2003); Spin dynamics in confined magnetic structures B. Hillebrands, K. Ounadjela (Eds.) (Springer, Berlin, 2001); Magnetic Nanostructures B. Aktas, L. Tagirov, F. Mikailov (Eds.), (Springer Series in Materials Science, Vol. 94) (Spring er, 2007) and references therein. [2] M. Vomir, L. H. F. Andrade, L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, Phys. Rev. Lett. 94, 237601 (2005). [3] J. Slonczewski, J. Magn. Magn. Mater., 159, L1, (1996). [4] L. Berger, Phys. Rev. B 54, 9353 (1996). [5] C. Thirion, W. Wernsdorfer, and D. Mailly, Nat. Mater. 2, 524 (2003). [6] Z. Z. Sun and X. R. Wang, Phys. Rev. B 74, 132401 (2006). [7] Z. Z. Sun and X. R. Wang, Phys. Rev. Lett. 97, 077205 (2006). [8] L. F. Zhang, C. Xu, Physics Letters A 349, 82-86 (2006). [9] C. Xu, P. M. Hui , Y. Q. Ma, et al., Solid State Communications 134, 625-629 (2005). [10] T. Moriyama, R. Cao, J. Q. Xiao, et al., Applied Physics Letters 90, 152503 (2007). [11] H. K. Lee, Z. M. Yuan, Journal of Applied Physics 101, 033903 (2007). [12] H. T. Nembach, P. M. Pimentel, S. J. Hermsdoerfer, et al., Physics Letters 90, 062503 (2007). [13] K. Rivkin, J. B. Ketterson, Applied Physics Letters 89, 252507 (2006). [14] R. W. Chantrell and K. O’Grady The Magnetic Properties of fine Particles in R. Gerber, C. D. Wright and G. Asti (Eds.), Applied Magnetism (Kluwer, Academic Pub., Dordrecht, 1994). [15] S. I. Denisov, T. V. Lyutyy, P. H¨ anggi, and K. N. Trohidou, Ph ys. Rev. B 74, 104406 (2006). [16] S. I. Denisov, T. V. Lyutyy, and P. H¨ anggi, Phys. Rev. Lett. 97, 227202 (2006). [17] E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. Londo n, Ser A 240, 599 (1948). [18] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). [19] W. F. Brown, Phys. Rev. 130, 1677 (1963). [20] J. L. Garcia-Palacios and F. Lazaro, Phys. Rev. B 58, 14937 (1998). [21]Algorithmen in der Quantentheorie und Statistischen Physi kJ. Schnakenberg (Zimmermann- Neufang, 1995). [22] U. Nowak, Ann. Rev. Comp. Phys. 9, 105 (2001). [23] K. D. Usadel, Phys. Rev. B 73, 212405 (2006). [24] C. Antoniak, J. Lindner, and M. Farle, Europhys. Lett. 70, 250 (2005). [25] I. Klik and L. Gunther, J. Stat. Phys. 60, 473 (1990). [26] C. Antoniak, J. Lindner, M. Spasova, D. Sudfeld, M. Acet, and M. Farle, Phys. Rev. Lett. 97, 117201 (2006). [27] S. Ostanin, S. S. A. Razee, J. B. Staunton, B. Ginatempo and E . Bruno, J. Appl. Phys. 93, 453 (2003).

2008-02-12

Recent experimental and theoretical studies show that the switching behavior of magnetic nanoparticles can be well controlled by external time-dependent magnetic fields. In this work, we inspect theoretically the influence of the temperature and the magnetic anisotropy on the spin-dynamics and the switching properties of single domain magnetic nanoparticles (Stoner-particles). Our theoretical tools are the Landau-Lifshitz-Gilbert equation extended as to deal with finite temperatures within a Langevine framework. Physical quantities of interest are the minimum field amplitudes required for switching and the corresponding reversal times of the nanoparticle's magnetic moment. In particular, we contrast the cases of static and time-dependent external fields and analyze the influence of damping for a uniaxial and a cubic anisotropy.

Temperature dependent magnetization dynamics of magnetic nanoparticles

0802.1740v1

arXiv:0807.3715v1 [quant-ph] 23 Jul 2008Damped driven coupled oscillators: entanglement, decoherence and the classical limit R. D. Guerrero Mancilla, R. R. Rey-Gonz´ alez and K. M. Fonseca-Romero Grupo de ´Optica e Informaci´ on Cu´ antica, Departamento de F´ ısica, Universidad Nacional de Colombia - Bogot´ a E-mail:rdguerrerom@unal.edu.co E-mail:rrreyg@unal.edu.co E-mail:kmfonsecar@unal.edu.co Abstract. The interaction of (two-level) Rydberg atoms with dissipative QED cavity fields can be described classically or quantum mechanically, eve n for very low temperatures and mean number of photons, provided the damp ing constant is large enough. We investigate the quantum-classical border, the e ntanglement and decoherence of an analytically solvable model, analog to the atom-ca vity system, in which the atom (field) is represented by a (driven and damped) harm onic oscillator. The maximum value of entanglement is shown to depend on the initial st ate and the dissipation-rate to coupling-constant ratio. While in the original model the atomic entropy never grows appreciably (for large dissipation rate s), in our model it reaches a maximum before decreasing. Although both models predic t small values of entanglement and dissipation, for fixed times of the order of the inv erse of the coupling constant and large dissipation rates, these quantities decrease f aster, as a function of the ratio of the dissipation rate to the coupling constant, in our mod el. ‡This research was partially funded by DIB and Facultad de Ciencias, U niversidad Nacional de Colombia. PACS numbers: 03.65.Ud, 03.67.Mn, 42.50.Pq, 89.70.CfDamped Driven Coupled Oscillators 2 1. Introduction One expects quantum theory to approach to the classical theory , for example in the singular limit of a vanishing Planck’s constant, /planckover2pi1→0, or for large quantum numbers. However, dissipative systems can bring forth some surp rises: for example, QED (quantum electrodynamics) cavity fields interacting with two-le vel systems, may exhibitclassicalorquantumbehavior, evenifthesystemiskeptatv erylowtemperatures and if the mean number of photons in the cavity is of the order of one [1, 2], depending on the strength of the damping constant. Classical behavior, in th is context refers to the unitary evolution of one of the subsystems, as if the other s ubsystem could be replaced by a classical driving. In QED cavities, the atom, which ente rs in one of the relevant Rydberg states (almost in resonance with the field sustain ed in the cavity), conserves its purity and suffers a unitary rotation inside the cavity – exactly as if it were controlled by a classical driving field – without entangling with the electromagnetic field. This unexpected behavior was analyzed in reference [1] emplo ying several short- time approximations, and it was found that in the time needed to rota te the atom, its state remains almost pure. Other driven damped systems, composed by two (or more) subsys tems can be readily identified. Indeed, in the last years there has been a fast de velopment of quite different physical systems and interfaces between them, including electrodynamical cavities [3, 4], superconducting circuits [5, 6], confined electrons [7, 8, 9] and nanoresonators [10, 11, 12], on which it is possible to explore genuine quantum effects at the level of a few excitations and/or in individual systems. For ins tance, the interaction atom-electromagnetic field is exploited in experiments wit h trapped ions [13, 14], cavity electrodynamics and ensembles of atoms interacting with coherent states of light [15], radiation pressure over reflective materials in experime nts coupling the mechanical motion of nanoresonators to light [12], and the coupling o f cavities with different quality factors in the manufacturing of more reliable Ramse y zones [16]. In many of these interfaces it is possible to identify a system which coup les strongly to the environment and another which couples weakly. For example, in the e xperiments of S. Haroche the electromagnetic field decays significantly faster [17] (or significantly slower [16]) than the atoms, the quality factor Qof the nanoresonators is much smaller than that of the cavity, and the newest Ramsey zones comprise two cou pled cavities of quite different Q. Several of these systems therefore, can be modelled as coupled harmonic oscillators, one which can be considered dissipationless. In this contribution we study an exactly solvable system, composed of two oscillators, which permits the analysis of large times, shedding additio nal light on the classical-quantum border. Entanglement and entropy, as measur ed by concurrence and linear entropy, are used to tell “classical” from quantum effects.Damped Driven Coupled Oscillators 3 2. The model The system that we consider in this manuscript comprises two oscillat ors of natural frequencies ω1andω2, coupled through an interaction which conserves the (total) number of excitations and whose coupling constant abruptly chang es from zero to g at some initial time, and back to zero at some final time. We take into a ccount that the second oscillator loses excitations at the rate γ, through a phenomenological Liouvillian of Lindblad form, corresponding to zero temperature, in the dyna mical equation of motion [18]. Lindblad superoperators are convenient because they preserve important characteristics of physically realizable states, namely hermiticity, c onservation of the trace and semi-positivity [19]. In order to guarantee the presence of excitations, the second oscillator is driven by a classical resonant field. The interaction can be considered to be turned on (off) in the remot e past (remote future) if it is always present (coupled Ramsey zones or nanoreson ators coupled to cavity fields), or can really be present for a finite time interval (for example in atoms travelling through cavities). The initial states of the coupled oscillat ors also depend on the experimental setup, varying from the base state of the co mpound system to a product of the steady state of the coupled damped oscillator with the state of the other oscillator. Since we want to make comparisons with Ramsey zon es, the choices in the formulation of this model have been inspired by the analogy with t he atom-cavity system, for a cavity –kept at temperatures of less than 1K– whos e lifetime is much shorter than the lifetime of Rydberg states, allowing us to ignore th e Lindblad operator characterizing the atomic decay process. The first oscillator ther efore is a cartoon of the atom, at least in the limit where only its first two states are signific antly occupied, while the second oscillator corresponds to the field. All the ingredients detailed before can be summarily put into the Liouv ille-von Neumann equation for the density matrix ˆ ρof the total system dˆρ dt=−i /planckover2pi1[ˆH,ˆρ]+γ(2ˆaˆρˆa†−ˆa†ˆaˆρ−ˆρˆa†ˆa) (1) whereˆHis the total Hamiltonian of the system and the second term of the rh s of (1) is the Lindblad superoperator which accounts for the loss of excita tions of the second oscillator. In absence of the coupling with the first oscillator, the inv erse of twice the dissipation rate γgives the mean lifetime of the second oscillator. The first two terms of the total Hamiltonian ˆH=/planckover2pi1ω1ˆb†ˆb+/planckover2pi1ω2ˆa†ˆa+/planckover2pi1g(Θ(t)−Θ(t+T))(ˆa†ˆb+ˆaˆb†)+i/planckover2pi1ǫ(e−iωDtˆa†−eiωDtˆa),(2) are the free Hamiltonians of the two harmonic oscillators; the next t erm, which is modulated by the step function Θ( t), is the interaction between them and the last is the driving. The bosonic operators of creation ˆb(ˆa) and annihilation ˆb†(ˆa†) of one excitation of the first (second) oscillator, satisfy the usual comm utation relations. From here on we focus on the case of resonance, ω1=ω2=ωD=ω. The interaction time Tis left undefinite until the end of the manuscript, where we compare our results with those of the atom-cavity system.Damped Driven Coupled Oscillators 4 3. Dynamical evolution The solution of the dynamical equation (1) can be written as ˆρ(t) =D(β(t),α(t))˜ρ(t)D†(β(t),α(t)), (3) whereD(β(t),α(t)) is the two-mode displacement operator, D(β(t),α(t)) =D1(β(t))D2(α(t)) =eβ(t)ˆb†−β∗(t)ˆbeα(t)ˆa†−α∗(t)ˆa, and ˜ρ(t) is the total density operator in the interaction picture defined by equation (3). By replacing (3) into (1), and employing the operator identities d dtD(α) =/parenleftbigg −α∗˙α−˙α∗α 2+ ˙αˆa†−˙α∗ˆa/parenrightbigg D(α) =D(α)/parenleftbiggα∗˙α−˙α∗α 2+ ˙αˆa†−˙α∗ˆa/parenrightbigg , with the dot designating the time derivative as usual, we are able to de couple the dynamics of the displacement operators, obtaining the following dyn amical equations for the labels αandβ d dt/parenleftigg α β/parenrightigg =/parenleftigg −γ−iω−ig −ig−iω/parenrightigg/parenleftigg α β/parenrightigg +/parenleftigg ǫe−iωt 0/parenrightigg , (4) for times between zero and T. On the other hand, the Ansatz (3) also provides the equation of motion for ˜ ρ(t), which turns out to be very similar to (1) but with the hamiltonian ˜H=ˆH(ǫ= 0), that is, without driving. The separation provided by our Ansatz is also appealing from the point of view of its possible physical in terpretation, because the effect of the driving has been singled out, and quantum (entangling and purity) effects are extracted from the displaced density operato r ˜ρ(t). The two oscillators interact after the second oscillator reaches its stationary coherent state ˆρ2(t) = tr1ˆρ(t) =/vextendsingle/vextendsingle/vextendsingle/vextendsingleǫ γe−iωt/angbracketrightbigg/angbracketleftbiggǫ γe−iωt/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5) as can be verified by solving (4) with the interaction turned off. If we want a mean number of excitations of the order of one then the driving amplitude must satisfy ǫ≈γ, and thereby the larger the dissipation is, the larger the driving is to b e chosen. At zero time, when the oscillators begin to interact, the state of the total system is separable with the second oscillator state given by (5). The first oscillator, on the other hand, begins in a pure state which we choose as a linear combination of its gro und and first excited states (again inspired on the analogy with the atom-cavity s ystem). Thus, the initial state ˆ ρ(0) given by D/parenleftbigg 0,ǫ γ/parenrightbigg (cos(θ)|0∝angbracketright+sin(θ)|1∝angbracketright)(cos(θ)∝angbracketleft0|+sin(θ)∝angbracketleft1|)⊗|0∝angbracketright∝angbracketleft0|/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright ˜ρ(0)D†/parenleftbigg 0,ǫ γ/parenrightbigg ,(6) corresponds to a state of the form described by equation (3) with β(0) = 0 and α(0) =ǫ/γ. At later times, the solution maintains the same structure, but –as canDamped Driven Coupled Oscillators 5 be seen from the solution of (4) – the labels of the displacement oper ators evolve as follows α(t) =ǫe−1 2(γ+2iω)t/braceleftbigg1 γcos(˜gt)+sin(˜gt) 2˜g/bracerightbigg , (7) β(t) =−ie−iωtǫ g+iǫ ge−1 2(γ+2iω)t/braceleftbigg cos(˜gt)+/parenleftbig −2g2+γ2/parenrightbigsin(˜gt) 2γ˜g/bracerightbigg , (8) where we have defined the new constant ˜ g=1 2/radicalbig 4g2−γ2.We employ ˜ g, which also appears in the solution of the displaced density operator, to define three different regimes: underdamped (˜ g2>0), critically damped (˜ g2= 0) and overdamped (˜ g2<0) regime. It is important to notice that there is no direct connection w ith the quality factorofthedampedoscillator: itispossibletohavephysical syste ms intheoverdamped regime defined here even with relatively large quality factors, if the in teraction constant gis much smaller than ω, the frequency of the oscillators. The inspection of the equations (7) and (8), allows one to clearly iden tify the time scale 2/γ, after which the stationary state is reached and the state of the first oscillator just rotates with frequency ωand have a mean number of excitations equal to ǫ2/g2. The doubling of the damping time of the second oscillator, from 1 /γin the absence of interaction to 2 /γ, in the underdamped regime, can be seen as an instance of the shelving effect [20]. The first oscillator, which in absence of interactio n, suffers no damping, it is now driven and damped. It can be thought that the exc itations remain half of the time on each oscillator, and that they decay with a damping constant γ, thereby leading to an effective damping constant of γ/2. An interesting feature of the solution is that the displacement of the second oscillator goes to zer o, in the stationary state. In the stationary state, the first oscillator evolves as if it w ere driven by a classical field−i/planckover2pi1ǫexp(−iωt) and damped with a damping rate g, without any interaction with a second oscillator. More generally speaking, we remark that from t he point of view of the first oscillator, the evolution of its displacement operator happ ens as if there were damping but no coupling, and the driving were of the form /planckover2pi1g(β−iα), or, in terms of the parameters of the problem, F(t) =−i/planckover2pi1ǫe−iωt−i/planckover2pi1ǫe−(γ/2+iω)t/parenleftbigg/parenleftbiggg γ−1/parenrightbigg cos(˜gt)+2g2+gγ−γ2 2γ˜gsin(˜gt)/parenrightbigg .(9) This behavior is particularly relevant in the following extreme case, wh ose complete solution depends only on the displacement operators. If the initial s tate of the first oscillator is the ground state then ˜ ρdoes not evolve in time, i. e. it remains in the state |00∝angbracketright, and the total pure and separable joint state is ρ(t) =|β(t)∝angbracketright∝angbracketleftβ(t)|⊗|α(t)∝angbracketright∝angbracketleftα(t)|. (10) Even in the more general case considered here, corresponding to the initial state (6), the solution of ˜ ρ(t) possesses only a few non-zero elements. If we write the total density operator as ˜ρ(t) =/summationdisplay i1i2j1j2˜ρj1j2 i1i2|i1i2∝angbracketright∝angbracketleftj1j2|, (11)Damped Driven Coupled Oscillators 6 we canarrangethe elements corresponding to zero and one excita tions ineach oscillator, as the two-qubit density matrix ˜ρ00 00(t) ˜ρ01 00(t) ˜ρ10 00(t) 0 ˜ρ00 01(t) ˜ρ01 01(t) ˜ρ10 01(t) 0 ˜ρ00 10(t) ˜ρ01 10(t) ˜ρ10 10(t) 0 0 0 0 0 . (12) If we measure time in units of gby defining t′=gtwe have only two free parameters Γ =γ gand Ω =ω g. The nonvanishing elements of the density matrix, written in the underdamped case ( |Γ|<2), are given by (hermiticity of the density operator yields the missing non-zero elements) ˜ρ00 00(t′) = 1−sin2θe−Γt′/parenleftbigg4−Γ2cos(√ 4−Γ2t′) 4−Γ2−Γsin(√ 4−Γ2t′)√ 4−Γ2/parenrightbigg ˜ρ01 01(t′) = 2sin2(θ)e−Γt′1−cos/parenleftbig√ 4−Γ2t′/parenrightbig 4−Γ2 ˜ρ10 10(t′) = sin2(θ)e−Γt′/parenleftigg (2−Γ2)cos/parenleftbig√ 4−Γ2t′/parenrightbig +2 4−Γ2−Γsin/parenleftbig√ 4−Γ2t′/parenrightbig √ 4−Γ2/parenrightigg ˜ρ00 01(t′) =isin(2θ)eiΩt′−Γt′ 2sin/parenleftbig√ 4−Γ2t′ 2/parenrightbig √ 4−Γ2 ˜ρ00 10(t′) =sin(2θ) 2eiΩt′−Γt′ 2/parenleftigg cos/parenleftbigg√ 4−Γ2t′ 2/parenrightbigg −Γsin/parenleftbig√ 4−Γ2t′ 2/parenrightbig √ 4−Γ2/parenrightigg ˜ρ01 10(t′) = 2isin2(θ)e−Γt′sin/parenleftbig√ 4−Γ2t′ 2/parenrightbig √ 4−Γ2/parenleftigg Γsin/parenleftbig√ 4−Γ2t′ 2/parenrightbig √ 4−Γ2−cos/parenleftbigg√ 4−Γ2t′ 2/parenrightbigg/parenrightigg The expressions of the elements of the density matrix in the critically damped case Γ = 2 and in the overdamped case Γ >2 can be obtained from those given in the text for the underdamped case Γ <2. 4. Entanglement Although quantities like quantum discord [21] have been proposed to extract the quantum content of correlations between two systems, we prese ntly quantify the quantum correlations between both oscillators employing a measure of entanglement. Due to the dynamics of the system, and the initial states chosen, t he whole system behaves as a couple of qubits and therefore its entanglement can b e measured by Wootters’ concurrence [22]. One of the most important characte ristics of the form of the solution given by (3) is that concurrence, as well as linear ent ropy, depend only on the displaced density operator ˜ ρ(t′). In our case the concurrence reduces to C(t′) =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalig ˜ρ01 10(t′)˜ρ10 01(t′)+/radicalig ˜ρ01 01(t′)˜ρ10 10(t′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/radicalig ˜ρ01 10(t′)˜ρ10 01(t′)−/radicalig ˜ρ01 01(t′)˜ρ10 10(t′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle = 2/radicalig ˜ρ01 10(t′)˜ρ10 01(t′) = 2|˜ρ10 01(t′)|, (13)Damped Driven Coupled Oscillators 7 where the positivity and hermiticity of the density matrix were used. The explicit expressions for the concurrence in the underdamped (UD), critic ally damped (CD) and overdamped (OD) regimes are CUD(t′) = 4sin2(θ)e−Γt′sin/parenleftig√ 4−Γ2t′ 2/parenrightig √ 4−Γ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓsin/parenleftig√ 4−Γ2t′ 2/parenrightig √ 4−Γ2−cos/parenleftbigg√ 4−Γ2t′ 2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, CCD(t′) = 2sin2(θ)e−2t′t′|t′−1|, (14) COD(t′) = 4sin2(θ)e−Γt′sinh/parenleftig√ Γ2−4t′ 2/parenrightig √ Γ2−4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓsinh/parenleftig√ Γ2−4t′ 2/parenrightig √ Γ2−4−cosh/parenleftbigg√ Γ2−4t′ 2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Allthedependence ontheinitialstateiscontainedonthesquared n ormofthecoefficient of the state |1∝angbracketrightof the displaced density operator. In all regimes the concurrence vanishes at zero time, because the initial state considered is separable. How ever, while in the underdamped case the concurrence vanishes periodically (see equ ation (15) below), in the other two cases it crosses zero once ( t >0) and reaches zero assymptotically as time grows. This shows a markedly different qualitative behavior (see figu res 1 and 2). In the underdamped regime the zeroes of the concurrence are fo und at times τ1n=2nπ√ 4−Γ2,andτ2n=2πn+2arccos/parenleftbigΓ 2/parenrightbig √ 4−Γ2, (15) wherenis a non-negative integer. In this contribution, the inverse sine and cosine functions are chosen to take values in the interval [0 ,π/2]. The time τ10corresponds to the initial state. The sequence of concurrence zeroes is thereby 0 =τ10< τ20< τ11< τ21...As the critical damping is approached, the time τ11is pushed towards infinity, whileτ20approaches the finite time 2 /Γ (see figure 2). For the initial states considered in this manuscript we do not observe the sudden death of the entan glement since the concurrence is zero only for isolated instants of time. If one writes the concurrence in the underdamped regime in the alte rnative form CUD(t′) =sin2θ 2(1−Γ2/4)e−Γt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΓ 2−sin/parenleftigg arcsin/parenleftbiggΓ 2/parenrightbigg +2/radicalbigg 1−Γ2 4t′/parenrightigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle(16) it is easy to verify that at the times τ±n, given by τ±n=1√ 4−Γ2/parenleftbigg (2n+1)π±arccos/parenleftbiggΓ2 4/parenrightbigg −2arcsin/parenleftbiggΓ 2/parenrightbigg/parenrightbigg >0, n= 0,1,...(17) the concurrence reaches the local maxima C±n= sin2θ/parenleftigg/radicalbigg 1+Γ2 4±Γ 2/parenrightigg exp −Γ/parenleftig (2n+1)π±arccos(Γ2 4)−2arcsin(Γ 2)/parenrightig 2/radicalig 1−Γ2 4 . We observe these maxima to lie on the curves sin2θK±exp(−Γt), where the constants K±=/radicalig 1+Γ2 4±Γ 2satisfy the inequalities√ 2−1≤K−≤1≤K+≤√ 2+1. Maxima of concurrence depend on both the initial state and the value of th e rescaled dampingDamped Driven Coupled Oscillators 8 constant, and reach the maximum available value of one only in the non -dissipative case for a particular initial state. In order to have negligible values of con currence (except for small time intervals around the zeroes of concurrence) it is necess ary to have times much larger than 1 /γ. From the point of view of classical-like behavior, the most favourab le scenario corresponds to zero or almost zero concurrence, which are obtained for short time intervals around τ1n, τ2nand for large values of time. In the overdamped regime, the concurrence presents two maxima ,τ−andτ+> τ− τ±=2arccosh(Γ /2)±arccosh(Γ2/4) 2/radicalbig 1−Γ2/4, (18) both of which go to zero as the rescaled dissipation rate grows, τ+→4ln(Γ)/Γ and τ−→ln(2)/Γ (see figure 2). The function arccosh( x) is chosen as to return nonnegative values for x≥1. Since the global maximum of concurrence, which corresponds to the later time, scales like 1 /(2Γ) for large values of Γ, in the highly overdamped regime quantum correlations are not developed at any time. Figure 1. Concurrence as a function of time and the rescaled damping consta nt in the (a) underdamped, (b) critically damped, and (c) overdamped c ase. The behavior of concurrence in the different regimes is shown in figur e 1. It is apparent that small values of concurrence are obtained for very small times and for large times in the underdamped case and for all times for a highly overdamp ed oscillator. In figure2we depict thetimesatwhichconcurrence attainsamaximum, andthemaximum values of concurrence, as a function of the rescaled damping cons tant. One can see how the first two times of maximum concurrence go to zero, while the oth er times diverge, as the critically damped regime is reached. The first two maxima of con currence vanish more slowly than the rest of maxima, which hit zero at Γ = 2. 5. Entropy The entropy is analized employing the linear entropy of the first oscilla tor, the system of interest. As remarked before, the first oscillator behaves like a tw o-level system, whereDamped Driven Coupled Oscillators 9 Figure 2. (a)Concurrence local maxima at times τ+(−)nin black (gray) color for n= 0 full line, n= 1 dashed line and n= 2 dashed-dot line and (b) times of maximum values of concurrence, as a function of the rescaled damping cons tant. the maximum value of the linear entropy, 0.5, is obtained when the pop ulationof each of the two states is one half. The type of “classical” behavior which allow s the interaction with Ramsey zones to be modelled like a classical driving force occurs w hen the linear entropy is very small, and hence the state of the first oscillator is (a lmost) pure and uncorrelated with the state of the second oscillator. The linear ent ropy for the first oscillator can be computed as δ1(t′) = 1−tr1(tr2ρtr2ρ) = 1−tr1(tr2˜ρtr2˜ρ) = 2det(tr 2˜ρ),(19) where thelast equality holdsfortwo-level systems. Inequation(1 9) thedensity operator of the first oscillator is assumed to be represented by a 2 ×2 matrix. Employing the expressions we have found for the elements of ˜ ρwe obtain δ1(t′) = 2sin4θ x(t′)(1−x(t′)), (20) wherex, in the underdamped regime, is given by xUD(t) =e−Γtsin2/parenleftig/radicalbig 1−Γ2/4t−arccos(Γ/2)/parenrightig 1−Γ2/4. (21) Surprisingly, as in the case of concurrence, the influence of the init ial state factors out in the expression of the linear entropy of the first oscillator, which t urns out to be proportional to the square of the population of |1∝angbracketrightin the initial displaced operator. As it is well known, in the limit of zero dissipation, the linear entropy of t he reduced density matrix is equal to one fourth of the square of the concurr ence. At times τ2n (see eq.(15)), when both concurrence and linear entropy vanish, the total state of the system is separable, ρ(gτ2n) =|β(gτ2n)∝angbracketright∝angbracketleftβ(gτ2n)| ⊗ρ2(gτ2n); that is, from the point of view of the first oscillator the evolution is unitary like. Since the linea r entropy begins at zero, because the initial state is pure and separable, the re is a maximum in the interval (0 ,τ20), which turns out to give a linear entropy of exactly 0.5 (we treat the case sin θ= 1, because —due to the scaling property discussed before— a simp leDamped Driven Coupled Oscillators 10 multiplication by sin4θgives the result for other cases). Indeed, as the function x(t) changes continuously from x(t= 0) = 1 to x(t=τ20) = 0, it crosses 0.5 at some time τ30in between, giving the maximum value possible of the linear entropy. Alt hough the exact value of τ30can be obtained only numerically, good analytical approximations can be readily obtained. For example, τ30≈π/(4+ 4g+ 2g2), gives an error smaller than 0.5%. For small values of the rescaled damping constant, Γ /lessorapproxeql0.237, there are several solutions to the equation x(t) = 0.5 in the interval (0 ,log(2)/Γ), which give absolute maxima of the linear entropy, while the times τ4n=2arccos(Γ /2)+nπ/radicalbig 1−Γ2/4, n= 0,1,2,··· (22) correspond to local minima. In the interval (log(2) /Γ,∞) the times τ4ngive local maxima. All of the local maxima and minima given by eq. (22) belong to th e curve 2e−Γt(1−e−Γt). The large time behavior of the local maxima of linear entropy and concurrence is, thereby, of the same form constant ×exp(−Γt′). For values of Γ >0.237 all times τ4ngive local maxima. The maxima of concurrence and linear entropy coin cide only in the weakly damped case, because concurrence and linear ent ropy are not independent for pure bipartite states. At times τ1n(see eq.(15)), where the total state ρ(gτ1n) =ρ1(gτ1n)⊗ |α(gτ1n)∝angbracketright∝angbracketleftα(gτ1n)|, is separable, the reduced state of the first oscillator is mixed. The linear entropy is small for short ( τ1n≪log(2)/Γ) and large ( τ1n≫log(2)/Γ) times. In the overdamped regime the function x(t), which appers on the expression for linear entropy (20) and is given by xOD(t′) =e−Γt′sinh2(/radicalbig Γ2/4−1t′−arccosh(Γ /2)) Γ2/4−1, (23) begins at one for t′= 0, and goes down to zero for large values of time. The time at which it crosses one half can be calculated to be τ0.5≈0.16557<1/6 for Γ = 2 and for large values of Γ it goes as τ0.5≈ln2/(2Γ). It is easy to find interpolating functions with small error for the time of crossing, ˜ρ10 10 t′=1 6+4 ln2sinh/parenleftbigg arccosh(Γ 2)tanh/parenleftbigg arccosh (Γ 2) 1.6/parenrightbigg/parenrightbigg = 0.5(1+∆) where|∆|<2.5%. It is interesting to notice that for large values of the damping th is time (ln(2) /(2Γ)) is half the time needed to obtain the maximum value of concurre nce, and that, at the later time, the linear entropy is 3/4 of the maximum v alue of entropy, a relatively large value. The state of the first oscillator always become s maximally mixed before becoming pure again, no matter how large the value of the da mping. We show the behavior of linear entropy in figure 3. In the underdamped regim e there are infinite maxima and minima, while for critical damping and for the overdamped r egime there are only two maxima. The first maximum always corresponds to a linear entropy of one half.Damped Driven Coupled Oscillators 11 Figure 3. Linear entropy of the first oscillator as a function of time and the re scaled damping constant in the (a) underdamped, (b) critically damped, an d (c) overdamped case. 6. Conclusions In the present contribution we have shown that the classical quan tum border in this model depends mainly on the initial state and on damping constant to interaction coupling ratio, and that quantum effects, characteristic of the un derdamped regime, can be seen in the other regimes for small times. In order to make co nnection with Ramsey zones we remember in that physical system ω≈1010Hz,Q≈104,g≈104 Hz andTR≈10−5s, which was chosen as to produce π/2 pulse, that is a pulse that can rotate the state of the two-level system, as represented in a Bloch sphere, by an angleπ/2. These numbers place thesystem into the highly overdamped (reg ime because Γ =ω/(Qg)≈102≫2) and give a rescaled evolution time of the order of gT≈10−1. Here we use the same values of ω,γandg, and an evolution time of order 1/g. Indeed, the hamiltonian /planckover2pi1ωˆb†ˆb+/planckover2pi1g(Θ(t)−Θ(t+T))(α0e−iωtˆb†+α∗ 0eiωtˆb), with∝bardblα0∝bardbl ≈1 — which would model the interaction of the first oscillator with a classica l driving field of an average number of excitations of the order of one — has a chara cteristic time 1 /g, corresponding to T′≈1. The dynamical behavior of the linear entropy obtained here, is quite different from that of ref. [1]: there the linear entropy was never large for t he relevant time interval, here it grows to the maximum possible for a two-level syste m, and then goes to zero very quickly. Therefore, in this model dissipation produces relaxation also, and a description obviating the second oscillator still needs a dissipation p rocess. Although, at the evolution time T, both models predict a small atomic entropy, in Ramsey zones it decreases as δ1(TR′≈0.1)≈4/Γ, while in the present model it goes to zero as δ1(T′≈1)∝1/Γ4. Qualitative and quantitative differences notwithstanding, at the evolution time the linear entropy is very small, in both cases, due to th e smallness of the ratio g/γ. As remarked before the quality factor of the damped oscillator do es not appear directly in either case; it can be perfectly possible to hav e a very weaklyDamped Driven Coupled Oscillators 12 damped oscillator and a highly overdamped interaction. However, as the first oscillator quality factor is improved, the damping constant will eventually be co mparable with the interaction constant, and there will be considerable entanglem ent between both oscillators. For the same physical system if the damping rate can be changed then classical or quantum behavior can be obtained. Acknowledgements This work was partially funded by DIB-UNAL and Facultad de Ciencias, Universidad Nacional (Colombia). References [1] J. I. Kim, K. M. Fonseca Romero, A. M. Horiguti, L. Davidovich, M. C. Nemes, and A. F. R. de Toledo Piza. Classical behavior with small quantum numbers: The p hysics of Ramsey interferometry of Rydberg atoms. Phys. Rev. Lett. , 82:4737, 1999. [2] J. M. Raimond, M. Brune, and S. Haroche. Colloquium: Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys. , 73:565, 2001. [3] J. M. Raimond, M. Brune, and S. Haroche. Manipulating quantum e ntanglement with atoms and photons in a cavity. Rev. Mod. Phys , 73:565, 2001. [4] H. Mabuchi and A. C. Doherty. Decoherence, chaos and the se cond law. Rev. Mod. Phys. , 298(5597):1372–1377, 2002. [5] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Gir vin, and R. J. Schoelkopf. Coupling superconducting qubits via a cavity bus. Nature (London) , 449:443–447, 2007. [6] A. Wallraff, D. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Maj er, S. Kumar, S. M. Girvin, and R. J. Schoelkopf. Circuit quantum electrodynamics: Coherent cou pling of a single photon to a cooper pair box. Nature (London) , 431:162, 2004. [7] M. Pioro-Ladriere, Y. Tokura, T. Obata, T. Kubo, and S. Taruc ha. Micro-magnets for coherent electron spin control in quantum dots. Appl. Phys. Lett. , 90:024105, 2007. [8] H.-A. Engel, L. P. Kouwenhoven, D. Loss, and C. M. Marcus. Con trolling spin qubits in quantum dots.Quantum Information Processing 3 , page 115, 2004. [9] S. Amasha, K. MacLean, D. Zumb¨ uhl, I. Radu, M. A. Kastner, M . P. Hanson, and A. C. Gossard. Toward the manipulation of a single spin in an algaas/gaas single-electr on transistor. Proc. of SPIE, 6244:624419, 2006. [10] K. Jacobs. Engineering quantum states of a nanoresonator v ia a simple auxiliary system. Phys. Rev. Lett. , 99:117203, 2007. [11] D. Vitali et. al. Entangling a nanomechanical resonator and a superconducting m icrowave cavity. Phys. Rev. A , 76:042336, 2007. [12] A. M. Jayich, J.C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson , S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris. Dispersive optomechanics: a me mbrane inside a cavity. arXiv:0805.3723 , 2008. [13] C. Monroe. Quantum information processing with atoms and pho tons.Nature, 416:238, 2002. [14] B. B. Blinov, D. L. Moehring, L. M. Duan, and C. Monroe. Observ ation of entanglement between a single trapped atom and a single photon. Nature, 428(6979), 2002. [15] J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Ham merer, I. Cirac, and E. S. Polzik. Quantum teleportation between light and matter. Nature, 443(7111):557–560, 2006. [16] S. Haroche, M. Brune, and J. M. Raimond. Measuring the photo n number parity in a cavity: fromDamped Driven Coupled Oscillators 13 light quantum jumps to the tomography of non-classical field state s.Nature, 54(13-15):p2101– 2114, 2007. [17] M. Brune, E. Hagley, J. Dreyer, X. Maˆ ıtre, A. Maali, C. Wunder lich, S. Haroche and J. M. Raimond. Observing the progressive decoherence of the meter in a quantum measurement. Phys. Rev. Lett. , 77:4887, 1996. [18] W. H. Louisell. Quantum Statistical Properties of Radiation . Wiley & Sons, New York, 1973. [19] G. Lindblad. On the generators of quantum dynamical semigrou ps.Commun. Math. Phys. , 48:119, 1976. [20] D. Wineland and H. Dehmelt. Proposed 1014Dν < ν laser fluorescence spectroscopy on Tl+ mono-ion oscillator III (side band cooling). Bull. Am. Phys. Soc. , 20:637, 1975. [21] H. Ollivier and W. H. Zurek. Quantum discord: A measure of the qu antumness of correlations. Phys. Rev. Lett. , 88(1):017901, Dec 2001. [22] W. K. Wootters. Entanglement of formation of an arbitrary st ate of two qubits. Phys. Rev. Lett. , 80(10):2245–2248, Mar 1998.

2008-07-23

The interaction of (two-level) Rydberg atoms with dissipative QED cavity fields can be described classically or quantum mechanically, even for very low temperatures and mean number of photons, provided the damping constant is large enough. We investigate the quantum-classical border, the entanglement and decoherence of an analytically solvable model, analog to the atom-cavity system, in which the atom (field) is represented by a (driven and damped) harmonic oscillator. The maximum value of entanglement is shown to depend on the initial state and the dissipation-rate to coupling-constant ratio. While in the original model the atomic entropy never grows appreciably (for large dissipation rates), in our model it reaches a maximum before decreasing. Although both models predict small values of entanglement and dissipation, for fixed times of the order of the inverse of the coupling constant and large dissipation rates, these quantities decrease faster, as a function of the ratio of the dissipation rate to the coupling constant, in our model.

Damped driven coupled oscillators: entanglement, decoherence and the classical limit

0807.3715v1

arXiv:0809.2611v1 [cond-mat.mes-hall] 16 Sep 2008Stochastic dynamics of magnetization in a ferromagnetic na noparticle out of equilibrium Denis M. Basko1and Maxim G. Vavilov2 1International School of Advanced Studies (SISSA), via Beir ut 2-4, 34014 Trieste, Italy 2Department of Physics, University of Wisconsin, Madison, W I 53706, USA (Dated: September 15, 2008) We consider a small metallic particle (quantumdot) where fe rromagnetism arises as a consequence of Stoner instability. When the particle is connected to ele ctrodes, exchange of electrons between the particle and the electrodes leads to a temperature- and b ias-driven Brownian motion of the direction of the particle magnetization. Under certain con ditions this Brownian motion is described by the stochastic Landau-Lifshitz-Gilbert equation. As an example of its application, we calculate the frequency-dependentmagnetic susceptibility of the pa rticle in a constant external magnetic field, which is relevant for ferromagnetic resonance measurement s. PACS numbers: 73.23.-b, 73.40.-c, 73.50.Fq I. INTRODUCTION The description of fluctuations of the magnetization in small ferromagnetic particles pioneered by Brown1is based on the Landau-Lifshitz-Gilbert (LLG) equation2,3 with a phenomenologically added stochastic term. This approach has been widely used: just a few recent appli- cations are a numerical study of the dynamic response of the magnetization to the oscillatory magnetic field,4 a numerical study of ferromagnetic resonance spectra,5 study of resistance noise in spin valves,6and a study of the magnetization switching and relaxation in the pres- ence of anisotropy and a rotating magnetic field.7 In equilibrium the statistics of stochastic term in the LLG equation can be simply written from the fluctuation-dissipation theorem.1However, out of equi- librium a proper microscopic derivation is required. Mi- croscopic derivations of the stochastic LLG equation out of equilibrium, available in the literature, use the model of a localized spin coupled to itinerant electrons,8,9,10,11 or deal with non-interacting electrons.12In contrast to this approach, we start from a purely electronic system where the magnetization arises as a consequence of the Stoner instability. Our derivation has certain similarity with that of Ref. 13 for a bulk ferromagnet, where the di- rection of magnetization is fixed and cannot be changed globally, so its local fluctuations are small and their de- scription by a gaussianaction is sufficient. This situation should be contrasted to the case of a nanoparticle where the direction ofthe magnetizationcan be completely ran- domized by the fluctuations, so that the effective action for the direction of the classicalmagnetization has a non- gaussian part. The bias-driven Brownian motion of the magnetization with a fixed direction (due to and easy- axis anisotropy and ferromagnetic electrodes) has been also studied in Ref. 14 using rate equations. We assume that the single-electron spectrum of the particle, which is also called a quantum dot in the litera- ture, to be chaotic and described by the random-matrix theory15,16. Totakeintoaccounttheelectron-electronin-teractions in the dot we use the universal Hamiltonian,17 with a generalized spin part, corresponding to a ferro- magnetic particle. Electrons occupy the quantum states of the full Hamiltonian and form a net spin of the par- ticle of order of S0≫1; throughout the paper we use ¯h= 1. The dot is coupled to two leads, see Fig. 1, which we assumed to be non-magnetic. The approach can be easily extended to the case of magnetic leads. The num- berNchof the transversechannels in the leads, which are well coupled to the dot, is assumed to be large, Nch≫1. Equivalently, the escape rate 1 /τof electrons from the dot into the leads is largecompared to the single-electron mean level spacing δ1in the dot. This coupling to the leads is responsible for tunnelling processes of electrons between states in the leads and in the dot with random spin orientation. As a result of such tunnelling events, the net spin of the particle changes. We show that this exchangeofelectronsgivesrisebothtotheGilbert damp- ing and the magnetization noise in the presented model, and under conditions specified below, the time evolution of the particle spin is described by the stochastic LLG equation. We study in detail the conditions for applicability of the stochastic approach. We find that these limits are set by three independent criteria. First, the contact re- sistance should be low compared to the resistance quan- tum, which is equivalent to Nch≫1. If this condi- tion is broken, the statistics of the noise cannot be con- sidered gaussian. Physically, this condition means that each channel can be viewed as an independent source of noise, so the contribution of many channels results in the gaussian noise by virtue of the central limit theo- rem ifNch≫1. Second, the system should not be too close to the Stoner instability: the mean-field value of the total spin S0≫√Nch. If this condition is violated, the fluctuations of the absolute value of the magneti- zation become of the order of the magnetization itself. Third,S2 0≫Teff/δ1, whereTeffis the effective tempera- ture of the system, which is the energy scale of the elec- tronicdistribution function determined bya combination2 NLNRB~ z B~ B0Drain Source µ+eVµFerromagnetic nanoparticle µ FIG. 1: (Color online). Device setup considered in this work : a small ferromagnetic particle (quantum dot) coupled to two non-magnetic leads (see text for details). oftemperature andbiasvoltage(the Boltzmannconstant kB= 1throughoutthe paper). Otherwise, theseparation of the degrees of freedom into slow (the direction of the magnetization) and fast (the electron dynamics and the fluctuations of the absolute value of the magnetization) is not possible. In the present model we completely neglect the spin- orbit interaction inside the particle, whose effect is as- sumedtobeweakascomparedtotheeffectoftheleads.18 The effects of the electron-electron interaction in the charge channel (weak Coulomb blockade) are suppressed forNch≫1,15so we do not consider it. As an application of the formalism, we consider themagnetic susceptibility in the ferromagnetic resonance measurements, which is a standardcharacteristicofmag- netic samples. Recently, a progress was reported in mea- surements of the magnetic susceptibility on small spatial scales in response to high-frequency magnetic fields.19 Measurements of the ferromagnetic resonance were also reportedfornanoparticles, connected to leadsfor asome- what different setup in Ref. 20. The paper is organized as follows. In Sec. II we intro- duce the model for electrons in a small metallic particle subject to Stoner instability. In Sec. III we analyze the effective bosonic action for the magnetization of the par- ticle. In Sec. IV we obtain the equation of motion for the magnetization with the stochastic Langevin term, which has the form of the stochastic Landau-Lifshitz-Gilbert equation, and derive the associated Fokker-Planck equa- tion. In Sec. V we discuss the conditions for the applica- bility of the approach. In Sec. VI we calculate the mag- netic susceptibility from the stochastic LLG equation. II. MODEL AND BASIC FORMALISM Within the random matrix theory framework, elec- trons in a closed chaotic quantum dot are described by the following fermionic action: S[ψ,ψ∗] =/contintegraldisplay dt N/summationdisplay n,n′=1ψ† n(t)(δnn′i∂t−Hnn′)ψn′(t)−E(S(t)) , Si≡N/summationdisplay n=1ψ† nˆσi 2ψn. (2.1) Hereψnis a two-component Grassmann spinor, truns along the Keldysh contour, as marked by/contintegraltext ; ˆσx,y,zare the Pauli matrices (we use the hat to indicate matrices in the spin space and us e the notation ˆ σ0for the 2 ×2 unit matrix). Hnn′is anN×Nrandom matrix from a gaussian orthogonal ensemble, described by the pair correlators: HmnHm′n′=Nδ2 1 π2[δmn′δnm′+δmm′δnn′]. (2.2) Hereδ1is the mean single-particle level spacing in the dot. The magnetization energy E(S) is the generalizationof the JsS2term in the universal Hamiltonian for the electron- electron interaction in a chaotic quantum dot.17Since we are going to describe a ferromagnetic state with a largevalu e of the total spin on the dot, we must go beyond the quadratic term ; in fact, all terms should be included. E(S) can be viewed as the sum of all irreducible many-particle vertices in the spin c hannel, obtained after integrating out degrees of freedom with high energies (above Thouless energy); the corre sponding term in the action is thus local in time, and can be written as the time integral of an instantaneous function E(S(t)). This functional can be decoupled using the Hubbard-Stratonovich transformation with a real vector field h(t), which we call below the internal magnetic field: exp/parenleftbigg −i/contintegraldisplay dtE(S)/parenrightbigg =/integraldisplay Dh(t) exp/parenleftbigg i/contintegraldisplay dt(2h·S−˜E(h))/parenrightbigg . (2.3) We rewrite the action S[ψ,ψ∗] in the form S[ψ,ψ∗,h] =/contintegraldisplay dt N/summationdisplay n,n′=1ψ† n(t)/parenleftBig ˆG−1/parenrightBig nn′ψn′(t)−˜E(h(t)) , (2.4)3 where the inverse Green’s function /parenleftBig ˆG−1/parenrightBig nn′= (iˆσ0∂t+h·ˆσ)δnn′−Hnn′ˆσ0(2.5) is a matrix in time variables t,t′, in orbital indices nand n′with 1≤n,n′≤N, in spin indices, and in forward(+) and backward ( −) directions on the Keldysh contour. Integration over fermionic fields ψn,ψ† nyields the purely bosonic action: S[h] =−iTr/braceleftBig ln(−iˆG−1)/bracerightBig −/contintegraldisplay dt˜E(h(t)),(2.6) where the trace is taken over allindices of the Green’s function, listed above. In the space of forward and backward directions on the Keldysh contour, we perform the standard Keldysh rotation, introducing the retarded ( GR), advanced ( GA), Keldysh (GK), and zero ( GZ) components of the Green’s function: /parenleftbiggˆGRˆGK ˆGZˆGA/parenrightbigg =1 2/parenleftbigg 1 1 1−1/parenrightbigg/parenleftbiggˆG++ˆG+− ˆG−+ˆG−−/parenrightbigg/parenleftbigg 1 1 −1 1/parenrightbigg , (2.7) as well as the classical ( hcl) and quantum ( hq) compo- nents of the field: /parenleftbigg hclhq hqhcl/parenrightbigg =1 2/parenleftbigg 1−1 1 1/parenrightbigg/parenleftbigg h+0 0−h−/parenrightbigg/parenleftbigg 1 1 1−1/parenrightbigg . (2.8) We will also write this matrix as h=hclτcl+hqτq, whereτclandτqare 2×2 matrices in the Keldysh space coinciding with the unit 2 ×2 matrix and the first Pauli matrix, ˆσx, respectively. The saddle point of the bosonic action Eq. (2.6) is found by the first order variation with respect to hcl,q(t), which gives the self-consistency equation: hq(t) = 0,−∂˜E(hcl(t)) ∂hcl j(t)=i 2Tr n,σ/braceleftBig ˆσjˆGK nn(t,t)/bracerightBig .(2.9) We also note that the right-hand side of this equation is proportional to the total spin of electrons of the particle for a given trajectory of hcl(t): S(t) =i 4Tr n,σ/braceleftBig ˆσjˆGK(t,t)/bracerightBig . (2.10) In Eqs. (2.9) and (2.10), the trace is taken over orbital and spin indices only. In the limit N→ ∞, one can obtain a closed equation for the Green’s function traced over the orbital indices:21 ˆg(t,t′) =iδ1 πN/summationdisplay n=1ˆGnn(t,t′). (2.11) The matrix ˆ g(t,t′) satisfies the following constraint: /integraldisplay ˆg(t,t′′)ˆg(t′′,t′)dt′′=τclˆσ0δ(t−t′),(2.12)where the right-hand side is just the direct product of unit matrices in the spin, Keldysh, and time indices. The Wigner transform of ˆ gK(t,t′) is related to the spin- dependent distribution function ˆf(ε,t) of electrons in the dot: ∞/integraldisplay −∞ˆgK(t+˜t/2,t−˜t/2)eiε˜td˜t= 2ˆf(ε,t).(2.13) In equilibrium, ˆf(ε) = ˆσ0tanh(ε/2T). The self-consistency condition (2.9) takes the form −∂˜E(hcl(t)) ∂hcl i(t)=π 2δ1lim t′→tTr σ/braceleftbig ˆσiˆgK(t,t′)/bracerightbig −2hcl i(t) δ1. (2.14) The last term takes care of the anomaly arising from non-commutativity of the limits N→ ∞andt′→t. In this paper we consider the dot coupled to two leads, identified as left ( L) and right ( R). The leads have NLandNRtransverse channels, respectively, see Fig 1. For non-magnetic leads and spin-independent coupling between the leads and the particle, we can characterize each channel by its transmission Tnwith 0< Tn≤1 and by the distribution function of electrons in the chan- nelFn(t−t′), assumed to be stationary. We consider the limit of strong coupling between the leads and the particle,/summationtextNch n=1Tn≫1. The coupling to the leads gives rise to a self-energy term, which should be included in the definition of the Green’s function, Eq. (2.5). Without going into details of the derivation, presented in Ref. 21, we give the final form of the equation for the Green’s function traced over the orbital states, Eq. (2.11): [∂t−ih·ˆσ,ˆg] =Nch/summationdisplay n=1Tnδ1 2π/parenleftbigg −FnˆgZˆgRFn−FnˆgA−ˆgK ˆgZ−ˆgZFn/parenrightbigg ×/bracketleftbigg ˆ1+Tn 2/parenleftbigg ˆgR−ˆ1+FnˆgZˆgRFn+FnˆgA 0 −ˆgA−ˆ1+ ˆgZFn/parenrightbigg/bracketrightbigg−1 . (2.15) Here the products of functions include convolution in time variables. This equation is analogous to the Usadel equation used in the theory of dirty superconductors.22 To conclude this section, we discuss the dependence ˜E(h). Deep in the ferromagnetic state, i.e.far from the Stoner critical point, we expect the mean-field approach to give a good approximation for the total spin of the dot. Namely, the mean field acting on the electron spins, is given by 2 h0=dE(S)/dS≡E′(S). We then require that the response of the system to this field gives the same average value for the spin: S0=2h0 2δ1=E′(S0) 2δ1. (2.16) Here we evaluated S0from Eq. (2.10) and applied the self-consistency equation (2.14) to equilibrium state with4 ˆgK∝ˆσ0, when the contribution of the first term in the right hand side of Eq. (2.14) vanishes. Not expecting strong deviations of the magnitude of the spin from the mean-field value, we focus on the form of˜E(h) when|h| ≈h0. The inverse Fourier transform of Eq. (2.3) and angular integration for the isotropic E(S) gives e−i˜E(h)∆t= const∞/integraldisplay 0sin2Sh∆t 2Sh∆te−iE(S)∆tS2dS,(2.17) where ∆tis the infinitesimal time increment used in the construction of the functional integral in Eq. (2.3). Expanding E(S) near the mean-field value S0, E(S)≈E(S0)+E′(S−S0)+E′′ 2(S−S0)2,(2.18) performing the integration in the stationary phase ap- proximation and using S0=h0/δ1=−E′/(2δ1), we ob- tain ˜E(h) =−2(h−h0−E′′S0/2)2 E′′+˜E0,(2.19) where˜E0ish−independent term. This expression for ˜E(h) defines the action S[h], Eq. (2.6). The energy E(S) does not contain the energy Eorb(S), associated with the orbital motion of electrons in the particle. Namely, to form a total spin Sof the par- ticle, we have to redistribute Selectrons over orbital states, which changes the orbital energy of electrons by Eorb(S)≃δ1S2. The total energy Etot(S) of the particle is the sum of two terms: Etot(S) =E(S)+Eorb(S). Sim- ilarly, we obtain the total energy of the system in terms of internal magnetic field ˜Etot(h) =˜E(h)−h2 δ1 =−2/parenleftbigg1 E′′+1 2δ1/parenrightbigg (h−h0)2+˜E1,(2.20) where˜E1does not depend on h. We notice that the extremumof ˜Etot(h)correspondsto h=h0anddescribes the expectation value of the internal magnetic field in an isolated particle. The energy cost of fluctuations of the magnitude of the internal magnetic field is characterized by the coefficient 1 /E′′+1/2δ1. III. KELDYSH ACTION In this Section we analyze the action Eq. (2.6) for the internal magnetic field hα. We expect that the classical component hcl(t) of this field contains fast and small os- cillations of its magnitude around the mean-field value h0. We further expect that the orientation of hcl(t) changes slowly in time, but is not restricted to smalldeviations from some specific direction. Based on this picture, we introduce a unit vector n(t), assumed to de- pend slowly on time, and write hcl(t) = (h0+hcl /bardbl(t))n(t), (3.1) wherehcl /bardbl(t) is assumed to be fast and small. We expand the action (2.6) to the second order in small fluctuations of the quantum component hq(t) and the radial classical component hcl /bardbl(t): S[h]≈ −2π δ1/integraldisplay dtgK(t,t)hq(t)+ +8 E′′/integraldisplay dthcl /bardbl(t)n(t)hq(t) −/integraldisplay dtdt′ΠR ij(t,t′)hq i(t)hcl /bardbl(t′)nj(t′) −/integraldisplay dtdt′ΠA ij(t,t′)hcl /bardbl(t)ni(t)hq j(t′) −/integraldisplay dtdt′ΠK ij(t,t′)hq i(t)hq j(t′).(3.2) The applicability of this quadratic expansion is discussed in Sec. VB. In Eq. (3.2) we introduced the polarization operator, defined as the kernel of the quadratic part of the action of the fluctuating bosonic fields: /parenleftbigg ΠZΠA ΠRΠK/parenrightbigg ≡/parenleftbigg Πcl,clΠcl,q Πq,clΠq,q/parenrightbigg ,(3.3a) Παβ ij(t,t′) =i 2δ2Tr/braceleftbig lnG−1/bracerightbig δhβ j(t′)δhα i(t),(3.3b) whereα,β=cl,qandi,j=x,y,z. The short time anomaly is explicitly taken into account in the definition of the polarization operators, see Eq. (3.14) below. ThefirsttermofEq.(3.2)containsthevector gKofthe Keldysh component of the Green function ˆ gK= ˆσ0gK 0+ ˆσ·gK. We emphasize that the Green’s function and the polarization operator in Eq. (3.2) are calculated at hcl /bardbl(t) = 0 and hq(t) = 0 for a given trajectory of the classical field h0n(t). A. Keldysh component of the Green function For the Green’s function in the classical field we have ˆgR(t,t′) =−ˆgA(t,t′) = ˆσ0δ(t−t′),(3.4) while the Keldysh component satisfies the equation /bracketleftbig ∂t−ih0n·ˆσ,ˆgK/bracketrightbig =Nch/summationdisplay n=1Tnδ1 2π/parenleftbig 2Fn−ˆgK/parenrightbig .(3.5) We introduce the notation 1 τ=Nch/summationdisplay n=1Tnδ1 2π=1 τL+1 τR, (3.6)5 Then the scalar gK 0and vector gKcomponents of ˆ gK= ˆσ0gK 0+ˆσ·gKsatisfy two coupled equations: /bracketleftbigg ∂t+∂t′+1 τ/bracketrightbigg gK 0(t,t′) =Nch/summationdisplay n=1Tnδ1 2π2Fn(t−t′) +ih0[n(t)−n(t′)]·gK(t,t′), (3.7a)/bracketleftbigg ∂t+∂t′+1 τ/bracketrightbigg gK(t,t′)+h0[n(t)+n(t′)]×gK(t,t′) =ih0[n(t)−n(t′)]gK 0(t,t′). (3.7b) As a zero approximation, we can consider the station- ary situation: gK 0(t,t′) =gK 0(t−t′) andn(t) = const. In this case, we have gK 0(t,t′) =τ τL2FL(t−t′)+τ τR2FR(t−t′) (3.8) andgK= 0. For an arbitrarytime dependence n(t), Eq. (3.7b) can- not be solved analytically. However, if the variation of n(t) is slow enough, we can make a gradient expansion: /parenleftbigg ∂t+1 τ/parenrightbigg gK+2h0n×gK=i˜th0˙ngK 0.(3.9) Here we introduced t= (t+t′)/2,˜t=t−t′,∂t+∂t′=∂t. The dependence on ˜tis split off and remains unchanged, while for the dependence on tthe solution is determined by a linear operator L+ n: L± n=/parenleftbigg1 τ±∂t±2h0n×/parenrightbigg−1 , (3.10a) L+ n(ω)X(ω) =n(n·X(ω)) −iω+1/τ+ +1 2/summationdisplay ±−n×[n×X(ω)]±i[n×X(ω)] −i(ω±2h0)+1/τ. (3.10b) Here we assume that the direction of the internal mag- netic field nchanges slowly in time, and |˙n|τ≪1. Thus, all perturbations of gKdecay with the charac- teristic time τ. In particular, the solution of Eq. (3.9) has the form gK=i˜th0L+ n˙ngK 0≈i˜th0τgK 0˙n+2h0τn×˙n (2h0τ)2+1.(3.11) Expression for the first term in Eq. (3.2) can be easily obtained from Eq. (3.11) by taking the limit ˜t→0 and taking into account that any fermionic distribution func- tion in the time representation has the following equal- time asymptote: gK 0(t,t′)≈2 iπ1 t−t′,(t→t′).(3.12) We have gK(t,t) =2h0τ π˙n+2h0τn×˙n (2h0τ)2+1. (3.13)We notice that n·gK(t,t) = 0, and therefore the first term in the action Eq. (3.2) is coupled only to the tan- gential fluctuations of hq(t)⊥n(t). B. Polarization operator We express the polarization operator in terms of the unit vector n(t). The polarization operator can be rep- resented as the response of the Green’s functions to a change in the field, as follows directly from the defini- tion (3.3b) and the expression (2.6) for the action: Παβ ij(t,t′) =π 2δ1Tr4×4/braceleftbig ταˆσiδˆg(t,t)/bracerightbig δhβ j(t′)−2 δ1τq αβδijδ(t−t′). (3.14) Here the Green function δˆg(t,t) can be calculated as the first-order response of the solution of Eq. (2.15) to small arbitrary (in all three directions) increments of δhcl(t) andδhq(t). The zero-order solution of Eq. (2.15) in the fieldhcl=h0nandhq= 0 is ˆg(t,t) = ˆσ0/parenleftbigg δ(t−t′)gK 0(t−t′) 0−δ(t−t′)/parenrightbigg .(3.15) First, we calculate δˆgZ, which responds only to δhq: /bracketleftbigg ∂t+∂t′−1 τ/bracketrightbigg δˆgZ(t,t′)−ih0ˆσ·n(t)δˆgZ(t,t′) +δˆgZ(t,t′)ih0ˆσ·n(t′) = 2iˆσ·δhq(t)δ(t−t′). (3.16) Since∂t+∂t′=∂t, the solution always remains propor- tional toδ(t−t′): δˆgZ(t,t′) =−2iˆσ(L− nδhq)(t)δ(t−t′).(3.17) GivenδˆgZ, components δˆgR,Acan be found either from Eq. (2.15), or, equivalently, using the constraint ˆ g2=ˆ1: ˆgδˆg+δˆgˆg= 0⇒δˆgR=−ˆgKδˆgZ 2, δˆgA=δˆgZˆgK 2. (3.18) We notice that both δˆgR,Arespond only to hq(t) and, therefore, ΠZ ij(t,t′)∝Tr{ˆσi(δˆgR(t,t)+δˆgA(t,t))} δhcl j(t′)≡0.(3.19) This equation ensures that the action along the Keldysh contour vanishes for hq≡0. To evaluate the remaining three components of the po- larization operator, we can apply the variational deriva- tives to the sum of δˆgK(t,t) +δˆgZ(t,t) with respect to either classical δhcl(t′) or quantum δhq(t′) field, which give ΠR ij(t,t′) and ΠK ij(t,t′), respectively. Then, the ad- vanced component ΠA ij(t,t′) = [ΠR ji(t′,t)]∗.6 The equation for δˆgK=ˆσ·δgKreads as /bracketleftbigg ∂t+∂t′+1 τ/bracketrightbigg δgK(t,t′) +h0/bracketleftbig n(t)×δgK(t,t′)−δgK(t,t′)×n(t′)/bracketrightbig =i/bracketleftbig δhcl(t)−δhcl(t′)/bracketrightbig gK 0(t−t′) −2iδhq(t)δ(t−t′)−Q(t,t′), (3.20a) where Q(t,t′) =Nch/summationdisplay n=1Tnδ1 2π/parenleftbigggK 0 2δgZgK 0 2+FnδgZFn/parenrightbigg −Nch/summationdisplay n=1Tn(1−Tn)δ1 2π/parenleftbigggK 0 2−Fn/parenrightbigg δgZ/parenleftbigggK 0 2−Fn/parenrightbigg (3.20b) andδgZ= Tr{ˆσδˆgZ}/2 withδˆgZgiven by Eq. (3.17). To calculate the retarded component ΠR ijof the polar- ization operator, we calculate the response of δgK(t,t′) toδhqin the limit t′→t. Using the asymptotic behavior of the Fermi function, Eq. (3.12), we obtain: δgK(t,t) =/integraldisplaydω 2π−2iω πL+ n(ω)δhcl(ω)e−iωt.(3.21)Substituting this expression for δgK(t,t) to Eq. (3.14), we obtain ΠR ij(ω) = ΠR /bardbl,ij(ω)+ΠR ⊥,ij(ω),(3.22a) with ΠR /bardbl,ij(ω) =−2 δ1ninj 1−iωτ, (3.22b) ΠR ⊥,ij(ω) =−2 δ1/summationdisplay ±δij−ninj±ieijknk 2(3.22c) ×(1±2ih0τ) 1−i(ω∓2h0)τ. Here we represented the polarization operator ΠR ij(ω) as a sum of the radial, ΠR /bardbl,ij(ω), and tangential, ΠR ⊥,ij(ω), terms. We note that the action Eq. (3.2) contains only theradialcomponentoftheretardedandadvancedpolar- ization operators because we do not perform expansion in terms of the tangential fluctuations of the classical component of the field hcl(t). In response to δhq, both corrections δgK(t,t′) and δgZ(t,t′) contain terms ∝δ(t−t′), However, their sum δgK(t,t′)+δgZ(t,t′) remains finite in the limit t→t′: δgK(t,t′)+δgZ(t,t′) =−2i/integraldisplay dt′′/integraldisplay dt1dt2L+ n(¯t−t1)Q(t1−t2+˜t/2;t2−t1+˜t/2)L− n(t2−t′′)δhq(t′′),(3.23a) Q(τ1;τ2) =2 τδ(τ1)δ(τ2)−Nch/summationdisplay n=1Tnδ1 2π/bracketleftbigggK 0(τ1) 2gK 0(τ2) 2+Fn(τ1)Fn(τ2)/bracketrightbigg (3.23b) −Nch/summationdisplay n=1Tn(1−Tn)δ1 2π/bracketleftbigggK 0(τ1) 2−Fn(τ1)/bracketrightbigg/bracketleftbigggK 0(τ2) 2−Fn(τ2)/bracketrightbigg with¯t= (t+t′)/2 and˜t=t−t′. From Eq. (3.23) we obtain the following expression for the Keldysh component of the polarization operator: ΠK ij(ω) = ΠK /bardbl,ij(ω)+ΠK ⊥,ij(ω), (3.24a) ΠK /bardbl,ij(ω) =−ininj ω2+1/τ2R(ω), (3.24b) ΠK ⊥,ij(ω) =−i 2/summationdisplay ±δij−ninj±ieijknk (ω∓2h0)2+1/τ2R(ω).(3.24c) Here function R(ω) coincides with the noise power of electric current through a metallic particle in the approx-imation of non-interacting electrons R(ω) =Nch/summationdisplay n=1/integraldisplaydε 8πTn ×/braceleftBig/bracketleftbig 8−gK 0(ε)gK 0(ε+ω)−4Fn(ε)Fn(ε+ω)/bracketrightbig +(1−Tn)/bracketleftbig gK 0(ε)−2Fn(ε)/bracketrightbig/bracketleftbig gK 0(ε+ω)−2Fn(ε+ω)/bracketrightbig/bracerightBig . (3.25) In principle, electron-electron interaction in the charge channel can be taken into account. The interaction mod- ifies the expression Eq. (3.25) for R(ω) to the higher order23inτδ1≪1 and we neglect this correction here. In this paper we consider a particle connected to electron leads at temperature Twith the applied bias V. In this case, FL,R(ε) = tanh(ε−µL,R)/(2T) with7 µL−µR=V, and the integration over εgives 2πτR(ω) = 4ωcothω 2T+ΞΥT(V,ω),(3.26) where ΥT(V,ω)≡/summationdisplay ±2(ω±V)cothω±V 2T−4ωcothω 2T (3.27) and Ξ is the ”Fano factor” for a dot Ξ =τ2 τLτR+τ3δ1 2πτ2 R/summationdisplay n∈LTn(1−Tn) +τ3δ1 2πτ2 L/summationdisplay n∈RTn(1−Tn).(3.28) At|V| ≫Tthe function Υ T(V,ω) has two scales of ω: (i)Tsmears the non-analyticity at ω→0, but the value of ΥT(V,ω) deviates from Υ T(V,0) at|ω| ∼ |V|. Thus, the typical time scale above which one can approximate ΠK(ω) by a constant is at least ω≪max{T,|V|}. In the limitω→0 we have ΠK ij(ω= 0) =−i8τTeff δ1/parenleftbigg ninj+δij−ninj (2h0τ)2+1/parenrightbigg .(3.29) The effective temperature Teffis given by Teff≡T+Ξ/parenleftbiggV 2cothV 2T−T/parenrightbigg .(3.30) C. Final form of the action We can rewrite the action for magnetization field h= {hcl;hq}withhclin the form of Eq. (3.1) as a sum of the radial and tangential terms: S[h] =S/bardbl[hcl /bardbl,hq /bardbl]+S⊥[n(t),hq ⊥]. (3.31) The radial term in the action has the form S/bardbl[hcl /bardbl,hq /bardbl] = (D−1 /bardbl)αβ(t,t′)hα /bardbl(t)hβ /bardbl(t′),(3.32) where the inverse function of the internal magnetic field propagator is given by (D−1 /bardbl)αβ(t,t′) =4 E′′/parenleftbigg 0 1 1 0/parenrightbigg δ(t−t′) −/parenleftBigg 0 ΠR /bardbl(t,t′) ΠA /bardbl(t,t′) ΠK /bardbl(t,t′)/parenrightBigg(3.33) and Παβ /bardbl(t,t′) =ninjΠαβ /bardbl,ij(t,t′). From this equation we find DR /bardbl(ω) =Dq,cl /bardbl(ω) =E′′ 4−iω+1/τ −iω+(δ1+E′′/2)/(τδ1), (3.34)andDA /bardbl(ω) = [DR /bardbl(ω)]∗. The Keldysh component is DK /bardbl(ω) =Dq,q /bardbl(ω) =DR /bardbl(ω)ΠK /bardbl(ω)DA /bardbl(ω).(3.35) The tangential term in the action is S⊥[n(t),hq ⊥] =−4h0τ δ1/integraldisplay dt(˙n+2h0τn×˙n)hq ⊥ (2h0τ)2+1 −4 δ1/integraldisplay dt[n(t)×[n(t)×B]]·hq ⊥(t) −/integraldisplay dtdt′hq ⊥,i(t)ΠK ⊥,ij(t−t′)hq ⊥,j(t′). (3.36) Here we recovered the external magnetic field B(t). The polarization operator ΠK ⊥,ijis given by Eq. (3.24c). IV. LANGEVIN EQUATION A. Langevin equation for the direction of the internal magnetic field In this section we consider evolution of the direction vectorn, described by the tangential terms in the action, Eq. (3.36). We neglect fluctuations of the magnitude of the internal magnetic field, h/bardbl, the conditions when these fluctuations can be neglected are listed in the next section. We decouple the quadratic in hq ⊥component of the actionin Eq. (3.36) by introducingan auxiliaryfield w(t) with the probability distribution P[w(t)]∝exp/braceleftbigg4i δ2 1/integraldisplay dtdt′(ΠK ⊥)−1 ij(t,t′)wi(t)wj(t′)/bracerightbigg , (4.1) and the correlation function ∝an}bracketle{twi(t)wj(t′)∝an}bracketri}ht=δ2 1 8iΠK ⊥,ij(t,t′).(4.2) The field w(t) plays the role of the gaussian random Langevin force. Integration of the tangential part of the action, Eq. (3.36), over hq ⊥produces a functional δ-function, whose argument determines the equation of motion: ˙n+2τh0[n×˙n] 4τ2h2 0+1−1 τh0(w−n×[n×B]) = 0.(4.3) The above equation can be resolved with respect to ˙n: ˙n=−2[n×(w+B)]−1 h0τ[n×[n×(w+B)]].(4.4) This equation is the Langevin equation for the direc- tionn(t) of the internal magnetic field in the presence of the external magnetic field B(t) and the Langevin stochastic forces w(t).8 B. The Fokker-Plank equation Next, we follow the standard procedure of derivation of the Fokker-Plank equation for the distribution P(n) of the probability for the internal magnetic field to point in the direction n. The probability distribution satisfies the continuity equation: ∂P ∂t+∂Ji ∂ni= 0, (4.5) where the probability current is defined as J=−/parenleftbigg 2n×B+1 h0τn×[n×B]/parenrightbigg P +1 2/angbracketleftBig ξ/parenleftbigg ξ·∂P ∂n/parenrightbigg/angbracketrightBig (4.6) and the stochastic velocity ξis introduced in terms of the field was ξ=−2[n×w]−1 h0τ[n×[n×w]].(4.7) The derivative ∂/∂nis understood as the differentiation withrespecttolocalEuclideancoordinatesinthetangent space. Performing averaging over fluctuations of win Eq. (4.6), we obtain ∂P ∂t=∂ ∂n/braceleftbigg(2h0τ)[n×B]+[n×[n×B]] h0τP/bracerightbigg +1 T0∂2P ∂n2,(4.8) where the time constant T0is defined as T0=2(h0τ)2 τTeffδ1. (4.9) Below we use the polar coordinates for the direction of the internal magnetic field, n= {sinθcosϕ,sinθsinϕ,cosθ}. In this case the Fokker- Plank equation can be rewritten in the form ∂P ∂t=1 sinθ∂ ∂ϕ/bracketleftbigg FϕP+1 T01 sinθ∂P ∂ϕ/bracketrightbigg +1 sinθ∂ ∂θ/bracketleftbigg sinθFθP+sinθ T0∂P ∂θ/bracketrightbigg ,(4.10) where Fϕ=Bxsinϕ−Bycosϕ h0τ + 2cosθ(Bxcosϕ+Bysinϕ)−2sinθBz,(4.11) Fθ= 2(Bxsinϕ−Bycosϕ) −cosθ h0τ(Bxcosϕ+Bysinϕ)+sinθ h0τBz.(4.12)It should be supplemented by the normalization condi- tion: 2π/integraldisplay 0dϕπ/integraldisplay 0sinθdθP(ϕ,θ) = 1, (4.13) which is preserved if the boundary conditions at θ= 0,π are imposed: lim θ→0,πsinθ2π/integraldisplay 0dϕ∂P ∂θ= 0. (4.14) Below we apply the Fokker Plank equation for calcu- lations of the magnetization of a particle M=/integraldisplaydΩn 4πnP(n) (4.15) V. APPLICABILITY OF THE APPROACH In this section we discuss the conditions of validity of the stochastic LLG equation, see Eq. (4.10), for the model of ferromagnetic metallic particle connected to leads at finite bias. We briefly listed these conditions in the Introduction. Here we present their more detailed quantitative analysis. A. Fluctuations of the radial component of the internal magnetic field We represented the classical component of the internal magnetic field hclin terms of a slowly varying direction n(t) and fast oscillations hcl /bardblof its magnitude around the average value h0. Now, we evaluate the amplitude of oscillations of the radial component hcl /bardblof the field, using the radial term in the action, see Eqs. (3.31) and (3.32). The typicalfrequencies fortime evolutionofsmallfluc- tuations of the internal magnetic field in the radial direc- tion are of order of ω∼δ1+E′′/2 δ11 τ(5.1) as one can conclude from the explicit form of the prop- agatorDR /bardbl(ω), Eq. (3.34), of these fluctuations. This scale has the meaning of the inverse RC-time in the spin channel. Deep in the ferromagnetic state (i. e., far from the Stoner critical point E′′+2δ1= 0) we estimate δ1+E′′/2∼δ1(which is equivalent to E′′∼h0/S0), so this spin-channel RC-time is of the same order as the escape time τ. This estimate for the frequency range is consistent with the simple picture, which describes the evolution of the internal magnetic field of the grain as a response to a changing value of the total spin of the particle due to random processes of electron exchange9 between the dot and the leads. The electron exchange happens with the characteristic rate 1 /τ. The correlation function ∝an}bracketle{thcl /bardbl(t)hcl /bardbl(t′)∝an}bracketri}htcan be evalu- ated by performing the Gaussian integration with the quadratic action in hcl /bardblandhq /bardbl. Using Eq. (3.35), we ob- tain the equal-time correlation function ∝an}bracketle{t(hcl /bardbl)2∝an}bracketri}ht=i 2/integraldisplaydω 2πDK /bardbl(ω) =(E′′)2 32τδ1/integraldisplaydω 2π2πR(ω) ω2+[1+E′′/(2δ1)]2/τ2.(5.2) This equation gives the value of fluctuations of the radial component of the internal magnetic field of the particle. These fluctuations survive even in the limit T= 0 and V= 0, when R(ω) = 2|ω|/πτ. We have the following estimate ∝an}bracketle{t(hcl /bardbl)2∝an}bracketri}ht=(E′′)2 16πτδ1lnETτ 1+E′′/(2δ1),(5.3) the upper cutoff ETis the Thouless energy, ET=vF/L for a ballistic dot with diameter Land electron Fermi velocityvF. The separation of the internal magnetic field into the radial and tangential components is justified, provided that the fluctuations/radicalBig ∝an}bracketle{t(hcl /bardbl)2∝an}bracketri}htof the radial component are much smaller than the average value of the field h0, i.e.∝an}bracketle{t(hcl /bardbl)2∝an}bracketri}ht ≪h2 0. Using the estimate Eq. (5.3), we obtain the necessary requirement for the applicability of equations for the slow evolution of the vector of the in- ternal magnetic field of a particle: S0≫/radicalbigg 1 τδ1ln(ETτ), (5.4) whereS0is the spin of a particle in equilibrium and we again used the estimate E′′∼h0/S0. Condition of Eq. (5.4) requires that the system is not close to the Stoner instability. B. Applicability of the gaussian approximation Let us discuss the applicability of the gaussian approx- imation for the action in hcl /bardblandhq. The coefficients in front of terms hq(t)h/bardbl(t1)...hcl /bardbl(tn) are obtained by tak- ing thenth variational derivative of δgK(t,t)+δgZ(t,t), or, equivalently, byiteratingthe Usadelequation ntimes. Since the typical frequencies of h/bardblareω∼1/τ, the left- hand side of the equation is ∼δ(n+1)gK/τ, while the right-hand side is hcl /bardblδ(n)gK. Since the only time scale hereisτ, allthe coefficientsofthe expansionofthe action inhcl /bardbl(ω) atω∼1/τare of the same order: Sn+1∼τn−1 E′′/integraldisplaydω1...dωn (2π)n× ×hcl /bardbl(ω1)...hcl /bardbl(ωn)hq(−ω1−...−ωn).(5.5)At the same time, the typical value of hcl /bardbl(ω∼1/τ), as determined by the gaussian part of the action, was estimated in the previous subsection to be of the order of/radicalBig τDK /bardbl(ω∼τ)∼√τδ1≪1, so the higher-order terms are indeed not important. For the quantum component of the field the quadratic and quartic terms in the action are estimated as Sn+1∼Teffτn δ1/integraldisplaydω1...dωn (2π)n× ×hq(ω1)...hq(ωn)hq(−ω1−...−ωn).(5.6) IfTeff≫1/τ, then the typicalfrequencyscaleis ω∼1/τ, sothe quadraticterm gives hq(ω∼1/τ)∼/radicalbig δ1/Teff, and Sn∼(δ1/Teff)n/2−1∼[τδ1/(τTeff)]n/2−1. IfTeff≪1/τ, at the typical scale ω∼Teffwe obtainhq(ω∼Teff)∼/radicalbig δ1/(T2 effτ), so again Sn∼(τδ1)n/2−1≪1 forn>2. Physically, the parameter 1 /(τδ1) =Nch(orTeff/δ1, if it is larger) can be identified with the number of the independent sources of the noise acting on the magne- tization field. Thus, the smallness of the non-gaussian part of the action is nothing but the manifestation of the central limit theorem. C. Applicability of the Fokker-Plank equation From the above analysis we found that evolution of the direction of the internal magnetic field in time is described by a characteristic time T0, introduced in Eq. (4.9). From the analysis of the fluctuations of the magnitude of the internal magnetic field, see Eq. (5.1), we obtain the following condition when the separation into slow and fast variables is legitimate. The criterium can be formulated as T0≫τ, which can be presented as Teff δ1≪/parenleftbiggh0 δ1/parenrightbigg2 =S2 0. (5.7) VI. MAGNETIC SUSCEPTIBILITY OF METALLIC PARTICLES OUT OF EQUILIBRIUM The LLG equation derived in this paper for a ferro- magnetic particle with finite bias between the leads can be applied to a number of experimental setups. More- over, the derivation of the equation can be generalized to spin-anisotropic contacts with leads or Hamiltonian of electron states in the particle. In this paper we apply the stochastic equation for spin distribution function to the analysis of the magnetic susceptibility at finite fre- quency. The susceptibility is the basic characteristic of magnetic systems, it can often be measured directly, and determines other measurable quantities. Below, we calculate the susceptibility of an ensemble of particles placed in constant magnetic field of an ar- bitrary strength and oscillating weak magnetic field, see10 Fig. 1. We consider the oscillating magnetic field with its components in directions parallel and perpendicular to the constant magnetic field. A. Solution at zero noise power AtTeff= 0 when w(t) = 0, and at fixed direction of the field, B(t) =ezB(t), equation of motion (4.4) is easily integrated for an arbitrary time dependence B(t): ϕ=ϕ0+t/integraldisplay 02B(t′)dt′, (6.1) tanθ 2= tanθ0 2exp −t/integraldisplay 0B(t′) h0τdt′ .(6.2) Here the direction of magnetic field correspondsto θ= 0. B. Constant magnetic field At finiteTeffin constant magnetic field B0the Fokker- Plank equation has a simple solution P0(θ) =b sinhbebcosθ 4π, (6.3) where the strength of constant magnetic field is written in terms of the dimensionless parameter b≡(2h0τ)B0 τδ1Teff. (6.4) Substituting this probability function to Eq. (4.15), we obtain the classical Langevin expression for the magne- tization of a particle in a magnetic field Mz= cothb−1 b, Mx=My= 0.(6.5) This expression for the magnetization coincides with the magnetization of a metallic particle in thermal equilib- rium, provided that the temperature is replaced by the effective temperature Teffdefined by Eq. (3.30). The differential dc susceptibility is equal to χdc /bardbl=dMz(b) db=1 b2−1 sinh2b. (6.6) C. Longitudinal susceptibility We now consider the response of the magnetization to weakoscillations ˜Bz(t)oftheexternalmagneticfieldwith frequencyωin direction parallel to the fixed magnetic fieldB0. We write the oscillatory component of the field in terms of the dimensionless field strength: b/bardble−iωt+b∗ /bardbleiωt=2h0˜Bz(t) δ1Teff. (6.7)The linear correction to the probability distribution can be cast in the form P(θ,t) =/bracketleftBig 1+b/bardblu/bardbl(θ)e−iωt+b∗ /bardblu∗ /bardbl(θ)eiωt/bracketrightBig P0(θ), (6.8) withP0(θ) defined by Eq. (6.3). The magnetic ac sus- ceptibility can be evaluated from Eq. (6.8) as χ/bardbl(ω,b) = 2ππ/integraldisplay 0u/bardbl(θ)P0(θ)cosθsinθdθ. (6.9) Theequationfor u/bardbl(θ) isobtainedfromEq.(4.10)with Bz=B0+˜Bz(t): ∂2u/bardbl ∂θ2+cosθ−bsin2θ sinθ∂u/bardbl ∂θ+iΩu/bardbl=bsin2θ−2cosθ, (6.10) where we introduced the dimensionless frequency Ω =ωT0, (6.11) and the time constant T0is defined in Eq. (4.9). Note the symmetry of Eq. (6.10) with respect to the simultaneous change b→ −bandθ→π−θ. Also, the normalization condition for the probability function re- quires that π/integraldisplay 0u/bardbl(θ)P0(θ)sinθdθ= 0. (6.12) The latter holds if the boundary conditions Eq. (4.14) are satisfied, which in the case of axial symmetry can be written as lim θ→0,π/braceleftbigg sinθ∂u/bardbl(θ) ∂θ/bracerightbigg = 0. (6.13) Thedifferentialequation(6.10)withtheboundarycon- dition Eq. (6.13) can be solved numerically and then the susceptibility is evaluated according to Eq. (6.9). The result is shown in Figs. 2 and 3, where the susceptibility is shown as a function of frequency ωor magnetic field b, respectively. We also consider various asymptotes for the acsusceptibility, obtainedfromthe solutionofEq.(6.10). At zero constant magnetic field, b= 0, we find the exact solution of Eq. (6.10) explicitly: u/bardbl(θ) =cosθ 1−iΩ/2. (6.14) This solution allows us to calculate the ac susceptibility in the form χ/bardbl(Ω,b= 0) =1 31 1−iΩ/2. (6.15) Forb≫1 only cosθ∼1/bmatter, and we can find a specific solution of the inhomogeneous equation: u/bardbl(θ) =1−b(1−cosθ) b−iΩ/2, b≫1. (6.16)11 0.000.100.200.30 02468100.000.040.080.120.16 PSfrag replacementsReχ/bardbl(Ω,b) Imχ/bardbl(Ω,b) Ωb= 0.1 b= 0.5 b= 1 b= 2 b= 5 FIG. 2: (Color online). Plot of the real and imaginary parts of the susceptibility χ/bardbl(Ω,b) as a function of the dimension- less frequency Ω = ωT0. The oscillatory field at frequency ω is parallel to the constant magnetic field with strength b. The real part of the susceptibility decreases monotonically fr om its dc value, Eq. (6.6), as frequency increases, while the im ag- inary part increases linearly at small Ω ≪1, see Eq. (6.20), and decreases at higher frequencies. The requirement of regularity at the opposite end can be replaced by the probability normalization condition, Eq. (6.12), which is satisfied by this solution. Substitut- ing this solution to Eq. (6.9), we obtain the strong field asymptote for the ac susceptibility χ/bardbl(Ω,b) =1 b(b−iΩ/2). (6.17) For Ω≫1 and Ω ≫b, we can neglect the derivatives in Eq. (6.10) and find the solution in the form u/bardbl(θ)≈bsin2θ−2cosθ iΩ, (6.18) This solution u/bardbl(θ) also satisfies Eq. (6.12). For the sus- ceptibility, Eq. (6.9), we obtain χ/bardbl(Ω→ ∞,b) =2i Ω/parenleftbiggcothb b−1 b2/parenrightbigg .(6.19) Finally, the low frequency limit can be also analyzed analytically. The real part of the susceptibility coincides with the differential susceptibility in dc magnetic field, Eq. (6.6), for the imaginary part to the first order in frequency we obtain, see Appendix, Imχ/bardbl(Ω,b) = Ωf/bardbl(b). (6.20) The function f/bardbl(b) has a complicated analyticalform and is not presented here, but its plot is shown in Fig. 4. In all considered four limiting cases, the asymptotic approximations hold regardless the order in which the limits are taken. Indeed, the asymptote of the expression for the susceptibility in the zero field, Eq. (6.15), has the asymptote at Ω → ∞consistent with Eq. (6.19) at b= 0.0.000.100.200.30 02468100.000.040.080.120.16 PSfrag replacementsReχ/bardbl(Ω,b) Imχ/bardbl(Ω,b) bΩ = 0 .1 Ω = 0 .5 Ω = 1 Ω = 2 Ω = 5 FIG. 3: (Color online). Plot of the real and imaginary parts of theacsusceptibility χ/bardbl(Ω,b) at several values of the dimen- sionless frequency Ω of the oscillating magnetic field along the constant magnetic field with strength b. In general, magnetic field suppresses both real and imaginary parts of the suscep- tibility. Similarly, the high frequency limit of Eq. (6.17) coincides with the limit b→ ∞of Eq. (6.19). Both limits of weak and strong magnetic field of the imaginary part of the susceptibilityatlowfrequencies, Eq.(6.20), coincidewith the imaginarypartof χ/bardbl(Ω,b), calculatedfromEq.(6.15) and Eq. (6.19), respectively. In general, we make a conjecture that the ac suscepti- bility is given by the following expression: χ/bardbl(Ω,b) =/summationdisplay nχ/bardbl n(b) 1−iΩ/Γ/bardbl n(b),(6.21) where functions χ/bardbl n(b) and Γ/bardbl n(b) are real and describe the degeneracy points of the homogeneous differential equation Eq. (6.10) with real iΩ. This expansion is re- lated to the expansion of time-dependent Fokker-Plank equations in the spherical harmonics, analyzed in Ref. 1. In particular, χ/bardbl n>1(b→0) =O(b) and Γ/bardbl n(b→0) = n(n+1)+O(b). For practical purposes, we found from a numerical analysis that even the single-pole approximation, χapp /bardbl(ω,b) =/parenleftbigg1 b2−1 sinh2b/parenrightbigg/bracketleftbig 1−iωT/bardbl(b)/bracketrightbig−1,(6.22) gives a very good estimate of the susceptibility for all values ofωandb. The analysis shows that the suscep- tibility, Eq. (6.9), obtained from a numerical solution of Eq. (6.10), is within a few per cent of the estimate given by Eq. (6.22). The characteristic time constant, T/bardbl(b), as a function of magnetic field bis chosen from the high frequency asymptote Eq. (6.19): T/bardbl(b) =T0 2sinhbsinh2b−b2 bcoshb−sinhb.(6.23) To evaluate the accuracy of the above approximation, Eq. (6.22), we consider the opposite limit of low frequen-12 012345670.00.050.100.15 PSfrag replacementsf/bardbl(b), fapp /bardbl(b) bf/bardbl(b) fapp /bardbl(b) Ω = 5 FIG. 4: (Color online). Dependence on magnetic field bof the slope of the imaginary part of the linear in frequency susceptibility χ/bardbl(Ω,b) at low frequencies Ω ≪1, calculated according toEq. (6.20). For comparison, we also plot functi on fapp /bardbl(b), see Eq. (6.24). cies, Ω≪1, and compare the exact result for the imagi- nary part of the susceptibility, Eq. (6.20), with Imχapp /bardbl(ω,b) =ωT0fapp /bardbl(b), fapp /bardbl(b) =1 2b2sinh3b(sinh2b−b2)2 bcoshb−sinhb.(6.24) For visual comparison of functions f/bardbl(b) andfapp /bardbl(b), we plot both functions in Fig. 4, where these curves are nearly indistinguishable. The difference between these two curves vanishes at b→0 andb→ ∞, and has a maximal difference at b≈2, which constitutes only tiny fraction off/bardbl(b). D. Transverse susceptibility Next, we consider the response of the magnetization to weak oscillations ˜B⊥(t) of the external magnetic field with frequency ωin direction perpendicular to the fixed magnetic field B0. We write the oscillatory component of the field in the form: ˜B⊥(t) =δ1Teff 2h0/bracketleftbig b⊥(ex+iey)e−iωt+b∗ ⊥(ex−iey)eiωt/bracketrightbig . (6.25) This field represents a circular polarization of an ac magnetic field in the ( x,y) plane, perpendicular to the fixed magnetic field in the z-direction: B= {B⊥cosωt;B⊥sinωt;B0}. We look for the linear cor- rection to the probability distribution in the form P(ϕ,θ,t) =P0(θ) ×/bracketleftbig 1+b⊥u⊥(θ)eiϕ−iωt+b∗ ⊥u∗ ⊥(θ)e−iϕ+iωt/bracketrightbig .(6.26) The equation for u⊥(θ) is obtained from the Fokker- Plank equation Eq. (4.10), linearized in the parame-terb⊥: ∂2u⊥ ∂θ2+cosθ−bsin2θ sinθ∂u⊥ ∂θ+/parenleftbigg iΩ⊥−1 sin2θ/parenrightbigg u⊥ =−sinθ(2+2ih0τb+bcosθ). (6.27) Here the dimensionless frequency is a difference between the drive frequency ωand the precession frequency in external field B0: Ω⊥= (ω−2B0)T0= Ω−2(h0τ)b, (6.28) whereT0is defined in Eq. (4.9) and the right equality is written in terms of dimensionless variables Ω, Eq. (6.11), andb, Eq. (6.4). Equation (6.27) is symmetric with re- spect to the simultaneous change θ→π−θ,b→ −b, i→ −i, Ω⊥→ −Ω⊥(“parity”). The function P(ϕ,θ,t) is single-valued at the poles θ= 0 andθ=π, only if u⊥(θ= 0) = 0, u⊥(θ=π) = 0.(6.29) The latter equations establish the boundary conditions for the differential equation (6.27). We also note that the normalization condition is satisfied for any function u⊥(θ). We define the susceptibility in response to the ac mag- netic field, Eq. (6.25), as χ⊥(Ω,b) = 2ππ/integraldisplay 0u⊥(θ)P0(θ)sin2θdθ. (6.30) This expression for the susceptibility can be used to cal- culate the magnetization of a particle M(t) =/integraldisplay n(ϕ,θ)P(ϕ,θ,t)sinθdθdϕ (6.31) tothe lowestorderin the acmagnetic field. Inparticular, Mx(t) = Re(χ⊥b⊥e−iωt), My(t) = Im(χ⊥b⊥e−iωt). (6.32) Solving numerically the differential equation (6.27) with the corresponding boundary conditions, Eq. (6.29), we obtain the transverse susceptibility, Eq. (6.30), shown in Figs. 5 and 6. Below we analyze several limiting cases. In zero fixed magnetic field, b= 0, we have the exact solution of Eq. (6.27): u⊥(θ) =sinθ 1−iΩ⊥/2. (6.33) This solution corresponds to the solution in the longitu- dinal case, rotated by 90◦, cf. Eq. (6.14). Atω= 0 and Ω ⊥=−2bh0τ, the solution of Eq. (6.27) has a simple form u/bardbl(θ) = sinθand corresponds to a tilt of the external field. The susceptibility due to such tilt is χ⊥(Ω = 0,b) =2 b2(bcothb−1).(6.34)13 t −0.20.00.20.40.6 −5 0 5 10 15−0.20.00.20.40.6PSfrag replacementsReχ⊥(Ω, b) Imχ⊥(Ω, b) Ωbb= 0.4;h0τ= 5 b= 1;h0τ= 2 b= 2.5;h0τ= 0.8 Ω = 2; h0τ= 2 Ω = 2; h0τ= 0.5 Ω = 0 .5;h0τ= 2 Ω = 0 .5;h0τ= 0.5 Ω = 2 Ω = 5 FIG. 5: (Color online). Plot of the real and imaginary parts of the transverse susceptibility χ⊥(Ω,b) as a function of the dimensionless frequency Ω. Negative frequency correspond s to the opposite sense of the circular polarization of the ac magnetic field in a plane, perpendicular to the constant mag- netic field with strength b. The parameters of the three shown curves are chosen so that h0τb= 2. In strong fixed magnetic field, b≫1, we need to con- sider small angles θ∼1/√ b, therefore, we can approxi- mate cosθ≈1 in Eq. (6.27) and obtain: u⊥(θ) =b+2+2ih0τb b+2−iΩ⊥sinθ. (6.35) The susceptibility in the limit b≫1 is given by χ⊥(Ω,b) =1+2ih0τ b(b(1+2ih0τ)−iΩ). (6.36) At Ω⊥≫1,bwe can disregard the terms in Eq. (6.27) with derivatives. Moreover, the contribution to the sus- ceptibility, Eq. (6.30), from the vicinity of θ= 0 and θ=πis suppressed as sin2θ. This observation allows us to write the solution in the form u⊥(θ) = sinθ2+2ih0τb+bcosθ −iΩ⊥, (6.37) Consequently, we obtain the following high frequency, Ω⊥≫1, asymptote for the susceptibility: χ⊥(Ω,b) =i Ω−2h0τb/bracketleftbiggbcothb−1 b2(2ih0τb−1)+1/bracketrightbigg . (6.38) We can use the approximate expression for the suscep- tibility in response to the transverse oscillating magnetic field χapp ⊥(ω) =bcothb−1 b2[1+2iB0T⊥(b)] ×[1+i(2B0−ω)T⊥(b)]−10.00.10.20.30.4 −10 −5 0 5 100.00.10.20.3PSfrag replacementsReχ⊥(Ω, b) Imχ⊥(Ω, b) Ω bΩ = 2; h0τ= 2 Ω = 2; h0τ= 0.5Ω = 0 .5;h0τ= 2 Ω = 0 .5;h0τ= 0.5 Ω = 2 Ω = 5 FIG. 6: (Color online). Plot of the real and imaginary parts of the transverse susceptibility χ⊥(Ω,b) as a function of the strength bof a constant magnetic field, shown for two values of frequency Ω and two values of the “damping factor” h0τ. Negative values of bcorresponds to the opposite sense of the circular polarization of the ac magnetic field in a plane, per - pendicular to the constant magnetic field. The real part of the susceptibility exhibits a strong non-monotonic behavi or at weak magnetic fields. Thecorrespondingcharacteristictimeconstant T⊥(b)can be found for any bfrom the asymptotic behavior of χ⊥(Ω,b) at Ω ⊥≫1,b: T⊥(b) =T0bcothb−1 b2+1−bcothb. (6.39) VII. CONCLUSIONS We have studied the slow dynamics of magnetization in a small metallic particle (quantum dot), where the fer- romagnetism has arisen as a consequence of Stoner insta- bility. Theparticleisconnectedto non-magneticelectron reservoirs. A finite bias is applied between the reservoirs, thus bringing the whole electron system away from equi- librium. The exchange of electrons between the reser- voirs and the particle results in the Gilbert damping3of the magnetization dynamics and in a temperature- and bias-driven Brownian motion of the direction of the par- ticle magnetization. Analysis of magnetization dynam- ics and transport properties of ferromagnetic nanoparti- cles is commonly performed4,5,6,7,11within the stochas- tic Landau-Lifshitz-Gilbert (LLG) equation2,3, which is an analogue of the Langevin equation written for a unit three-dimensional vector. We derived the stochastic LLG equation from a mi- croscopicstarting point and established conditions under which the description ofthe magnetizationofaferromag- netic metallic particle by this equation is applicable. We concluded that the applicability of the LLG equation for14 a ferromagneticparticle is set by three independent crite- ria. (1)Thecontactresistanceshouldbelowcomparedto the resistance quantum, which is equivalent to Nch≫1. Otherwise the noise cannot be consideredgaussian. Each channel can be viewed as an independent source of noise and only the contribution of many channels results in the gaussian noise by virtue of the central limit theorem forNch≫1. (2) The system should not be too close to the Stoner instability: the mean-field value of the to- tal spinS2 0≫Nch. Otherwise, the fluctuations of the absolute value of the magnetization become of the or- der of the magnetization itself. (3) S2 0≫Teff/δ1, where Teff≃max{T,|eV|}is the effective temperature of the system, which is the energy scale of the electronic dis- tribution function. Otherwise, the separation into slow (the direction of the magnetization) and fast (the elec- tron dynamics and the magnitude of the magnetization) degrees of freedom is not possible. Under the above conditions, the dynamics of the mag- netization is described in terms of the stochastic LLG equation with the power of Langevin forces determined by the effective temperature of the system. The effective temperature is the characteristic energy scale of the elec- tronic distribution function in the particle determined by a combination of the temperature and the bias voltage. In fact, for a considered here system with non-magnetic contacts between non-magnetic reservoirs and a ferro- magnetic particle the power of the Langevin forces is proportional to the low-frequency noise of total charge current through the particle. We further reduced the stochastic LLG equation to the Fokker-Planck equation for a unit vector, corresponding to the direction of the magnetization of the particle. The Fokker-Plank equa- tion can be used to describe time evolution of the distri- bution of the direction of magnetization in the presence of time-dependent magnetic fields and voltage bias. As an example of application of the Fokker-Plank equation for the magnetization, we have calculated the frequency-dependent magnetic susceptibility of the par- ticle in a constant external magnetic field (i. e., linear response of the magnetization to a small periodic mod- ulation of the field, relevant for ferromagnetic resonance measurements). We have not been able to obtain an ex- plicit analytical expression for the susceptibility at ar- bitrary value of the applied external field and frequency; however, analysisofdifferent limiting caseshas lead us to a simple analytical expression which gives a good agree- ment with the numerical solution of the Fokker-Planck equation. Acknowledgements We acknowledge discussions with I. L. Aleiner, G. Catelani, A. Kamenev and E. Tosatti. M.G.V. is grate- ful to the International Centre for Theoretical Physics(Trieste, Italy) for hospitality. APPENDIX A: LONGITUDINAL SUSCEPTIBILITY AT LOW FREQUENCIES We find the linearin frequency Ω ≪1 correctionto the dc susceptibility. For this purpose, we look for a solution to Eq. (6.10) in the form u/bardbl(θ) =u(0) /bardbl(θ)+u(1) /bardbl(θ), (A1) whereu(0) /bardbl(θ) is the solution of Eq. (6.10) at Ω = 0 and u(1) /bardbl(θ)∝Ω. We choose u(0) /bardbl(θ) =1 b−cothb+cosθ, (A2) sincethis formof u(0) /bardbl(θ) preservesthe normalizationcon- dition (6.12). This function can be found directly as a solution of Eq. (6.10) with Ω = 0 or as a variational derivative of function P0(θ), defined in Eq. (6.3), with respect tob. The linear in Ω correction u(1) /bardbl(θ) is the solution to the differential equation ∂2u(1) /bardbl(θ) ∂θ2+cosθ−bsin2θ sinθ∂u(1) /bardbl(θ) ∂θ=−iΩu(0) /bardbl(θ).(A3) From this equation, we can easily find ∂u(1) /bardbl(θ) ∂θ=−iΩ bsinθ/bracketleftbigg cothb−cosθ−e−bcosθ sinhb/bracketrightbigg .(A4) We notice that the solution to the latter equation will automatically satisfy the boundary conditions, given by Eq. (6.13). Integrating Eq. (A4) once again, we obtain the following expression for function u(1) /bardbl(θ): u(1) /bardbl(θ) =C(b) −iΩ b/integraldisplayθ 0/bracketleftBigg cothb−cosθ′−e−bcosθ′ sinhb/bracketrightBigg dθ′ sinθ′.(A5) Here the integration constant C(b) has to be chosen to satisfy the normalization condition, Eq. (6.12), which re- sults in complicated expression for the final form of the functionu(1) /bardbl(θ). To obtain function f/bardbl(b), introduced in Eq. (6.20), we have to perform the final integration f/bardbl(b) =2π Ω/integraldisplayπ 0u(1) /bardbl(θ)P0(θ)sinθcosθdθ. (A6) The result of integration is shown in Fig. 4.15 1W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). 2L. Landau and E. Lifshitz, Phys. Z. Sowietunion 8, 153 (1935). 3T. Gilbert, Phys. Rev. 100, 1243 (1955). 4J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Phys. Rev. B 58, 14937 (1998). 5K. D. Usadel, Phys. Rev. B 73, 212405 (2006). 6J. Foros, A. Brataas, G. E. W. Bauer, and Y. Tserkovnyak, Phys. Rev. B 75, 092405 (2007). 7S. I. Denisov, K. Sakmann, P. Talkner, and H¨ anggi, Phys. Rev. B75, 184432 (2007). 8A. Rebei and M. Simoniato, Phys. Rev. B 71, 174415 (2005). 9H. Katsura, A. V. Balatsky, Z. Nussinov, and N. Nagaosa, Phys. Rev. B 73, 212501 (2006). 10A. S. N´ u˜ nez and R. A. Duine, Phys. Rev. B 77, 054401 (2008). 11A. L. Chudnovskiy, S. Swiebodzinski, and A. Kamenev, Phys. Rev. Lett. 101, 066601 (2008). 12J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,Phys. Rev. Lett. 95, 016601 (2005). 13R.A.Duine, A.S.N´ u˜ nez, J.Sinova, andA.H.MacDonald, Phys. Rev. B 75, 214420 (2007). 14X. Waintal and P. W. Brouwer, Phys. Rev. Lett. 91, 247201 (2003). 15I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Reports 358, 309 (2002). 16C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). 17I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, Phys. Rev. B62, 14886 (2000). 18Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 19S. Tamaru et al., J. Appl. Phys. 91, 8034 (2002). 20J. C. Sankey et al., Phys. Rev. Lett. 96, 227601 (2006). 21Y. Ahmadian, G. Catelani, and I. L. Aleiner, Phys. Rev. B72, 245315 (2005). 22K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970). 23G. Catelani and M. G. Vavilov, Phys. Rev. B 76, 201303 (2007).

2008-09-16

We consider a small metallic particle (quantum dot) where ferromagnetism arises as a consequence of Stoner instability. When the particle is connected to electrodes, exchange of electrons between the particle and the electrodes leads to a temperature- and bias-driven Brownian motion of the direction of the particle magnetization. Under certain conditions this Brownian motion is described by the stochastic Landau-Lifshitz-Gilbert equation. As an example of its application, we calculate the frequency-dependent magnetic susceptibility of the particle in a constant external magnetic field, which is relevant for ferromagnetic resonance measurements.

Stochastic dynamics of magnetization in a ferromagnetic nanoparticle out of equilibrium

0809.2611v1

arXiv:0809.4311v1 [cond-mat.other] 25 Sep 2008The theory of magnetic field induced domain-wall propagatio n in magnetic nanowires X. R. Wang, P. Yan, J. Lu, and C. He Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China A global picture of magnetic domain wall (DW) propagation in a nanowire driven by a magnetic field is obtained: A static DW cannot exist in a homogeneous ma gnetic nanowire when an external magnetic field is applied. Thus, a DW must vary with time under a static magnetic field. A moving DW must dissipate energy due to the Gilbert damping. As a resu lt, the wire has to release its Zeeman energy through the DW propagation along the field dire ction. The DW propagation speed is proportional tothe energy dissipation rate that is deter mined bythe DW structure. Anoscillatory DW motion, either the precession around the wire axis or the b reath of DW width, should lead to the speed oscillation. Magnetic domain-wall (DW) propagation in a nanowire due to a magnetic field[1, 2, 3, 4, 5] reveals many interesting behaviors of magnetization dynamics. For a tail-to-tail (TT) DW or a head-to-head (HH) DW (shown in Fig. 1) in a nanowire with its easy-axis along the wire axis, the DW will propagate in the wire un- der an external magnetic field parallel to the wire axis. The propagation speed vof the DW depends on the field strength[3, 4]. There exists a so-called Walker’s break- down field HW[6].vis proportional to the external field HforH < H WandH≫HW. The linear regimes are characterized by the DW mobility µ≡v/H. Ex- periments showed that vis sensitive to both DW struc- tures and wire width[1, 2, 3]. DW velocity vdecreases as the field increases between the two linear H-dependent regimes, leading to the so-called negative differential mo- bility phenomenon. For H≫HW, the DW velocity, whose time-average is linear in H, oscillates in fact with time [3, 6]. III II IMθ=0 θ=πH zxy∆ ADW FIG. 1: Schematic diagram of a HH DW of width ∆ in a magnetic nanowire of cross-section A. The wire consists of three phases, two domains and one DW. The magnetization in domains I and II is along +z-direction ( θ= 0) and -z- direction ( θ=π), respectively. III is the DW region whose magnetization structure could be very complicate. /vectorHis an external field along +z-direction. It has been known for more than fifty years that the magnetization dynamics is govern by the Landau- Lifshitz-Gilbert (LLG)[7] equation that is nonlinear and can only be solved analytically for some special problems[6, 8]. The field induced domain-wall (DW)propagation in a strictly one-dimensional wire has also been known for more than thirty years[6], but its exper- imental realization in nanowires was only achieved[1, 2, 3, 4, 5] in recent years when we are capable of fabricating various nano structures. Although much progress[9, 10] has been made in understanding field-induced DW mo- tion, it is still a formidable task to evaluate the DW propagation speed in a realistic magnetic nanowire even when the DW structure is obtained from various means like OOMMF simulator and/or other numerical software packages. A global picture about why and how a DW propagates in a magnetic nanowire is still lacking. In this report, we present a theory that reveals the origin of DW propagation. Firstly, we shall show that no static HH (TT) DW is allowed in a homogeneous nanowire in the presence of an external magnetic field. Secondly, energy conservation requires that the dissi- pated energy must come from the energy decrease of the wire. Thus, the origin of DW propagation is as follows. A HH (TT) DW must move under an external field along the wire. The moving DW must dissipate energy because of various damping mechanisms. The energy loss should be supplied by the Zeeman energy released from the DW propagation. This consideration leads to a general re- lationship between DW propagation speed and the DW structure. It is clear that DW speed is proportional to the energy dissipation rate, and one needs to find a way to enhance the energy dissipation in order to increase the propagation speed. Furthermore, the present theory at- tributes a DW velocity oscillation for H≫HWto the periodic motion of the DW, either the precession of the DW or oscillation of the DW width. In a magnetic material, magnetic domains are formed in order to minimize the stray field energy. A DW that separates two domains is defined by the balance between the exchangeenergy and the magnetic anisotropyenergy. The stray field plays little role in a DW structure. To describe a HH DW in a magnetic nanowire, let us con- sider a wire with its easy-axis along the wire axis (the shape anisotropy dominates other magnetic anisotropies and makes the easy-axis along the wire when the wire is small enough) which is chosen as the z-axis as illustrated2 in Fig. 1. Since the magnitude of the magnetization /vectorM does not change in the LLG equation[8], the magnetic state of the wire can be conveniently described by the polar angle θ(/vector x,t) (angle between /vectorMand the z-axis) and the azimuthal angle φ(/vector x,t). The magnetization energy is mainly from the exchange energy and the magnetic anisotropy because the stray field energy is negligible in this case. The wire energy can be written in general as E=/integraldisplay F(θ,φ,/vector∇θ,/vector∇φ)d3/vector x, F=f(θ,φ)+J 2[(/vector∇θ)2+sin2θ(/vector∇φ)2]−MHcosθ,(1) wherefis the energydensity due to all kinds ofmagnetic anisotropies which has two equal minima at θ= 0 and π(f(θ= 0,φ) =f(θ=π,φ)),J−term is the exchange energy,Mis the magnitude of magnetization, and His the external magnetic field along z-axis. In the absence ofH, a HH static DW that separates θ= 0 domain and θ=πdomain (Fig. 1) can exist in the wire. Non-existence of a static HH (TT) DW in a magnetic field-In order to show that no intrinsic static HH DW is allowed in the presence of an external field ( H/negationslash= 0), one only needs to show that following equations have no solution with θ= 0 at far left and θ=πat far right, δE δθ=J∇2θ−∂f ∂θ−HMsinθ−Jsinθcosθ(/vector∇φ)2= 0, δE δφ=J/vector∇·(sin2θ/vector∇φ)−∂f ∂φ= 0. (2) Multiply the first equation by ∇θand the second equa- tion by∇φ, then add up the two equations. One can show a tensor Tsatisfying ∇·T= 0 with T=[f−HMcosθ+J 2(|∇θ|2+sin2θ|∇φ|2)]1− J(∇θ∇θ+sin2θ∇φ∇φ), where1is 3×3 unit matrix. A dyadic product ( ∇θ∇θ and∇φ∇φ) between the gradient vectors is assumed in T. If a HH DW exists with θ= 0 in the far left and θ=πin the far right, then it requires −f(0,φ)+HM= −f(π,φ)−HMthat holds only for H= 0 since f(0,φ) = f(π,φ). In other words, a DW in a nanowire under an external field must be time dependent that could be ei- ther a local motion or a propagation along the wire. It should be clear that the above argument is only true for a HH DW in a homogeneous wire, but not valid with de- fect pinning that changes Eq. (2). Static DWs exist in fact in the presence of a weak field in reality because of pinning. What is the consequence of the non-existence of a static DW? Generally speaking, a physical system un- der a constant driving force will first try a fixed point solution[11]. It goes to other types of more complicatedsolutionsifafixedpointsolutionisnotpossible. Itmeans that a DW has to move when an external magnetic field is applied to the DW along the nanowire as shown in Fig. 1. It is well known[10] that a moving magnetiza- tion must dissipate its energy to its environments with a rate,dE dt=αM γ/integraltext+∞ −∞(d/vector m/dt)2d3/vector x,where/vector mis the unit vector of /vectorM,αandγare the Gilbert damping constant and gyromagnetic ratio, respectively. Following the simi- lar method in Reference 12 for a Stoner particle, one can also show that the energy dissipation rate of a DW is related to the DW structure as dE dt=−αγ (1+α2)M/integraldisplay+∞ −∞/parenleftBig /vectorM×/vectorHeff/parenrightBig2 d3/vector x,(3) where/vectorHeff=−δF δ/vectorMis the effective field. In regions I and II or inside a static DW, /vectorMis parallel to /vectorHeff. Thus no energy dissipation is possible there. The energy dissipa- tion can only occur in the DW region when /vectorMis not parallel to /vectorHeff. DW propagation and energy dissipation- Foramagnetic nanowire in a static magnetic field, the dissipated energy must come from the magnetic energy released from the DW propagation. The total energy of the wire equals the sum of the energies of regions I, II, and III (Fig. 1), E=EI+EII+EIII.EIincreases while EIIdecreases when the DW propagates from left to the right along the wire. The net energy change of region I plus II due to the DW propagation is d(EI+EII) dt=−2HMvA, (4) wherevis the DW propagating speed, and Ais the cross section of the wire. This is the released Zeeman energy stored in the wire. The energy of region III should not change much because the DW width ∆ is defined by the balanceofexchangeenergyand magneticanisotropy,and is usually order of 10 ∼100nm. A DW cannot absorb or release too much energy, and can at most adjust tem- porarily energy dissipation rate. In other words,dEIII dt is either zero or fluctuates between positive and nega- tive values with zero time-average. Since energy release from the magnetic wire should be equal to the energy dissipated (to the environment), one has −2HMvA+dEIII dt=−αγ (1+α2)M/integraldisplay III/parenleftBig /vectorM×/vectorHeff/parenrightBig2 d3/vector x. (5) or v=αγ 2(1+α2)HA/integraldisplay III/parenleftBig /vector m×/vectorHeff/parenrightBig2 d3/vector x+1 2HMAdEIII dt. (6) Velocity oscillation- Eq. (6) is our central result that relates the DW velocity to the DW structure. Obviously, the right side of this equation is fully determined by the3 DW structure. A DW can have two possible types of motion under an external magnetic field. One is that a DW behaves like a rigid body propagating along the wire. Thiscaseoccursoftenatsmallfield, anditisthebasicas- sumption in Slonczewski model[9] and Walker’s solution forH < H W. Obviously, both energy-dissipation and DW energy is time-independent,dEIII dt= 0. Thus,and the DW velocity should be a constant. The other case is that the DW structure varies with time. For example, theDWmayprecessaroundthewireaxisand/ortheDW width may breathe periodically. One should expect both dEIII dtandenergydissipationrateoscillatewith time. Ac- cording to Eq. (6), DW velocity will also oscillate. DW velocity should oscillate periodically if only one type of DW motion (precession or DW breathing) presents, but it could be very irregular if both motions are present and the ratio of their periods is irrational. Indeed, this os- cillation was observed in a recent experiment[3]. How can one understand the wire-width dependence of the DW velocity? According to Eq. (6), the velocity is a functional of DW structure which is very sensitive to the wire width. For a very narrow wire, only transverse DW is possible while a vortex DW is preferred for a wide wire (large than DW width). Different vortexes yield differ- ent values of |/vector m×/vectorHeff|, which in turn results in different DW propagation speed. /s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s32/s32 /s68/s87 /s32/s118 /s32 /s40/s109/s47/s115/s41 /s72/s32/s40/s79/s101/s41/s32/s118/s40/s109/s47/s115/s41 /s116/s32/s40/s110/s115/s41/s118/s40/s109/s47/s115/s41/s32 /s116/s32/s40/s110/s115/s41 FIG. 2: The time-averaged DW propagation speed versus the applied magnetic field for a biaxial magnetic nanowire of cross section 4 nm×20nm. The wire parameters are K1=K2= 105J/m3,J= 4.×10−11J/m,M= 106A/m, andα= 0.1. Cross are for the calculated velocities from Eq. (7), and the open circles are for the simulated average veloc i- ties. The dashed straight line is the fit to the small H < H W results, and solid curve is the fit to a(H−H0)2/H+b/H. Insets: the instantaneous DW speed calculated from Eq. (6) forH= 50Oe < H W(left) and H= 1000Oe > H W(right). Time averaged velocity is ¯v=αγ 2(1+α2)HA/integraldisplay III/parenleftBig /vector m×/vectorHeff/parenrightBig2 d3/vector x,(7) wherebardenotestimeaverage. Itsaysthattheaveraged velocity is proportional to the energy dissipation rate. In order to show that both Eqs. (6) and (7) are useful in evaluating the DW propagation speed from a DW struc-ture. We use OOMMF package to find the DW struc- tures and then use Eq. (7) to obtain the averagevelocity. Figure 2 is the comparison of such calculated velocities (cross) and numerical simulation (open circles with their error bars smaller than the symbol sizes) for a magnetic nanowire of cross-section dimension 4 nm×20nmwith a biaxial magnetic anisotropy f=−K1 2M2 z+K2 2M2 x. The system parameters are K1=K2= 105J/m3, J= 4.×10−11J/m,M= 106A/m, andα= 0.1. The good overlap between the cross and open circles confirm the correctness of Eq. (7). The ¯ v−Hcurve for H > H W can be fit well by a∆(H−H0)2/H+b/H(see discus- sion later). The insets are instantaneous DW propaga- tion velocities for both H < H WandH > H W, by Eq. (6) from the instantaneous DW structures obtained from OOMMF. The left inset is the instantaneous DW speed atH= 50Oe < H W, reaching its steady value in about 1ns. The right inset is the instantaneous DW speed at H= 1000Oe > H W, showingclearlyanoscillation. They confirm that the theory is capable of capturing all the features of DW propagation. The right side of Eq. (7) is positive and non-zero since a time dependent DW requires /vector m×/vectorHeff/negationslash= 0, implying a zero intrinsic critical field for DW propa- gation. If the DW keep its static structure, then the first term in the right side of Eq. (6) shall be pro- portional to a∆AH2, where ais a numerical number of order of 1 that depends on material parameters and the DW structure. This is because the effective field due to the exchange energy and magnetic anisotropy is parallel to /vectorM, and does not contribute to the en- ergy dissipation. Thus, in this case, v=aαγ∆ 1+α2H withµ=aαγ∆ 1+α2. Consider the Walker’s 1D model[6] in which f=−K1 2M2cos2θ+K2 2M2sin2θcos2φ,here K1andK2describe the easy and hard axes, respec- tively. From Walker’s trial function of a DW of width ∆, lntanθ(z,t) 2=1 ∆(t)/bracketleftBig z−/integraltextt 0v(τ)dτ/bracketrightBig andφ(z,t) =φ(t), one has (from Eq. (3)) the energy dissipation rate dE dt=−2αγA∆ 1+α2/bracketleftbig K2 2M3sin2φcos2φ+H2M/bracketrightbig ,(8) and DW energy change rate is dEIII dt=d dt/integraldisplay IIIF(θ,φ,/vector∇θ,/vector∇φ)d3/vector x=−4JA·˙∆ ∆2.(9) Substituting Eqs. (8) and (9) into Eq. (6), one can easily reproduceWalker’sDWvelocityexpressionfor both H < HWand≫HW. For example, for H < H W=αK2M/2 and ∆ = const., Eq. (6) gives v=α∆γ 1+α2/bracketleftBigg 1+/parenleftbiggK2Msinφcosφ H/parenrightbigg2/bracketrightBigg H.(10) This velocity expression is the same as that of the Slonczewski model[9] for a one-dimensional wire. In4 Walker’s analysis, φis fixed by K2andHthrough K2Msinφcosφ=H α. Using this φin the above ve- locity expression, Walker’s mobility coefficient µ=γ∆ α is recovered. This inverse damping relation is from the particularpotentiallandscape in φ-direction. One should expect different result if the shape of the potential land- scape is changed. Thus, this expression should not be used to extract the damping constant[1, 3]. A DW may precess around the wire axis as well as be substantially distorted from its static structure when H > H Was it was revealed in Walker’s analysis. Ac- cording to the minimum energy dissipation principle[13], a DW will arrange itself as much as possible to satisfy Eq. (2). Thus, the distortion is expected to absorb part ofH. The precession motion shall induce an effective fieldg(φ) in the transverse direction, where gdepends on the magnetic anisotropy in the transverse direction. One may expect /vector m×/vectorHeff≃(H−H0)sinθˆz+sinθg(φ)ˆy, whereH0istheDWdistortionabsorbedpartof H. Using |/vector m×/vectorHeff|2= (H−H0)2sin2θ+g2sin2θin Eq. (7), the DWpropagatingspeed takesthe followingh-dependence, v=aαγ∆(H−H0)2/H(1+α2)+bαγ∆/[H(1+α2)], linear in both ∆ and HforH≫H0, but a smaller DW mobil- ity. This field-dependence is supported by the excellent fit in Fig. 2 for H > H W. The reasoning agrees also with the minimum energy dissipation principle[13] since |/vector m×Heff|=Hsinθwhen/vectorMforH= 0 is used, and any modification of /vectorMshould only make |/vector m×Heff|smaller. The smaller mobility at H≫HW,H0leads naturally to a negative differential mobility between H < H Wand H≫HW! In other words, the negative differential mo- bility is due to the transition of the DW from a high energy dissipation structure to a lower one. This picture tells us that one should try to make a DW capable of dissipating as much energy as possible if one wants to achieve a high DW velocity. This is very different from what people would believefrom Walker’sspecial mobility formula of inverse proportion of the damping constant. To increase the energy dissipation, one may try to re- duce defects and surface roughness. The reason is, by minimum energy dissipation principle, that defects are extra freedoms to lower |/vector m×/vectorHeff|because, in the worst case, defects will not change |/vector m×/vectorHeff|when/vectorMwithout defects are used. The correctness of our central result Eq. (6) depends only on the LLG equation, the general energy expres- sion of Eq. (1), and the fact that a static magnetic field can be neither an energy source nor an energy sink of a system. It does not depend on the details of a DW struc- ture aslong asthe DWpropagationis induced by a static magnetic field. In this sense, our result is very general and robust, and it is applicable to an arbitrary magnetic wire. However, it cannot be applied to a time-dependent field orthe current-inducedDWpropagation,atleastnotdirectly. Also, it may be interesting to emphasize that there is no inertial in the DW motion within LLG de- scription since this equation contains only the first order time derivative. Thus, there is no concept of mass in this formulation. In conclusion, a global view of the field-induced DW propagation is provided, and the importance of energy dissipation in the DW propagation is revealed. A gen- eral relationship between the DW velocity and the DW structure is obtained. The result says: no damping, no DW propagation along a magnetic wire. It is shown that the intrinsic critical field for a HH DW is zero. This zero intrinsic critical field is related to the absence of a static HH or a TT DW in a magnetic field parallel to the nanowire. Thus, a non-zero critical field can only come from the pinning of defects or surface roughness. The observed negative differential mobility is due to the transition of a DW from a high energy dissipation struc- ture to a low energy dissipation structure. Furthermore, the DW velocity oscillation is attributed to either the DW precession around wire axis or from the DW width oscillation. This work is supported by Hong Kong UGC/CERG grants (# 603007 and SBI07/08.SC09). [1] T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and T. Shinjo, Science 284, 468 (1999). [2] D. Atkinson, D.A. Allwood, G. Xiong, M.D. Cooke, C. Faulkner, and R.P. Cowburn, Nat. Mater. 2, 85 (2003). [3] G.S.D. Beach, C. Nistor, C. Knutson, M. Tsoi, and J.L. Erskine, Nat. Mater. 4, 741 (2005); J. Yang, C. Nistor, G.S.D. Beach, and J.L. Erskine, Phys. Rev. B 77, 014413 (2008). [4] M. Hayashi, L. Thomas, Y.B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and S.S.P. Parkin, Phys. Rev. Lett. 96, 197207 (2006). [5] G.S.D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J.L. Erskine, Phys. Rev. Lett. 97, 057203 (2006). [6] N.L. Schryer and L.R. Walker, J. of Appl. Physics, 45, 5406 (1974). [7] T. L. Gilbert, Phys. Rev. 100, 1243 (1955). [8] Z.Z. Sun and X.R. Wang, Phys. Rev. Lett. 97, 077205 (2006); X.R. Wang and Z.Z. Sun, ibid98, 077201 (2007). [9] A.P.Malozemoff andJ.C. Slonczewski, Magnetic Domain Walls in Bubble Material (Academica, New York, 1979). [10] A. Thiaville and Y. Nakatani in Spin Dynamics in Con- fined Magnetic Structures III Eds. B. Hillebrands and A. Thiaville, Springer 2002. [11] X. R. Wang, and Q. Niu, Phys. Rev. B 59, R12755 (1999); Z.Z. Sun, H.T. He, J.N. Wang, S.D. Wang, and X.R. Wang, Phys. Rev. B 69, 045315 (2004). [12] Z.Z. Sun, and X.R. Wang, Phys. Rev. B 71, 174430 (2005);73, 092416 (2006); 74132401 (2006). [13] T. Sun, P. Meakin, and T. Jssang, Phys. Rev. E 51, 5353 (1995).

2008-09-25

A global picture of magnetic domain wall (DW) propagation in a nanowire driven by a magnetic field is obtained: A static DW cannot exist in a homogeneous magnetic nanowire when an external magnetic field is applied. Thus, a DW must vary with time under a static magnetic field. A moving DW must dissipate energy due to the Gilbert damping. As a result, the wire has to release its Zeeman energy through the DW propagation along the field direction. The DW propagation speed is proportional to the energy dissipation rate that is determined by the DW structure. An oscillatory DW motion, either the precession around the wire axis or the breath of DW width, should lead to the speed oscillation.

The theory of magnetic field induced domain-wall propagation in magnetic nanowires

0809.4311v1

Transverse spin diusion in ferromagnets Yaroslav Tserkovnyak,1E. M. Hankiewicz,2and Giovanni Vignale3 1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 2Institut f ur Theoretische Physik und Astrophysik, Universit at W urzburg, 97074 W urzburg, Germany 3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA (Dated: October 29, 2018) We discuss the dissipative diusion-type term of the form mr2@tmin the phenomenological Landau-Lifshitz equation of ferromagnetic precession, which describes enhanced Gilbert damping of nite-momentum spin waves. This term arises physically from itinerant-electron spin ows through a perturbed ferromagnetic conguration and can be understood to originate in the ferromagnetic spin pumping in the continuum limit. We develop a general phenomenology as well as provide microscopic theory for the Stoner and s-dmodels of ferromagnetism, taking into account disorder and electron-electron scattering. The latter is manifested in our problem through the Coulomb drag between the spin bands. The spin diusion is identied in terms of the transverse spin conductivity, in analogy with the Einstein relation in the kinetic theory. PACS numbers: 75.30.Ds,72.25.-b,76.50.+g,75.45.+j I. INTRODUCTION The problem of spin diusion through conducting fer- romagnetic medium attracted much attention over sev- eral decades.1,2,3,4,5,6Semiclassical spin transport in the presence of a weak magnetic eld Hcan be captured by the conventional diusion equation (neglecting spin re- laxation): @tS=HS+Dr2S; (1) whereDis the diusion coecient and His the total eective eld (omitting the gyromagnetic ratio), includ- ing the applied and exchange contributions. The rst term on the right-hand side describes spin precession in the local eld while the second term stands for the ordi- nary diusion of spin density S. Equation (1) is, how- ever, not applicable to most realistic ferromagnets, whose spin interactions are characterized by a large exchange energy
xc. In particular, when
xcis comparable to the Fermi energy (which is the case in transition met- als), the spin precession in the exchange eld cannot be treated in the diusive transport framework. Further- more, the time-dependent exchange eld induces spin- pumping currents7,8inside the ferromagnet with spatially inhomogeneous magnetization dynamics, which can con- siderably modify the self-consistent magnetic equation of motion. Here, we wish to elucidate the central role of such self-consistent dissipative spin currents, which gov- ern the diusion-like terms in the magnetic equation of motion in the limit of strong ferromagnetic exchange cor- relations. This paper is a follow up to our previous work,9pro- viding additional technical details and oering a broader phenomenological base. Apart from assuming strong exchange correlations limit, our phenomenological ap- proach and the main results of the paper should not be sensitive to the microscopic details and do not rely on the specic model of the ferromagnetic material (suchas the Stoner or an s-dmodel, for example). The main goals of this paper are as follows: (i) to put the results of Ref. 9 into a broader phenomenological perspective, (ii) to explicitly show that two quite dierent models|the spin-polarized itinerant electron liquid (treated in Ref. 9) and thes-dmodel|lead to the same phenomenology and can be treated in parallel, and (iii) to make direct contact with the spin-pumping theory.7,8 To be specic, let us consider a continuous ferromag- netic medium, with the eective eld and spin density initially pointing along the zaxis. For weak excitations close to this state, we may try expanding the ensuing transverse spin-current density as2,6 ji=D0z@iSD00@iS; (2) which enters in the continuity equation: @tS=HSX i=x;y;z@iji: (3) In the limit of vanishing ferromagnetic correlations, we recover Eq. (1) by setting D0!0 andD00!Din Eq. (2). Hereafter, we are focusing exclusively on the transverse spin dynamics and spin currents. The longitudinal spin ows are conventionally described in terms of the ordi- nary diusion for spin-up and spin-down electrons with spin-dependent diusion coecients and spin- ip scatter- ing between the up and down spin bands.10Understand- ing the transverse spin ows and dynamics requires more care, in part due to the inherently quantum-mechanical behavior in the case of a strong exchange eld. When the magnetic excitation is driven by the self-consistent transverse eld h=zHz, there should also be eld- driven contributions to the transverse spin current (2), such as ji/@ih. The problem in fact simplies in the limit of strong exchange correlations. We will in the following employ a mean-eld view of ferromagnetism, where the collec- tive spin dynamics are driven by the exchange eld, H=arXiv:0810.1340v2 [cond-mat.mes-hall] 17 Mar 20092
xcm(r;t) (setting h= 1 throughout), parametrized by the local and instantaneous spin-density orientation, m=S=S, which has to be solved for self-consistently. Since we are only interested in the transverse spin dy- namics, we set the magnitude of the spin density Sto be spatially and time independent. In the limit of large
xc, the spin currents can be parametrized by m(r;t). We can thus proceed phenomenologically and expand jiin spa- tial and time derivatives of m(r;t). For a static magnetic prole m(r), we have the familiar exchange spin ow j0 i=Am@im (4) (whereAis the material-dependent exchange-stiness constant), which is the only rst-order form allowed by spin-rotational and time-reversal symmetries. To avoid unnecessary complications, we will assume isotropic fer- romagnet throughout this paper. Dynamics allow for dissipative spin-current contributions that break time- reversal symmetry: j00 i=m@i@tm: (5) Focusing on linear deviations of mfrom the equilibrium, m(0)=z, we omit terms such as @im@tm. According to the time-reversal property, the spin- current density (4) corresponds to the D0term in Eq. (2), while the spin-current density (5) is analogous to the D00 term, although the latter two are certainly not identi- cal. In fact, we wish to emphasize the striking dier- ence between the diusive picture for the spin currents, Eq. (2), on one side and Eqs. (4) and (5) on the other side, where we expand spin currents phenomenologically in terms of the time-dependent magnetic texture m(r;t). The latter approximation is specic to the limit of strong exchange correlations, where the nondissipative spin cur- rent, Eq. (4), is determined by the instantaneous mag- netic prole, while the dissipative spin current, Eq. (5), can be interpreted as quasiparticle spin pumping by the collective magnetic dynamics,8rather than ordinary spin diusion. It is also instructive to draw analogy between coecients Aandin Eqs. (4), (5) and the shear modu- lus and shear viscosity, respectively, in elasticity theory. In the next section, we develop further the phenomeno- logical grounds for Eqs. (4) and (5), before proceeding with microscopic calculations for the dissipative coe- cientin Secs. III and IV. In Sec. V, we discuss a spin- pumping interpretation of dissipative spin current (5), before summarizing our work in Sec. VI. II. PHENOMENOLOGY A. Landau-Lifshitz theory The conventional starting point for studying ferro- magnetic precession is the nondissipative Landau-Lifshitz (LL) equation11 @tmjLL=Hm; (6)where we dene the eective eld Has the functional derivative of the free energy: H@mF[m]=S: (7) In this Landau-Lifshitz phenomenology, which is applica- ble well below the Curie temperature, only the position- dependent direction of the magnetization is taken to be a dynamic variable, parametrizing the Free energy F[m(r)]. The angular-momentum density S=Smis assumed to be related to the magnetization by a con- stant conversion factor, the eective gyromagnetic ratio. (Abusing terminology, we say spin density synonymously with angular-momentum density .) Since in the common transition-metal ferromagnets the gyromagnetic ratio is negative, we wrote Eq. (7) with an extra minus sign in comparison to the standard denition, where mis taken to be the direction of the magnetization rather than the spin density. The right-hand side of Eq. (6) is the phe- nomenological reactive torque on the spatially-resolved magnetic precession, which generalizes the simple Larmor precession of Eq. (1). Note that the dissipation power P[m(r;t)]SZ d3rH
@tm (8) clearly vanishes according to Eq. (6). We also eas- ily verify that the time reversal (under which t!t, m!m, and H!H) leaves Eq. (6) unchanged, as it should in the absence of dissipation. The only dis- sipative term we can write in the quasistationary limit (i.e., up to the rst order in @t), assuming spatially uni- form and isotropic ferromagnet, is the so-called Gilbert damping:12 @tmjLLG=Hmm@tm; (9) whereis a material-dependent dimensionless (Gilbert) constant. A typical experimental value for turns out to be often of the order of 102in various metallic ferro- magnets, which means that it takes roughly 2 =10 precession cycles for an out-of-equilibrium magnetiza- tion to relax to a static equilibrium direction along H. The Gilbert damping breaks time-reversal symmetry and causes a nite dissipation power: P[m(r;t)] =SZ d3r(@tm)2: (10) As a side comment, we note that an alternative, so-called Landau-Lifshitz damping term mHmis mathemati- cally identical to the Gilbert damping m@tmin Eq. (9), up to an extra factor of (1 + 2) on the left-hand side of the equation. The eective eld His in practice dominated by the applied magnetic eld, magnetic crystal anisotropies, and magnetostatic (dipole-dipole) interactions. In the pres- ence of spatial inhomogeneities, there is also exchange contribution to the free energy, which to the leading3 (quadratic) order in magnetic inhomogeneities can be written as11 Fxc=A 2Z d3r (@xm)2+ (@ym)2+ (@zm)2 :(11) The corresponding eective eld is Hxc=(A=S)r2m; (12) and the associated term in LL Eq. (6) is @tmjxc= (A=S)mr2m: (13) This equation can also be formally written as S@tmjxc=X i=x;y;z@ij0 i; j0 i=Am@im; (14) which simply recovers our equilibrium spin current (4). We emphasize that this spin current does not depend on magnetic dynamics. To summarize these preliminary considerations, the phenomenological LL equation describes collective mag- netic precession driven by local eective elds as well as equilibrium spin currents. At this point, there is, how- ever, a conspicuous asymmetry in the treatment of the dissipative correction to the LL equation, i.e., Gilbert damping (9), which depends only on the local magnetic dynamics and thus does not involve spin currents. To overcome this \discrepancy," we expand the dissipative terms to second order in spatial derivatives, generalizing Gilbert term to @tmjdiss=m@tm+ (=S)mr2@tm;(15) whereis a new phenomenological parameter, charac- terizing spin-wave damping. Assuming spatial-inversion symmetry (under which @i!@iandm!m), pre- vents us from writing any phenomenological terms linear in spatial derivatives. Recall also that we are always as- suming small perturbations with respect to a uniform equilibrium magnetization, so that all spatial and time derivatives must hit a single m(for example, a dissipa- tive term of the formP i[@im
(m@tm)]@imis dis- regarded since it is higher order in small deviations of m). Additional quadratic terms would be allowed phe- nomenologically if, e.g., we developed our linearized the- ory with respect to an equilibrium magnetic texture, such as a domain wall or magnetic spiral. Some of such terms were discussed in Ref. 13, which is beyond our present scope. Finally, we note that we wrote Eq. (15) with no direct coupling to the eective eld H. We justify this by assuming that the ferromagnetic correlations are char- acterized by a very large energy scale
xc, so that mi- croscopic processes responsible for dissipation are driven by the collective variable m, rather than directly by H. In transition-metal ferromagnets, the internal exchange energy is of the order of eV, while the eective eld Hcorresponds to microwave frequencies (i.e., at least three orders of magnitude smaller than the exchange energy). This means that when we excite magnetic dynamics by an external eld, the microscopic degrees of freedom re- spond not to the small driving eld but rather the much larger self-consistent exchange eld parametrized by the time-dependent m. For the same reason, the spin current in Eq. (14) depends only on the magnetic prole m(r), irrespective of how it is created by applied elds. The total dissipation power corresponding to Eq. (15) now becomes P[m(r;t)] =Z d3r S(@tm)2+(@i@tm)2 :(16) Similarly to Eq. (14), we can also write the term in Eq. (15) in the form of the divergence of the spin-current density j00 i=@i(m@tm); (17) reproducing Eq. (5). We thus identied two contribu- tions to the spin-current density: usual exchange spin current (14) and dissipative spin current (17), which we will later interpret as the dynamically-driven spin pumping.7,8Spin current (17) can thus damp down spin-wave excitations even in the absence of any spin- relaxation scattering.23The latter is, however, believed to be the culprit for a nite Gilbert damping ,14which relaxes uniform magnetic precession by transferring its angular momentum to the atomic lattice. In the presence of dissipative currents (17), the relative linewidth of the spin-wave resonance15is proportional to + (=S)q2, for the wave vector q. In the absence of the Gilbert damping , thus, the spectral width of the spin-wave excitation would vanish in the long-wavelength limit.1 B. Mermin ansatz for spin current We now wish to establish a microscopic procedure for evaluating the dissipative component of the spin current, Eq. (17). Ref. 9 adapted Mermin ansatz16for this pur- pose, which we will reproduce below. Microscopically, the spin-current density jiis carried by conducting elec- trons responding to the mean-eld exchange interaction ^Hxc=
xcm(r;t)
^=2 (18) in the self-consistent single-electron Hamiltonian (which could stem, e.g., either from the coupling to the localized delectrons in the sdmodel or the itinerant electron Stoner/LDA exchange). ^is the vector of Pauli matrices, which denes the electron spin operator. Let us for the moment view exchange interaction (18) as an external parametric driving eld, not concerning with a self-consistent determination of m(r;t). In par- ticular, we may allow for an instantaneous deviation of4 the electron spin density sfrom the exchange-eld direc- tionm. This will allow us for a trick to nd the ensuing spin ows, which is what we are after. The spin-density continuity equation corresponding to Hamiltonian (18) is @ts=
xcz(sms)@iji: (19) The equilibrium orientation of mis taken to be along the zaxis and we assume small-angle excitations, which do not modulate the magnitude of the spin density, s=jsj. shere is the spin density of the conducting electrons, which in, e.g., the sdmodel has to be distinguished from the total spin density Sthat enters Eq. (3). We next use the Mermin ansatz to relate the spin- current density jito the spin density s: ji=?
xc@i(ms=s); (20) where?is the transverse spin conductivity, to be evalu- ated later by the Kubo formula. Eq. (20) is analogous to Ohm's law for electric current density, with the expres- sion on the right-hand side reminiscent of the gradient of the electrochemical potential. The physical reasoning behind ansatz (20) is simple: there should be no dis- sipative spin currents in the static conguration, which corresponds to s(r) =sm(r). The advantage in writing the spin current in this form is that ?will now have to be evaluated in the limit of ( q;!)!0. Combining Eqs. (19) and (20) will then give us the spin current to the linear order in qand!: exactly what we need to re- late it to Eq. (17) and read out . In fact, it is sucient to nd
xc(ms=s)z@tmfrom Eq. (19), which is valid to the linear order in !and zeroth order in q, before putting it into Eq. (20) to nally nd ji=?@i(z@tm): (21) Comparing this with Eq. (17), we immediately identify  with the transverse spin conductivity: =?: (22) Equation (22) can be interpreted as an analog of the Ein- stein relation for transverse spin diusion in strong fer- romagnets. C. Transverse spin conductivity As is the case with the charge conductivity, it is con- venient to evaluate the transverse spin conductivity in the velocity gauge. Namely, we eliminate the spin \po- tential," corresponding to small magnetization deviations m=mzin Eq. (18), by the SU(2) gauge transfor- mation ^ (r;t)!ei
xcRt 1dt0m(r;t0)
^=2^ 0(r;t); (23) at the expense of introducing the SU(2) vector potential ^Ai=
xcZt 1dt0@im(r;t0)
^=2; (24)which enters the kinetic part of the single-particle Hamil- tonian as ^Hk=X i(pi^Ai)2=2m; (25) wherepi=i@iandmis the electron's eective mass (assuming exchange-split parabolic bands). It is easy to verify that the eective eld driving the spin current in velocity gauge (25), ^Ei=@t^Ai, is the same as the c- titious eld ^Ei=@i^Vin original length gauge (18). One caveat is in order: Eqs. (23)-(25) are only valid for an Abelian exchange potential, which would be the case if only one vector component of m(r;t) was modulated (e.g.,mxormy) in space and time. Such scenario is sucient for our purpose, in order to establish the trans- verse spin conductivity entering Eq. (20). Fourier transforming the electric eld ^Eiin time,R dtei!t, the usual relationship is obtained: ^Ei(!) = i!^Ai(!). We now proceed to construct the semiclassi- cal transport equation for the spin current driven by a spatially homogeneous ctitious eld Ei= Tr[ ^Ei^] =
xc@im, to deduce the long-wavelength conductivity de- ned by Ohm's law24 ji=?Ei: (26) The semiclassical spin-current response, in the presence of the exchange splitting
xc, with disorder and electron- electron scattering is given by17 @tji+
xczji=nEi 4mji1 dis ?+1 ee ? ;(27) wherenis the total equilibrium (conducting) electron density. The second term on the right-hand side of Eq. (27) describes spin-current relaxation, due to dis- order and electron-electron scattering. Note that even in Galilean-invariant systems, spin-independent Coulomb interaction between electrons causes relaxation of a ho- mogeneous spin current, in contrast to the ordinary cur- rent. Solving Eq. (27) at low frequencies, we recover Eq. (26) for the current component along Ei, with3,9 ?=n 4m? 1 + (?
xc)2; (28) where the total transverse spin scattering rate is dened by 1 ?=1 dis ?+1 ee ?: (29) In particular, in the limit of weak spin polarization and no electron-electron interactions, ?should reduce to the ordinary momentum scattering time , and?to the quarter of the Drude conductivity n=m.5 III. MICROSCOPIC CALCULATION A. Spin-current autocorrelator In order to substantiate the preceding phenomenology, we need to establish the microscopic expressions for the involved scattering times, dis ?andee ?. In the velocity gauge discussed in the previous section, the transverse spin conductivity is given, according to the Kubo for- mula, by the spin-current autocorrelation function:9 ?=1 4m2Vlim !!0=mhhP l^xlpxl;P l^xlpxlii! !;(30) where the summation is over all electrons in volume V and hh^A;^Bii!=iZ1 0dtei(!+i0+)th[^A(t);^B(0)]i(31) represents the Fourier-transformed retarded (Kubo) linear-response function for the expectation value of the observable ^Aunder the action of a classical eld that couples linearly to the observable ^B.=min Eq. (30) is inserted out of convenience, since the linear in !response function is guaranteed to be imaginary. (The zeroth- order in!correlator includes also the omitted \diamag- netic piece" of the spin current in the velocity gauge.) Assuming isotropic disorder (and for the moment no electron-electron interactions), the ladder vertex correc- tions to the conductivity vanish and we only need to eval- uate the bubble diagram dened by the (single-particle) spin-dependent Green's functions GR;A (p;!) =1 !p2=2m+i=2; (32) where=";#(=) is the spin index along the zaxis, =+
xc=2 is the spin- electron Fermi energy, is the chemical potential, and is the spin-dependent dis- order scattering time. In the Born approximation for dilute white-noise disorder, the scattering rate is pro- portional to the electron density of states, and we can write==, whereparametrizes the strength of the scattering potential, is the spin-band density of states, and = ("+#)=2. A straightforward calcula- tion then leads to9 ?=n 4m1 dis ?
2xc; (33) in the strong exchange coupling limit, where 1 dis ?=4 3"+# n(1 "+1 #)(34) identies the disorder contribution to eective transverse spin scattering rate (29).B. Spin-force autocorrelator In the presence of electron-electron interactions, it is convenient to express the spin- current autocorrelator (30) in terms of the spin- force autocorrelator. To this end, we use the equation of motion for the operators dening Kubo formula (30) to nd ?=1 4m2
2xcVlim !!0=mhhP l^xlFxl;P l^xlFxlii! !; (35) whereFxl= _pxl=i[pxl;^H] is the force operator along thexaxis for the lth electron. Evaluated with respect to a uniform magnetization, m=z, the force operator Fxl consists of two pieces: the disorder force and the electron- electron interaction force. Evaluating correlator (35) in the clean limit to second order in Coulomb interactions, one nds for the transverse spin scattering rate:5,9 1 ee ?= (p)ma2 Br4 s(kBT)2; (36) whereaBis the Bohr radius, Ttemperature, kBBoltz- mann constant, rsthe dimensionless Wigner-Seitz radius, and (p) is a dimensionless function of the spin polariza- tionp= (n"n")=n(nsbeing spin-selectron density), which was discussed in Refs. 5,9. Notice that scattering rate (36) has the Landau quasiparticle scaling with tem- perature. The nite-frequency modication of scatter- ing rate (36) is, furthermore, accomplished by replacing (2kBT)2!(2kBT)2+!2. C. Spin-density autocorrelator It is also possible to calculate the transverse spin dif- fusion directly, as a linear spin-density response to the transverse magnetic eld. We will carry that out in Sec. IV for two popular mean-eld models of ferromag- netism in metals: the Stoner and the sdmodels. In ad- dition to oering an alternative approach to the problem, this derivation provides a justication for the preceding heuristic utilization of the Mermin ansatz. Starting with the mean-eld Hamiltonian for itinerant electrons ^H=p2 2m+U(r)
xc^z=2; (37) and directly solving for the self-consistent spin-density response to a small driving magnetic eld, we will derive in the next section the following general relation: =
2 xc q2lim !!0=m~+(q;!) !; (38) valid at long wavelengths, q!0. The axially-symmetric (Kubo) spin-response function is dened by ~+(q;!) =1 2hhs+(r;t);s(r0;0)iiq;!; (39)6 wheres=sxisyis the transverse spin density of itinerant electrons. The disorder potential U(r) entering Eq. (37) is, as before, taken to obey the Gaussian white- noise correlations: hU(r)U(r0)i=1 2(rr0); (40)where= ("+#)=2 is the spin-averaged density of states at the Fermi level and is the characteristic scat- tering time. Writing the spin density s(r) = Tr [ ^^(r)]=2 in terms of the electron density matrix (r) = y (r) (r) in spin space, we proceed to evaluate ~ +in the standard imaginary-time formalism. At temperature T, we have: ~+(q;i n) =T 2VX pp0;mG#(p+q;p0+q;i!m+i n)G"(p0;p;i!m); (41) where G(p;p0;i!m) =1 VZ d3rd3r0Z1=T 0deip
r+ip0
r0+i!m (r;) y (r0;0) (42) is the nite-temperature single-particle Matsubara Green's function. n= 2nT is the bosonic and !m= (2m+1)T fermionic Matsubara frequencies, where nandmare integer indices. The disorder-averaged Green's function is given by hG(p;p0;i!m)i=pp0 i!m"p+isign(!m)=2; (43) where"p=p2=2m
xc=2. The analytic continuation of the Matsubara Green's functions into the retarded (advanced) Green's functions is accomplished by replacing i!m!!i0+and sign(!m)! . According to our convention (40), ==. Taking into account the vertex ladder corrections (as shown in Fig. 1), we obtain for the disorder-averaged response function: ~+(q;i n) =T 2VX mP pG#(p+q;i!m+i n)G"(p;i!m) 1(=V)P pG#(p+q;i!m+i n)G"(p;i!m); (44) where= 1=2 and by the Green's functions with a single wave-vector argument here we understand disorder- averaged propagators (43). Inserting Eq. (43) into Eq. (44) and performing an analytic continuation onto the real frequencies, it is straightforward to calculate ~ +(q;!). Setting the temperature to zero and taking the !!0 limit, we nd: =m~+(q;!) =! 4<e~RA~AA (1~RA)(1~AA); (45) where ~XY(q) =R dpGX #(p)GY "(pq) andR dpR d3p=(2)3in three dimensions. All energies entering these Green's functions are set at the Fermi level. To the lowest order in 1 =
, we now obtain: =m~+(q;!) =! 8Z dpA#(p)A"(pq) + 4=mZ dpGA "(pq)A#(p)Z dpGA "(pq)<eGR #(p) ; (46) whereA=2=mGR is the spectral function. The second term in Eq. (46) is the vertex ladder correc- tion, which is necessary for Eq. (46) to give a meaningful result. In particular, the vertex correction cancels the spuriousq= 0 contribution of the rst term, which would give1=
xc. Finally, in the limit of q
xc=vF,we arrive at: =m~+(q;!) ="+# 3m(1 "+1 #)
4xc!q2: (47) Using Eq. (38), this nally gives: ="+# 3m(1 "+1 #)
2xc; (48)7 which agrees with Eqs. (22), (28), and (34) in the relevant here limit of 1 ?
xc. IV. MEAN-FIELD FERROMAGNETISM A. Time-dependent LDA In a spin-density-functional theory (s-DFT),5,18the many-body problem of itinerant ferromagnetism is re- duced to the single-electron Hamiltonian ^H(t) =p2 2m+U(r) [
xcm(r;t) +!0z+h(r;t)]
^=2:(49) !0
xcis the ferromagnetic Larmor precession fre- quency in the presence of a uniform magnetic eld ap- plied along the zaxis.
xcm(r;t) is the self-consistent exchange eld, such that Hamiltonian (49) produces the correct spin-density response. Since we are ultimately in- terested in the equation of motion for the collective ferro- magnetic dynamics, the spin-density response is all that is needed. In the local-density approximation (LDA) of the s-DFT, the exchange eld follows the local and in- stantaneous magnetization direction m(r;t).h(r;t) is the external rf driving eld, which we will treat pertur- batively. The time-dependent portion of the Hamiltonian is thus given by ^H0(t) =Z d3r[
xcm(r;t) +h(r;t)]
^=2:(50) Sincem=s=S(wheredenotes small deviations from equilibrium), we have for the transverse spin component s+=sx+isy: s+(q;!) = ~+(q;!) h+(q;!) +
xc Ss+(q;!) :(51) The self-consistent response function to the rf eld, +=s+=h+, is thus 1 +(q;!) = ~1 +(q;!)
xc S: (52) In the LDA approximation, the problem thus trivially reduces to calculating the spin-spin response function for a noninteracting Hamiltonian with a xed exchange eld. Let us in general write ~+(q;!) =S [!r(q;!) +
xc!]i(q;!)!;(53) in terms of functions !randthat are to be determined. The self-consistent response function then becomes: +(q;!) =S [!r(q;!)!]i(q;!)!: (54)Atq= 0, obviously !r(!)!0and(!)0. This fol- lows in general from the spin conservation in the presence of Coulomb interactions and arbitrary spin-independent potentialU(r). In this paper, we are most interested in theq-dependent damping function (q;!), which can be identied by a microscopic evaluation of ~ +(q;!). In the limit of strong exchange correlations,
xc!r, we immediately obtain from Eq. (53): (q;!!0)
2 xc Slim !!0=m~+(q;!) !: (55) In inversion-symmetric systems, the leading in qspin- wave contribution to Gilbert damping is (q;!!0) = (=S)q2, so that self-consistent response function (54) corresponds to the dissipative term @tmjdiss= (=S)mr2@tm (56) in Landau-Lifshitz Eq. (6) of motion for the magnetic spin direction m(r;t). This is the desirable result and, according to Eq. (55), the microscopic expression for  gives Eq. (38) of the previous section. In the next section, we will demonstrate that Eq. (55) is generic to mean-eld treatment of conducting ferromagnets. B.sdmodel in RPA It is also instructive to pursue a more basic descrip- tion starting with a ferromagnetic lattice of localized d electrons exchange-coupled to itinerant selectrons. The corresponding Hamiltonian is ^H(t) =^H0X i[JSi
s(ri;t) +Si
h(ri;t)];(57) where Siare localdspins and ^H0consists of the de- coupled Hamiltonian for itinerant electrons, dc Zeeman Hamiltonian of the delectrons, as well as the ddex- change and possible dipolar interactions. his the applied rf eld, which we take for simplicity to couple to the lo- calized spin only. As long as the average exchange eld experienced by the selectrons is suciently strong and the magnetization is dominated by the delectrons, we can disregard their direct rf coupling for our purpose. If also the Fermi wavelength is long in comparison to thedlattice spacing, we will treat the electronic band structure in the eective-mass approximation, and also coarse grain the local spins:P iSi!R d3rS(r) and Si!(V=N)S(r), whereN=V is the density of dsites. Let us compute the spin-density response function for thedlattice: +(r;r0;t) =1 2hhS+(r;t);S(r0;0)ii: (58) For this purpose, it is convenient to dene bosonic magnon operators: ap=1p 2DNX ieip
riSi+; (59)8 8 F: q,iΩn= q,iΩn+ p+q p/prime+qp/primep ↓i(ωm+Ωn)↑iωm + +··· FIG. 1: In this RPA summation for the magnon propagator, Eq. (64), the magnon self-energy is provided by the s-electron spin-response bubble, Eq. (41). Straight double lines denote disorder-averaged s-electron Green’s functions and dashed lines describe vertex ladder corrections. Each bubble contains ladder corrections to the vertex function. Wavy lines are the d-electron spin propagators. Eq. (61) with ˆH0for the selectrons, resulting in the longitudinal mean-field exchange field ∆ xc=JS, where S=DN/V is the averaged d-electron spin density. The spin-flop term of Eq. (61) can be expanded in terms of the electronic field operators Ψ σas: ˆH/prime xc=−J V/radicalbigg DN 2/summationdisplay pp/prime/bracketleftBig a† pΨ† ↑(p/prime)Ψ↓(p/prime+p) +apΨ† ↓(p/prime)Ψ↑(p/prime−p)/bracketrightBig . (62) In the imaginary-time (Matsubara) formalism, the re- sponse function (58) can be written as: χ+−(q,τ)=−SF(q,τ), (63) in terms of the magnon Green’s function F(q,τ)=−T/angbracketleftbig aq(τ)a† q(0)/angbracketrightbig , (64) using the definition (59). Here, Tis the time-ordering symbol. For the noninteracting Hamiltonian (60), F0(q,iΩn)=1 iΩn−εq, (65) where F(q,iΩn)=/integraltext1/T 0dτeiΩnτF(q,τ) and Ω n=2nπT, as previously, is the bosonic Matsubara frequency. In or- der to calculate Fin the presence of the s−dexchange (62), we will sum up the bubble diagrams (which con- stitute an RPA approximation) shown in Fig. 1. It is easy to recognize that each bubble (which is the magnon self-energy in this approximation) is just the itinerant electron spin-density response function that was already calculated in Sec. III C. Summing up the diagrams in Fig. 1, we thus obtain F−1(q,iΩn)=F−1 0(q,iΩn)−Σ(q,iΩn), (66) where the self-energy (the s-electron spin-response func- tion) Σ(q,iΩn)=−J2S˜χ+−(q,iΩn) (67) follows from Eq. (44). Combining Eqs. (63), (66), and (67), we finally obtain (after analytic continuation onto real frequencies): χ−1 +−(q,ω)=εq−ω S−J2˜χ+−(q,ω). (68)Eq. (68) is actually quite trivial: It can be also obtained by treating the d-orbital magnetization dynamics and the associated s−dtorque in a mean-field approxima- tion analogous to the preceding discussion of the Stoner model. Inverting Eq. (68), we can write it in the form (54), after identifying ωr(q,ω)=εq−SJ2/Rfracture˜χ+−(q,ω) (69) and α(q,ω)=SJ2/Ifracturm˜χ+−(q,ω) ω, (70) which is, in fact, exactly the same as Eq. (55) in the low-frequency limit, using ∆ xc=JS. We should em- phasize that although χ+−is defined in this section for a different physical system than χ+−in Sec. IV A (and thus, not surprisingly, is found to be somewhat different), the itinerant-electron response ˜ χ+−is the same through- out the paper. The reason why the q2magnetic damping α(q,ω) is identified in terms of the same quantity ˜ χ+−in the two different models of ferromagnetism can be traced to our phenomenological identification of this damping in terms of the conducting-electron transverse conductivity, Eq. (22). The latter is governed by the mean-field struc- ture of the exchange field, irrespective of the microscopic origin of the ferromagnetic order. Let us also note in the passing that, unlike the idealized Stoner model considered in the previous section, the s−d magnetic damping may have a finite q= 0 value even in the absence of any additional spin-dependent terms in the Hamiltonian. When the gyromagnetic ratios of the two electron species differ (ultimately stemming from some form of spin-orbit interaction), the total spin does no longer precess undamped in the uniform field, and the uniform transverse spin component can decohere in the presence of ordinary scalar disorder. Since Eq. (70) cor- responds to the magnetic field coupled to the delectrons only, we implicitly set the selectron gfactor to zero. V. SPIN-PUMPING INTERPRETATION It is illuminating to interpret the key result of this paper for the transverse spin diffusion of the form (17) FIG. 1: In this RPA summation for the magnon propagator, Eq. (64), the magnon self-energy is provided by the s-electron spin-response bubble, Eq. (41). Straight double lines denote disorder-averaged s-electron Green's functions and dashed lines describe vertex ladder corrections. Each bubble contains ladder corrections to the vertex function. Wavy lines are the d-electron spin propagators. which obey the canonical commutation relations, [ap;ay p0] =pp0, close to the fully-magnetized ground state. To be specic, let us take the Heisenberg model for exchange coupling, so that in the ground state, Siz=D, thed-orbital spin [assuming the applied dc magnetic eld to point along the zdirection, as in Eq. (49)]. The d- orbital Hamiltonian for magnon excitations close to the ground state can thus be written as: ^H0=X p"pay pap: (60) In terms of the magnon operators, we, furthermore, rewrite the sdexchange interaction as ^Hxc=J Vr DN 2X p ay ps+(p) +aps(p) JX iSizsz(ri); (61) wheres(p) =R d3reip
rs(r) is the Fourier- transformed transverse s-electron spin density. Approx- imatingSizD, we can combine the second term in Eq. (61) with ^H0for theselectrons, resulting in the longitudinal mean-eld exchange eld
xc=JS, where S=DN=V is the averaged d-electron spin density. The spin- op term of Eq. (61) can be expanded in terms of the electronic eld operators as: ^H0 xc=J Vr DN 2X pp0h ay p y "(p0) #(p0+p) +ap y #(p0) "(p0p)i : (62) In the imaginary-time (Matsubara) formalism, re- sponse function (58) can be written as: +(q;) =SF(q;); (63) in terms of the magnon Green's function F(q;) =T aq()ay q(0) ; (64) using denition (59). Here, Tis the time-ordering sym- bol. For noninteracting Hamiltonian (60), F0(q;i n) =1 i n"q; (65)whereF(q;i n) =R1=T 0dei nF(q;) and n= 2nT, as previously, is the bosonic Matsubara frequency. In or- der to calculateFin the presence of sdexchange (62), we will sum up the bubble diagrams (which constitute an RPA approximation) shown in Fig. 1. It is easy to recog- nize that each bubble (which is the magnon self-energy in this approximation) is just the itinerant electron spin- density response function that was already calculated in Sec. III C. Summing up the diagrams in Fig. 1, we thus obtain F1(q;i n) =F1 0(q;i n)(q;i n); (66) where the self-energy (the s-electron spin-response func- tion) (q;i n) =J2S~+(q;i n) (67) follows from Eq. (44). Combining Eqs. (63), (66), and (67), we nally obtain (after analytic continuation onto real frequencies): 1 +(q;!) ="q! SJ2~+(q;!): (68) Equation (68) is actually quite trivial: it can be also ob- tained by treating the d-orbital magnetization dynamics and the associated sdtorque in a mean-eld approxima- tion analogous to the preceding discussion of the Stoner model. Inverting Eq. (68), we can write it in form (54), after identifying !r(q;!) ="qSJ2<e~+(q;!) (69) and (q;!) =SJ2=m~+(q;!) !; (70) which is, in fact, exactly the same as Eq. (55) in the low-frequency limit, using
xc=JS. We should em- phasize that, although +is dened in this section for a dierent physical system than +in Sec. IV A (and thus, not surprisingly, is found to be somewhat dierent), the itinerant-electron response ~ +is the same through- out the paper. The reason why the q2magnetic damping (q;!) is identied in terms of the same quantity ~ +in9 the two dierent models of ferromagnetism can be traced to our phenomenological identication of this damping in terms of the conducting-electron transverse conductivity, Eq. (22). The latter is governed by the mean-eld struc- ture of the exchange eld, irrespective of the microscopic origin of the ferromagnetic order. Let us also note in the passing that, unlike the idealized Stoner model considered in the previous section, the sd magnetic damping may have a nite q= 0 value even in the absence of any additional spin-dependent terms in the Hamiltonian. When the gyromagnetic ratios of the two electron species dier (ultimately stemming from some form of spin-orbit interaction), the total spin no longer precesses undamped in the uniform eld, and the uniform transverse spin component can decohere in the presence of ordinary scalar disorder. Since Eq. (70) corresponds to the magnetic eld coupled to the delectrons only, we implicitly set the selectrongfactor to zero. V. SPIN-PUMPING INTERPRETATION It is illuminating to interpret the key result of this paper for the transverse spin diusion of form (17) in terms of the spin pumping associated with a nonuni- form magnetic dynamics in ferromagnetic bulk.8The ferromagnetic spin pumping was originally proposed in the context of magnetic multilayers with sharp normal- metaljferromagnetic interfaces. This paper shows that analogous processes also take place in the continuous fer- romagnetic medium. To illustrate the direct connection between the trans- verse spin diusion and the spin pumping, we consider a periodic stack of alternating F and N layers forming a two-component superlattice in the xdirection.8We treat the model depicted in Fig. 2, in which an F jN bilayer forms the unit cell with thickness b=L+d, where the normal-metal spacer of width Lseparates the magnetic lms of thickness dsc=vF=
xc: (71) The latter approximation allows us to neglect the trans- verse spin-current coherence between two interfaces of the same magnetic layer.8Translational invariance is as- sumed for simplicity in the lateral directions. We con- sider here collective spin-wave excitations, taking both the static and dynamic exchange couplings into account.7 The static (RKKY-like) exchange interaction between neighboring ferromagnetic layers is mediated by the dis- sipationless spin currents owing through the normal- metal spacer.19We will parametrize the strength of this coupling by the corresponding precession frequency !xc of a single ferromagnetic lm that is exchange-coupled to a pinned lm. In the presence of the magnetic dynam- ics, additional dissipative spin currents set in. Their ori- gin lies in the spin pumping by the individual magnetic layers into the adjacent normal spacers, which at low frequencies is given by8Ipump s = (h=4)~g"# NjFm@tm. Lb=L+d d xFIG. 2: A schematic view of the superlattice considered in the text: an FjN bilayer is repeated along the xaxis, with either ferromagnetic or antiferromagnetic alignment of the consecu- tive magnetic layers. The system is translationally invariant along the two remaining axes. ~g"# NjFis the dimensionless spin-mixing conductance per unit area of the F jN interface (which is assumed to be real-valued, for simplicity). This interfacial spin pump- ing induces nonlocal spin transfer in magnetoelectronic circuits, which can in general be treated as a source term entering spin transport equations in normal and mag- netic layers. In a collinear superlattice of Fig. 2, the problem simplies considerably, because the spin-current vector Ipump s/m@tmis transverse with respect to the magnetic alignment (in both ferromagnetic and anti- ferromagnetic cases), within the linear-response regime. This means that the spin current pumped by one fer- romagnetic layer is either scattered back by the normal spacer and reabsorbed, or transmitted and absorbed by a neighboring layer, with no possibility to reach more dis- tant neighbors, subject to condition (71). Spin relaxation in normal spacers would only cause an overall increase in the eective Gilbert damping parameter of a uniform magnetic precession, and will thus be omitted, since our primary interest here is nonlocal damping eects. The problem of dynamic exchange between two adjacent fer- romagnetic layers thus eectively reduces to the analo- gous eect in magnetic bilayers, which was studied in de- tail in Ref. 7. In particular, the net spin pumping through a given normal spacer is /m1@tm1m2@tm2, which re ects the dynamic spin injection in the opposite direc- tions by the adjacent magnetic layers m1andm2. Notice that the total pumping vanishes in the case of a perfectly synchronous precession, m1(t) =m2(t). Let us now put the static and dynamic exchange inter- actions into the equation of motion for small-angle spin dynamics of a multilayer with respect to an all-parallel conguration, ui(t) =mi(t)z. For long-wavelength ex- citations, it may be approximated as a continuous func- tionu(x;t) of the coordinate xnormal to the interfaces. For the uniaxial eective eld H=!0z, the spin-wave dynamics obey the dierential equation @tu= !0u!xcb2@2 xu+@tu0b2@2 x@tu z;(72) where we made the following denitions: !xc=Jxc=Sd and0=<eA"# FjNjF=4Sd. Here,Jxcis the exchange10 coupling between two consecutive magnetic layers and 1=A"# FjNjF= 2=~g"# NjF+e2L= (73) is the eective pumping resistance of the spacer. The rst term on the right-hand side of Eq. (73) parametrizes the pumping strength of the individual interfaces, as was dis- cussed above, and the second term is the ordinary ohmic resistance of the normal spacer (neglecting any spin relax- ation), which backscatters the pumped currents and thus suppresses the dynamic exchange. The second spatial derivatives in Eq. (72) re ect simply the dierence of the static and dynamic exchange spin currents through two consecutive normal spacers (which themselves require a nite misalignment of the adjacent magnetic layers) in the continuum limit. The static Heisenberg coupling can be interpreted as the superlattice equivalent of the bulk exchange-stiness parameter Aof Eq. (11), which for the superlattice becomes A=Jxcb. Both!xcand0are sen- sitive to the normal-interlayer thickness L, vanishing in the limitL! 1 . It follows from Eq. (72) that the small-momentum, qb1, spin-wave excitations of the superlattice, propagating perpendicular to the interfaces, u/exp[i(qx!t)], obey the dispersion relation !(q) =!0+ (bq)2!xc 1 +i[+ (bq)20]: (74) Whenq!0,!(q) reduces to the Larmor frequency !0 of the individual magnetic layers because the static and dynamic exchange couplings vanish when the consecutive magnetic layers move coherently in phase. Equation (74) holds up to momenta comparable to b1, whenbqhas to be replaced by 2 sin( bq=2). The matters are quite dierent for an antiferromagnetically-aligned superlattice, which is the ground state when, for example, Jxc<0 and H= 0. In this case, we have a more complex dispersion: !(q) =!xcp (bq)2(1 +2)42+i[2+ (bq)20] 1 +2+ 40+ (bq)202; (75) where plus and minus signs refer, respectively, to the modes with antisymmetric and symmetric dynamics in the adjacent layers for overdamped motion, and to the right- and left-propagating modes when the real part of!(q) is signicant. Note that now !xc<0, so that =m! > 0, as required for a stable conguration. In the absence of bulk magnetization damping, = 0, Eq. (75) reduces to !(q) =(bq)!xc 1i(bq)0; (76) with linear dispersion and damping at small q. Equa- tions (75) and (76) can also be generalized to large mo- menta by replacing bqwith 2 sin(bq=2). Notice that in Eqs. (72), (74), and (76), the dynamic coupling modi- es the damping similarly to the way the static couplingaects the excitation frequency of the magnetic superlat- tice. Crystal and shape anisotropies on top of the simple eective elds assumed above might become important in real structures, and their inclusion is straightforward. Let us now compare the damping ( bq)20in Eq. (74) with(q) = (?=S)q2corresponding to Eq. (33), which is the analogous quantity for the bulk. Keeping only the mixing conductance contribution to Eq. (73) and approximating8~g"#p2 F=2in terms of the character- istic Fermi momentum pFin the normal metal, we have for theq-dependent part of the damping: (q) = (bq)20(b=F)2 Sdq2; (77) up to a numerical constant. At the same time, the bulk (q), corresponding to Eq. (33), can be written as (q)(sc=F)2 Slq2; (78) which establishes a loose formal correspondence between the two results. Here, l=vFis the mean free path, F the Fermi wavelength, and the ferromagnetic coherence lengthscwas dened in Eq. (71). Comparing Eqs. (77) and (78), we interpret the length scaleb$scto describe the longest distance over which ferromagnetic regions can communicate via spin trans- fer. The length scale d$characterizes momentum scattering relevant for spin transfer, which in the case of the superlattice with sharp interfaces corresponds to the magnetic lm width d: Approximating ~ g"#p2 F=2 above, we eectively took the normal spacers to be bal- listic and, because of Eq. (71), the spin transfer does not penetrate deep into the ferromagnetic layers, mak- ing possible disorder scattering there irrelevant for our problem. VI. DISCUSSION AND OUTLOOK Estimating the numerical value of the dimensionless q2 damping, according to Eq. (28), (q) =?q2 SF=
xc pF=q2?
xc 1 + (?
xc)2; (79) we can see that it will most likely be at most compa- rable or smaller than the typical q= 0 Gilbert damp- ing102, in metallic ferromagnets. Damping (79) may, however, become dominant in weak ferromagnets, such as diluted magnetic semiconductors. We are not aware of systematic experimental investigations of the q2damping in metallic ferromagnets. q2scaling of rel- ative linewidth was reported in Ref. 20 for the iron-rich amorphous Fe 90xNixZr10alloys. However, we are not certain whether the strong damping observed there can be attributed to the mechanism discussed in our paper. Another intriguing context where the physics discussed here can play out to be important is the current-driven11 nonlinear ferromagnetic dynamics in mesoscopic as well as bulk magnetic systems. The q2magnetic damping described by Eq. (15) can be physically thought of the viscous-like spin transfer between magnetic regions pre- cessing slightly out-of-phase. The obvious consequence of this is the enhanced damping of the inhomogeneous dynamics and thus the synchronization of collective mag- netic precession. This phenomenon was predicted in Ref. 7 and unambiguously observed in Ref. 21, in the case of the coupled dynamics of a magnetic bilayer: When the two layers are tuned to similar resonance conditions, only the symmetric mode corresponding to the synchronized dynamics produces a strong response, while the antisym- metric mode is strongly suppressed. It is thus natural to suggest that the q2viscous magnetic damping in the con- tinuum limit may have far-reaching consequences for the current-driven nonlinear power spectrum as that mea- sured in Ref. 22. This needs a further investigation. The role of electron-electron interactions was mani- fested in our theory through the spin Coulomb drag, which enhances the eective transverse spin scattering rate (29). This becomes particularly important, in com- parison to the disorder contribution to the transverse spin scattering, in the limit of weak magnetic polarization.9We nally emphasize that the study in this paper was limited exclusively to weak linearized perturbations of the magnetic order with respect to a uniform equilib- rium state. When the equilibrium or out-of-equilibrium magnetic state is macroscopically nonuniform, as is the case with, e.g., the magnetic spin spirals, domain walls, vortices, and other topological states, the longitudinal as well as transverse spin currents become relevant for the magnetic dynamics. The longitudinal spin currents lead to additional contributions to the spin-transfer torques, modifying the magnetic equation of motion. Such spin torques leading to the dissipative q2damping terms were discussed in Ref. 13. These latter contributions to the magnetic damping are likely to dominate in strongly- textured magnetic systems. Acknowledgments We are grateful to Gerrit E. W. Bauer and Arne Brataas for stimulating discussions. This work was sup- ported in part by the Alfred P. Sloan Foundation (YT) and NSF Grant No. DMR-0705460 (GV). 1B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898 (1969). 2A. J. Leggett, J. Phys. C: Sol. State Phys. 3, 448 (1970). 3A. Singh, Phys. Rev. B 39, 505 (1989); A. Singh and Z. Te sanovi c, ibid.39, 7284 (1989). 4V. L. Sobolev, I. Klik, C. R. Chang, and H. L. Huang, J. Appl. Phys. 75, 5794 (1994); A. E. Meyerovich and K. A. Musaelian, Phys. Rev. Lett. 72, 1710 (1994); D. I. Golosov and A. E. Ruckenstein, ibid.74, 1613 (1995); Y. Takahashi, K. Shizume, and N. Masuhara, Phys. Rev. B 60, 4856 (1999). 5Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404 (2002). 6V. P. Mineev, Phys. Rev. B 69, 144429 (2004); ibid.72, 144418 (2005). 7Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 67, 140404(R) (2003). 8Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 9E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B 78, 020404(R) (2008). 10T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993). 11E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox- ford, 1980), 3rd ed. 12T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 13J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. B 78, 140402(R) (2008). 14E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B 75, 174434 (2007). 15B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523 (1993).16N. D. Mermin, Phys. Rev. B 1, 2362 (1970). 17I. D'Amico and G. Vignale, Phys. Rev. B 62, 4853 (2000). 18K. Capelle, G. Vignale, and B. L. Gy ory, Phys. Rev. Lett. 87, 206403 (2001). 19J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989). 20J. A. Fernandez-Baca, J. W. Lynn, J. J. Rhyne, and G. E. Fish, J. Appl. Phys. 61, 3406 (1987). 21B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003). 22I. N. Krivorotov, D. V. Berkov, N. L. Gorn, N. C. Emley, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B76, 024418 (2007). 23It is natural to also wonder about a possible additional spin current of the form ji/@i@tm, which does not break time-reversal symmetry. Such spin current leads to a wave-vector-dependent correction to the eective gyromag- netic ratio, which is very small in practice. It parallels the structure of the spin pumping in magnetic nanostructures, which consists of the dominant dissipative piece of the form m@tmand a smaller piece of the form @tm. The latter merely causes a slight rescaling of the gyromagnetic ratio. While the dissipative piece of the spin pumping has been unambiguously established in a number of experiments,8 the small correction to the gyromagnetic ratio is yet to be observed. 24We need to remark here that the above gauge transfor- mation does not aect the transverse spin current in the linearized theory.

2008-10-08

We discuss the dissipative diffusion-type term of the form $\mathbf{m}\times\nabla^2\partial_t\mathbf{m}$ in the phenomenological Landau-Lifshitz equation of ferromagnetic precession, which describes enhanced Gilbert damping of finite-momentum spin waves. This term arises physically from itinerant-electron spin flows through a perturbed ferromagnetic configuration and can be understood to originate in the ferromagnetic spin pumping in the continuum limit. We develop a general phenomenology as well as provide microscopic theory for the Stoner and s-d models of ferromagnetism, taking into account disorder and electron-electron scattering. The latter is manifested in our problem through the Coulomb drag between the spin bands. The spin diffusion is identified in terms of the transverse spin conductivity, in analogy with the Einstein relation in the kinetic theory.

Transverse spin diffusion in ferromagnets

0810.1340v2

arXiv:0810.2870v1 [physics.class-ph] 16 Oct 2008 /C1/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D3/CU /D6/CT/CT/CS /CP/D2/CS /CP /D3/D9/D7/D8/CX /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CX/D2 /D0/CP/D6/CX/D2/CT/D8/B9/D0/CX/CZ /CT /D7/DD/D7/D8/CT/D1/D7/BY /CP/CQ /D6/CX /CT /CB/CX/D0/DA/CP/B8 /C2/CT/CP/D2 /C3/CT/D6/CV/D3/D1/CP /D6/CS/B8 /CP/D2/CS /BV/CW/D6/CX/D7/D8/D3/D4/CW/CT /CE /CT/D6/CV/CT/DE /CP/B5/C4 /CP/CQ /D3/D6 /CP/D8/D3/CX/D6 /CT /CS/CT /C5/GH /CP/D2/CX/D5/D9/CT /CT/D8 /CS/B3/BT /D3/D9/D7/D8/CX/D5/D9/CT /CD/C8/CA /BV/C6/CA/CB /BJ/BC/BH/BD/B8 /BD/BF/BG/BC/BE /C5/CP/D6/D7/CT/CX/D0 /D0/CT /CT /CS/CT/DC /BE/BC/B8 /BY /D6 /CP/D2 /CT/C2/D3 /GJ/D0 /BZ/CX/D0/CQ /CT/D6/D8 /CQ/B5/C4 /CP/CQ /D3/D6 /CP/D8/D3/CX/D6 /CT /CS/B3/BT /D3/D9/D7/D8/CX/D5/D9/CT /CS/CT /D0/B3/CD/D2/CX/DA/CT/D6/D7/CX/D8/GH /CS/D9 /C5/CP/CX/D2/CT /CD/C5/CA /BV/C6/CA/CB /BI/BI/BD/BF/B8 /BJ/BE/BC/BK/BH /C4 /CT /C5/CP/D2/D7 /CT /CS/CT/DC /BL/B8 /BY /D6 /CP/D2 /CT/B4/BW/CP/D8/CT/CS/BM /C7 /D8/D3/CQ /CT/D6 /BF/BD/B8 /BE/BC/BD/BK/B5/CB/D3/D9/D2/CS 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/D1/CT/D8/CW/D3 /CS/D7/D3/CU /D0/CX/D2/CT/CP/D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CP/D0/DD/D7/CX/D7 /BF/BC/CP/D6/CT /D9/D7/CT/CS /CX/D2 /D8/CW/CX/D7 /D7/D8/D9/CS/DD /B8 /CP/D2/CS/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BE/D7/D3/D0/D9/D8/CX/D3/D2/D7 /CW/CP /DA/CX/D2/CV /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2 /CT exp(jωt) /CP/D6/CT /D7/D3/D9/CV/CW /D8/BA/BV/CP/D2 /CT/D0/D0/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8 /D3/CUω /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /CP/D2/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/CP/D8 /CX/D7 /D2/CT/CX/D8/CW/CT/D6 /CS/CP/D1/D4 /CT/CS /D2/D3/D6 /CP/D1/D4/D0/CX/AS/CT/CS/BM /CX/D8 /CW/CP/D6/B9/CP /D8/CT/D6/CX/DE/CT/D7 /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3/CU /D7/D8/CP/D8/CX /D6/CT/CV/CX/D1/CT/BA /BT /D8/B9/D8/CT/D2 /D8/CX/D3/D2 /CX/D7 /CS/D6/CP /DB/D2 /D8/D3 /D8/CW/CT /CU/CP /D8 /D8/CW/CP/D8 /D8/CW/CX/D7 /D5/D9/CP/D2 /D8/CX/D8 /DD /D1/CP /DD /CS/CX/AR/CT/D6/CU/D6/D3/D1 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CS/CT/D4 /CT/D2/CS/CX/D2/CV /D3/D2 /D8/CW/CT /D2/CP/D8/D9/D6/CT/D3/CU /D8/CW/CT /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2/B8 /D7/D8/D9/CS/CX/CT/CS /CX/D2 /CB/CT /BA /C1/CE/BA/BU/BA /BT/D7 /CP /D0/CP/D2/CV/D9/CP/CV/CT/CP/CQ/D9/D7/CT /D3/D7 /CX/D0 /D0/CP/D8/CX/D3/D2 /D8/CW/D6 /CT/D7/CW/D3/D0/CS /CX/D7 /D3/CU/D8/CT/D2 /D9/D7/CT/CS /CX/D2/D7/D8/CT/CP/CS /D3/CU /CX/D2/D7/D8/CP/B9/CQ/CX/D0/CX/D8/DD /D8/CW/D6 /CT/D7/CW/D3/D0/CS /BA/BT/D7/D7/D9/D1/CX/D2/CV /D7/D1/CP/D0/D0 /DA/CX/CQ/D6/CP/D8/CX/D3/D2/D7 /CP/D6/D3/D9/D2/CS /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT/B4/D1/CT/CP/D2 /DA /CP/D0/D9/CT/D7 /D3/CUy /CP/D2/CSp /CP/D6/CTy0−Pm/K /CP/D2/CS0 /B8 /D6/CT/D7/D4 /CT /B9/D8/CX/DA /CT/D0/DD/B5/B8 /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /B4/BG/B5 /CX/D7 /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS/BA /BW/CX/B9/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7 /CP/D6/CT /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS /CW/CT/D6/CT/BMθ /B8Ye /B8 /CP/D2/CSD/CP/D6/CT /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CU/D6/CT/D5/D9/CT/D2 /DD /B8 /CX/D2/D4/D9/D8 /CP/CS/D1/CX/D8/D8/CP/D2 /CT /CP/D2/CS/D8/CW/CT /D6/CT/CT/CS /D8/D6/CP/D2/D7/CU/CT/D6 /CU/D9/D2 /D8/CX/D3/D2/B8 /D6/CT/D7/D4 /CT /D8/CX/DA /CT/D0/DD /BA θ=ω ωr,Ye(θ) =Zc Ze(θ) /CP/D2/CSD(θ) =1 1+jqrθ−θ2. /B4/BH/B5/CC/CW/CT/D6/CT /CP/D6/CT /D8 /DB /D3 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D3/D2 /D8/D6/D3/D0 /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BM γ=Pm Ky0 /CP/D2/CSζ=ZcW/radicalbigg2y0 Kρ. /B4/BI/B5 γ /CX/D7 /D8/CW/CT /D6/CP/D8/CX/D3 /CQ /CT/D8 /DB /CT/CT/D2 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D2/CS /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT/D6/CT/D5/D9/CX/D6/CT/CS /D8/D3 /D3/D1/D4/D0/CT/D8/CT/D0/DD /D0/D3/D7/CT /D8/CW/CT /D6/CT/CT/CS /CW/CP/D2/D2/CT/D0 /CX/D2 /D7/D8/CP/D8/CX /D6/CT/CV/CX/D1/CT/B8 /DB/CW/CX/D0/CTζ /D1/CP/CX/D2/D0/DD /CS/CT/D4 /CT/D2/CS/D7 /D3/D2 /D1/D3/D9/D8/CW/D4/CX/CT /CT /D3/D2/D7/D8/D6/D9 /B9/D8/CX/D3/D2 /CP/D2/CS /D0/CX/D4 /D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS /CP/D2/CS /CX/D7 /D0/CX/D2/CZ /CT/CS /D8/D3 /D8/CW/CT /D1/CP/DC/B9/CX/D1 /D9/D1 /AT/D3 /DB /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /D6/CT/CT/CS /CW/CP/D2/D2/CT/D0 /B4ζ /CT/D5/D9/CP/D0/D7 /D5/D9/CP/D2 /D8/CX/D8 /DD 2β /CX/D2 /CA/CT/CU/BA /BE/B5/BA/C4/CX/D2/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BG/B5 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /CW/CP/D6/CP /B9/D8/CT/D6/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2/BM Ye(θ) =ζ√γ/braceleftbigg D(θ)−1−γ 2γ/bracerightbigg , /B4/BJ/B5/DB/CW/CX /CW /CP/D2 /CQ /CT /D7/D4/D0/CX/D8 /CX/D2 /D8/D3 /D6/CT/CP/D0 /CP/D2/CS /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8/D7/BM/C1/D1(Ye(θ)) =ζ√γ /C1/D1(D(θ)), /B4/BK/B5/CA/CT(Ye(θ)) =ζ√γ/parenleftbigg/CA/CT(D(θ))−1−γ 2γ/parenrightbigg . /B4/BL/B5/BT /D8 /D0/CP/D7/D8/B8 /CP /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D0/CT/D2/CV/D8/CWkrL=ωrL/c /CX/D7 /CX/D2 /D8/D6/D3/B9/CS/D9 /CT/CS/BA/BV/BA /C6/D9/D1/CT/D6/CX /CP/D0 /D8/CT /CW/D2/CX/D5/D9/CT/D7/CC/CW/CT /D9/D2/CZ/D2/D3 /DB/D2/D7 θ, γ∈R+/D7/CP/D8/CX/D7/CU/DD/CX/D2/CV /BX/D5/BA /B4/BJ/B5 /CP/D6/CT /D2 /D9/D1/CT/D6/B9/CX /CP/D0/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CU/D3/D6 /CP /D6/CP/D2/CV/CT /D3/CU /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7/B8 /D4/CP/D6/CP/D1/CT/B9/D8/CT/D6/D7(qr,ζ,ωr) /CQ /CT/CX/D2/CV /D7/CT/D8/BA /CC/CW/CT/DD /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CU/D6/CT/D5/D9/CT/D2 /DD/CP/D2/CS /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D8 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3/CU /D8/CW/CT /D7/D8/CP/D8/CX /D6/CT/CV/CX/D1/CT/BA /CF/CW/CT/D2 /DA /CP/D6/CX/D3/D9/D7 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /CT/DC/CX/D7/D8 /CU/D3/D6 /CP /CV/CX/DA /CT/D2 /D3/D2/B9/AS/CV/D9/D6/CP/D8/CX/D3/D2 /CS/D9/CT /D8/D3 /D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT/DB/CX/D8/CW /D8/CW/CT /D7/CT/DA /CT/D6/CP/D0 /CQ /D3/D6/CT /D6/CT/D7/D3/D2/CP/D2 /CT/D7/B8 /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3/CQ/D7/CT/D6/DA /CT/CS/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D0/DD /CQ /DD /CX/D2 /D6/CT/CP/D7/CX/D2/CV /D8/CW/CT /CQ/D0/D3 /DB/CX/D2/CV /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /D8/CW/CT/D3/D2/CT /CW/CP /DA/CX/D2/CV /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /DA /CP/D0/D9/CT /D3/CUγ /BA/CC/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2 /CX/D7 /D8/D6/CP/D2/D7 /CT/D2/CS/CT/D2 /D8/CP/D0 /CP/D2/CS /D1/CP /DD/CW/CP /DA /CT /CP/D2 /CX/D2/AS/D2/CX/D8/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /D7/D3/D0/D9/D8/CX/D3/D2/D7/BA /CI/CT/D6/D3 /AS/D2/CS/CX/D2/CV /CX/D7 /CS/D3/D2/CT/D9/D7/CX/D2/CV /D8/CW/CT /C8 /D3 /DB /CT/D0/D0 /CW /DD/CQ/D6/CX/CS /D1/CT/D8/CW/D3 /CS /BF/BD/B8 /DB/CW/CX /CW /D3/D1 /CQ/CX/D2/CT/D7 /D8/CW/CT/CP/CS/DA /CP/D2 /D8/CP/CV/CT/D7 /D3/CU /CQ /D3/D8/CW /C6/CT/DB/D8/D3/D2 /D1/CT/D8/CW/D3 /CS /CP/D2/CS /D7 /CP/D0/CT/CS /CV/D6/CP/CS/CX/CT/D2 /D8/D3/D2/CT/BA /BT /D3/D2 /D8/CX/D2 /D9/CP/D8/CX/D3/D2 /D8/CT /CW/D2/CX/D5/D9/CT /CX/D7 /CP/CS/D3/D4/D8/CT/CS /D8/D3 /D4/D6/D3 /DA/CX/CS/CT /CP/D2 /CX/D2/CX/D8/CX/CP/D0 /DA /CP/D0/D9/CT /D8/D3 /D8/CW/CT /CP/D0/CV/D3/D6/CX/D8/CW/D1/BM /D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /CS/D3/D2/CT/CU/D3/D6 /DA /CT/D6/DD /CW/CX/CV/CW /DA /CP/D0/D9/CT/D7 /D3/CUL /B4krL≃30 /B5/B8 /CX/BA/CT/BA/B8 /CU/D3/D6 /CP /D2/CT/CP/D6/D0/DD/D2/D3/D2 /D6/CT/D7/D3/D2/CP/D2 /D8 /D6/CT/CT/CS /B4/CX/BA/CT/BA/B8 /DB/CX/D8/CW /D2/CT/CX/D8/CW/CT/D6 /D1/CP/D7/D7 /D2/D3/D6 /CS/CP/D1/D4/B9/CX/D2/CV/B8 /CW/CT/D6/CT/CP/CU/D8/CT/D6 /CS/CT/D2/D3/D8/CT/CS /D8/CW/CT /D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS /CP/D7/CT/B5/B8 /DB/CW/CT/D6/CT f≃(2n−1)c/4L /CP/D2/CSγ≃1/3 /B4/DB/CX/D8/CWn∈N∗/B5/BA /BU/D3/D6/CT/D0/CT/D2/CV/D8/CW /CX/D7 /D8/CW/CT/D2 /D4/D6/D3/CV/D6/CT/D7/D7/CX/DA /CT/D0/DD /CS/CT /D6/CT/CP/D7/CT/CS /CP/D2/CS /DE/CT/D6/D3 /AS/D2/CS/CX/D2/CV/CU/D3/D6 /CP /CV/CX/DA /CT/D2 /DA /CP/D0/D9/CT /D3/CUL /CX/D7 /CX/D2/CX/D8/CX/CP/D0/CX/DE/CT/CS /DB/CX/D8/CW /D8/CW/CT /D4/CP/CX/D6(θ,γ)/D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/D3/D0/DA/CX/D2/CV /B4/CQ /D3/D6/CT /D7/D0/CX/CV/CW /D8/D0/DD /D0/D3/D2/CV/CT/D6/B5/BA/BW/CT/D4 /CT/D2/CS/CX/D2/CV /D3/D2 /D8/CW/CT /CX/D2/CX/D8/CX/CP/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D0/D9/D8/CX/D3/D2/B4krL= 30 /B5/B8 /CX/D8 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /CT/DC/D4/D0/D3/D6/CT /D8/CW/CT /CQ/D6/CP/D2 /CW/CT/D7 /CP/D7/D7/D3/B9 /CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /D7/D9 /CT/D7/D7/CX/DA /CT /D6/CT/D7/D3/D2/CP/D2 /CT/D7 /D3/CU /D8/CW/CT /CQ /D3/D6/CT/BA /CF/CW/CT/D2/D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT /CP/D2/CS /CQ /D3/D6/CT /CP/D2 /D8/CX/D6/CT/D7/D3/D2/CP/D2 /CT /CV/CT/D8 /D0/D3/D7/CT/D6 /D8/D3 /CT/CP /CW/D3/D8/CW/CT/D6 /B4krL→nπ(n∈N) /B5/B8 /CU/CP/D7/D8 /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT/D8/CW/D6/CT/D7/CW/D3/D0/CS /D6/CT/D5/D9/CX/D6/CT/D7 /CP/CS/CY/D9/D7/D8/D1/CT/D2 /D8 /D3/CU /D8/CW/CT /D7/D8/CT/D4 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/D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CP/D2/CS /D4/D6/CT/D7/D7/D9/D6/CT/CU/D3/D6 /CP /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/BMqr= 0.008 /B8fr= 700 Hz /B8β= 0.05 /BA /CA/CT/D7/D9/D0/D8/D7 /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7/B5 /CP/D2/CS /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /B4/D7/D5/D9/CP/D6/CT/D7/B5 /CU/D6/D3/D1/CA/CT/CU/BA /BE/BN /D3/D9/D6 /D2 /D9/D1/CT/D6/CX /CP/D0 /D6/CT/D7/D9/D0/D8/D7 /B4/D7/D3/D0/CX/CS /D0/CX/D2/CT/D7/B5/BA /CP/D2 /CQ /CT /D4 /D3/CX/D2 /D8/CT/CS /D3/D9/D8/BA /C1/D2 /CP /DA /CP/D0/CX/CS/CP/D8/CX/D3/D2 /D4/CW/CP/D7/CT /D3/CU /D3/D9/D6 /D2 /D9/D1/CT/D6/B9/CX /CP/D0 /CP/D0/CV/D3/D6/CX/D8/CW/D1/D7/B8 /D3/D9/D6 /D6/CT/D7/D9/D0/D8/D7 /B4/D7/CT/CT /BY/CX/CV/BA /BD /CP/D2/CS /BE/B5 /DB /CT/D6/CT /D3/D1/B9/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT/CX/D6 /D3/D2/CT/D7 /B4/D8/CW/CT /CS/CP/D8/CP /CQ /CT/CX/D2/CV /CT/DC/D8/D6/CP /D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT/CX/D6/CP/D6/D8/CX /D0/CT/B5/B8 /D7/D3/D0/DA/CX/D2/CV /CT/DC/CP /D8/D0/DD /D8/CW/CT /D7/CP/D1/CT /CT/D5/D9/CP/D8/CX/D3/D2/BA /BW/CX/AR/CT/D6/CT/D2 /CT/D7/CP/D4/D4 /CT/CP/D6 /CQ /CT/D8 /DB /CT/CT/D2 /D2 /D9/D1/CT/D6/CX /CP/D0 /D6/CT/D7/D9/D0/D8/D7 /D3/D2 /CT/D6/D2/CX/D2/CV /D8/CW/D6/CT/D7/CW/D3/D0/CS/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7/B8 /CP/D4/D4/D6/D3/CP /CW/CX/D2/CV /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT /DB/CW/CT/D2 /D8/CW/CT /D0/CT/D2/CV/D8/CW/CS/CT /D6/CT/CP/D7/CT/D7/BA /C1/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/D7 /D4 /D3/CX/D2 /D8/CT/CS /D3/D9/D8 /D1/D3 /CS/CT/D0 /D0/CX/D1/CX/D8/D7 /D8/CW/CT/CP/D9/D8/CW/D3/D6/D7 /CS/D3 /D2/D3/D8 /D8/CP/CZ /CT /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8/BM /D0/CX/D2/CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /D3/CU /AT/D3 /DB/D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4 /CX/D7 /DA /CP/D0/CX/CS /D3/D2/D0/DD /DB/CW/CX/D0/CT /D6/CT/CT/CS /CW/CP/D2/D2/CT/D0 /CX/D7 /D2/D3/D8 /D0/D3/D7/CT/CS/CP/D8 /D6/CT/D7/D8/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2Pm< Ky 0 /BA /CD/D7/CX/D2/CV /D8/CW/CT /D0/CX/D2/CT/CP/D6 /CU/D3/D6/D1/CU/D3/D6 /CW/CX/CV/CW/CT/D6 /DA /CP/D0/D9/CT/D7 /D3/CUPm /DB /D3/D9/D0/CS /CQ /CT /D1/CT/CP/D2/CX/D2/CV/D0/CT/D7/D7/B8 /CT/DA /CT/D2 /CU/D3/D6/CU/D6/CT/CT /D6/CT/CT/CS /CP/CT/D6/D3/D4/CW/D3/D2/CT/D7/BM /D8/CW/CT /D3/D4 /CT/D2/CX/D2/CV /CU/D9/D2 /D8/CX/D3/D2 /B4/D0/CX/D2/CZ /CT/CS /D8/D3/D8/CW/CT /D6/CT/CT/CS /CS/CX/D7/D4/D0/CP /CT/D1/CT/D2 /D8/B5 /D8/CP/CZ/CX/D2/CV /D4/CP/D6/D8 /CX/D2 /AT/D3 /DB /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /CP/D2 /D2/CT/DA /CT/D6 /CQ /CT /D2/CT/CV/CP/D8/CX/DA /CT/BA /BY /D3/D6 /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /CU/D3/D6 /DB/CW/CX /CW /D6/CT/CT/CS/CQ /CT/CP/D8/D7 /CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D1/D3/D9/D8/CW/D4/CX/CT /CT/B8 /D6/CT/CT/CS /CW/CP/D2/D2/CT/D0 /CX/D7 /D3/D1/D4/D0/CT/D8/CT/D0/DD /D0/D3/D7/CT/CS /CP/D2/CS /D8/CW/CT/D2 /D7/D9/D7/D8/CP/CX/D2/CT/CS /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /CP/D2/D2/D3/D8 /D3 /D9/D6 /CU/D3/D6/CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7 /DB/CW/CT/D6/CT /CF/BU /D8/CW/CT/D3/D6/DD /D4/D6/CT/CS/CX /D8/D7 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS /CP/CQ /D3 /DA /CT /D7/D8/CP/D8/CX /CQ /CT/CP/D8/CX/D2/CV /D6/CT/CT/CS /D4/D6/CT/D7/D7/D9/D6/CT /CQ /DD /CT/DC/D8/CT/D2/CS/CX/D2/CV /D0/CX/D2/B9 /CT/CP/D6/CX/DE/CP/D8/CX/D3/D2 /CQ /CT/DD /D3/D2/CS /D1/D3 /CS/CT/D0 /D0/CX/D1/CX/D8/D7/BA/CC/CW/CT /CU/CP /D8 /D8/CW/CP/D8 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /D6/CT/B9/D1/CP/CX/D2/D7 /D0/D3 /DB /CT/D6 /D8/CW/CP/D2 /BD /CX/D1/D4/D0/CX/CT/D7 /D8/CW/CT /CT/DC/CX/D7/D8/CT/D2 /CT /D3/CU /CP /D1/CP/DC/CX/D1 /D9/D1/D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2 /DD /CU/D3/D6 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/BA /CF /CT /CP/D7/D7/D9/D1/CT /D8/CW/CP/D8/D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1 /DA /CP/D0/D9/CTθ/D1/CP/DC /CX/D7 /CP/D7/D7/D3 /CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /D1/CP/DC/CX/D1 /D9/D1/DA /CP/D0/D9/CT /D3/CUγ /B4/CX/BA/CT/BA/B8/BD/B5/BA /C6/D3 /D8/CW/CT/D3/D6/CT/D8/CX /CP/D0 /D4/D6/D3 /D3/CU /CX/D7 /CV/CX/DA /CT/D2 /CW/CT/D6/CT/B8 /CQ/D9/D8/D8/CW/CX/D7 /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /CX/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2 /CP/D0/D0 /D2 /D9/D1/CT/D6/CX /CP/D0 /D6/CT/D7/D9/D0/D8/D7 /D3/CQ/B9/D8/CP/CX/D2/CT/CS/BA /CC/CW/CT/D2 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2 θ/D1/CP/DC /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD Ye(θ/D1/CP/DC) =ζD(θ/D1/CP/DC), /B4/BD/BC/B5/DB/CW/CX /CW /D0/CT/CP/CS/D7 /D8/D3 /D6/CT/D0/CP/D8/CX/D3/D2 1−θ2/D1/CP/DC=ζ /CA/CT(Ze(θ/D1/CP/DC))>0 /B4/BD/BD/B5/CP/D7 /D8/CW/CT /CQ /D3/D6/CT /CX/D7 /CP /D4/CP/D7/D7/CX/DA /CT /D7/DD/D7/D8/CT/D1/BA /BT/D7 /CP /D3/D2 /D0/D9/D7/CX/D3/D2/B8 /CX/D8 /CX/D7 /D2/D3/D8/D4 /D3/D7/D7/CX/CQ/D0/CT /D8/D3 /D4/D0/CP /DD /D7/CW/CP/D6/D4 /CT/D6 /D8/CW/CP/D2 /CP /CU/D6/CT/D5/D9/CT/D2 /DD /D7/D0/CX/CV/CW /D8/D0/DD /AT/CP/D8/B9/D8/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D6/CT/CT/CS/B9/D0/CX/D4/B9/D1/D3/D9/D8/CW/D4/CX/CT /CT /D7/DD/D7/D8/CT/D1 /D6/CT/D7/D3/D2/CP/D2 /CT /CU/D6/CT/B9/D5/D9/CT/D2 /DD /B8 /D8/CW/CT /CS/CT/DA/CX/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CX/D7 /D0/CP/D8/D8/CT/D6 /CQ /CT/CX/D2/CV /D0/CX/D2/CZ /CT/CS /D8/D3 /D8/CW/CT/D0/D3/D7/D7/CT/D7 /CX/D2 /D8/CW/CT /CQ /D3/D6/CT/BA/BY/BA /C5/CX/D2/CX/D1/D9/D1 /D4 /D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/BM /C1/D1/D4 /D6/D3/DA/CT/CS /D4/D0/CP /DD /CP/CQ/CX/D0/CX/D8 /DD /CU/D3 /D6/CX/D2/D8/CT/D6/CP /D8/CX/D2/CV /D6/CT/D7/D3/D2/CP/D2 /CT/D7/BY /D3/D6 /CT/CP /CW /DA /CP/D0/D9/CT /D3/CUqr /B8 /D8/CW/CT/D6/CT /CT/DC/CX/D7/D8/D7 /D3/D2/CT /D3/D6 /D1/D3/D6/CT /D6/CP/D2/CV/CT/D7 /D3/CU/CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW/D7 /DB/CW/CT/D6/CT /D4/D0/CP /DD /CP/CQ/CX/D0/CX/D8 /DD /CX/D7 /CV/D6/CT/CP/D8/D0/DD /CX/D1/D4/D6/D3 /DA /CT/CS/BA /C1/D2/B9/CS/CT/CT/CS/B8 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D9/D6/DA /CT/D7 /D7/CW/D3 /DB /CP /D1/CX/D2/CX/D1 /D9/D1 /CU/D3/D6 /CP /CT/D6/D8/CP/CX/D2 /DA /CP/D0/D9/CT /D3/CUkrL /B8 /CS/CT/D2/D3/D8/CX/D2/CV /CP/D2 /CX/D2 /D6/CT/CP/D7/CT/CS /CT/CP/D7/CX/D2/CT/D7/D7 /D8/D3/D4/D6/D3 /CS/D9 /CT /D8/CW/CT /D2/D3/D8/CT /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CX/D7 /D0/CT/D2/CV/D8/CW/BA /BT/D7/D7/D3 /CX/B9/CP/D8/CX/D2/CV /CP /D0/CP/D6/CX/D2/CT/D8 /D1/D3/D9/D8/CW/D4/CX/CT /CT /DB/CX/D8/CW /CP /D8/D6/D3/D1 /CQ /D3/D2/CT /D7/D0/CX/CS/CT/B8 /CX/D2/B9/CU/D3/D6/D1/CP/D0 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D3/D2/AS/D6/D1 /D8/CW/CP/D8 /CX/D8 /CX/D7 /CT/CP/D7/CX/CT/D6 /D8/D3 /D4/D6/D3 /CS/D9 /CT/D7/D3/D1/CT /D2/D3/D8/CT/D7 /D8/CW/CP/D2 /D3/D8/CW/CT/D6 /D3/D2/CT/D7/BA /BT/D2/CP/D0/DD/D8/CX /CP/D0 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /CT/DC/B9/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CU/D3/D6 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CW/CP /DA /CT /CQ /CT/CT/D2 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS/BA /CD/D2/B9/CS/CT/D6 /D8/CW/CT /CP/D7/D7/D9/D1/D4/D8/CX/D3/D2 /D8/CW/CP/D8 /D8/CW/CX/D7 /D1/CX/D2/CX/D1/CP/D0 /DA /CP/D0/D9/CT /CX/D7 /D3/CQ/D8/CP/CX/D2/CT/CS/CU/D3/D6 /CP/D2 /CT/D1/CT/D6/CV/CX/D2/CV /CU/D6/CT/D5/D9/CT/D2 /DD /D0/D3 /CP/D8/CT/CS /D0/D3/D7/CT /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/B9/D2/CP/D2 /CT/B8 /CP/D2/CS /D8/CW/CT/D6/CT/CU/D3/D6/CT /CX/D7 /D1/CP/CX/D2/D0/DD /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /D6/CT/CT/CS /CS/CP/D1/D4/B9/CX/D2/CV/B8 /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /CP/D2 /CQ /CT /CX/CV/D2/D3/D6/CT/CS /CA/CT(Ye(ω)) = 0 /B8 /BX/D5/BA /B4/BL/B5/D0/CT/CP/CS/CX/D2/CV /D8/CW /D9/D7 /D8/D3/BM γ=1 1+2 /CA/CT(D(θ)). /B4/BD/BE/B5/C1/D2 /D8/CW/CT /D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS /D1/D3 /CS/CT/D0 /B4D(θ) = 1 /B5/B8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /D4/D6/CT/D7/B9/D7/D9/D6/CT /CX/D7 /CT/D5/D9/CP/D0 /D8/D31/3 /CP/D2/CS /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CU/D6/CT/B9/D5/D9/CT/D2 /CX/CT/D7 /CU/D3/D6 /DB/CW/CX /CW /D8/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8 /D3/CU /D8/CW/CT /CQ /D3/D6/CT /CX/D2/D4/D9/D8/CX/D1/D4 /CT/CS/CP/D2 /CT /DA /CP/D2/CX/D7/CW/CT/D7/B8 /DB/CW/CX /CW /CX/D7 /D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D6/CT/D7/D9/D0/D8/D7 /CP/D0/B9/D6/CT/CP/CS/DD /D4/D9/CQ/D0/CX/D7/CW/CT/CS /BF/BG/BA /CC/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D3 /B9 /D9/D6/D7 /CP/D8 /CP /D1/CP/DC/CX/D1 /D9/D1 /D3/CU /CA/CT(D(θ)) /BM/CA/CT(D(θ)) =1−θ2 (1−θ2)2+(qrθ)2, /B4/BD/BF/B5/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6θ=√1−qr /B4/DB/CW/CX /CW /CX/D7 /D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 θ≃1 /B5/B8 /D8/CW /D9/D7/B8 γ0=qr(2−qr) 2+qr(2−qr), /B4/BD/BG/B5/D2/CT/CP/D6/D0/DD /D4/D6/D3/D4 /D3/D6/D8/CX/D3/D2/CP/D0 /D8/D3qr /CU/D3/D6 /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/D7/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /CT/DA /CP/D0/D9/CP/D8/CT /D8/CW/CT /CT/AR/CT /D8 /D3/CU /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /D3/D2 /D8/CW/CX/D7/D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CP /D3/D2/CT/B9/D1/D3 /CS/CT /D6/CT/D7/D3/D2/CP/D8/D3/D6 /DB/CX/D8/CW/D0/D3/D7/D7/CT/D7 /CX/D7 /D2/D3 /DB /D3/D2/D7/CX/CS/CT/D6/CT/CS/BM Ye(ω) =Yn/parenleftbigg 1+jQn/parenleftbiggω ωn−ωn ω/parenrightbigg/parenrightbigg , /B4/BD/BH/B5/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BG/DB/CW/CT/D6/CTYn /CX/D7 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /CP/D1/D4/D0/CX/D8/D9/CS/CT /D3/CU /D8/CW/CT /CP/CS/D1/CX/D8/D8/CP/D2 /CT/CP/D2/CSQn /CX/D7 /D8/CW/CT /D5/D9/CP/D0/CX/D8 /DD /CU/CP /D8/D3/D6/BN /BX/D5/D7/BA /B4/BK/B5 /CP/D2/CS /B4/BL/B5 /CQ /CT /D3/D1/CT Yn+ζ1−γ 2√γ=ζ√γ1−θ2 (1−θ2)2+(qrθ)2, /B4/BD/BI/B5 YnQn/parenleftbiggθ θn−θn θ/parenrightbigg =−ζ√γqrθ (1−θ2)2+(qrθ)2. /B4/BD/BJ/B5/C1/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CUYn /DB/CX/D8/CW /D8/CW/CT /D0/CT/D2/CV/D8/CW /D3/CU /D8/CW/CT /CQ /D3/D6/CT/B8/D8/CW/CT /CS/CT/D6/CX/DA /CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BD/BI /B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3krL /D0/CT/CP/CS/D7 /D8/D3/CP /D1/CX/D2/CX/D1 /D9/D1 /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT /CU/D9/D2 /D8/CX/D3/D2 γ=f(krL) /CU/D3/D6θ2/D1/CX/D2= 1−qr /CP/D2/CSγ/D1/CX/D2 /D7/D3/D0/D9/D8/CX/D3/D2 /D3/CU /CT/D5/D9/CP/D8/CX/D3/D2 Yn ζ√γ/D1/CX/D2+1−γ/D1/CX/D2 2γ/D1/CX/D2=1 qr(2−qr), /B4/BD/BK/B5/DB/CW/CX /CW /D8/CW/CT /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D7/D3/D0/D9/D8/CX/D3/D2 /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD γmin≃γ0/parenleftbigg 1+2Yn ζ√γ0/parenrightbigg/B4/BD/BL/B5/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6 ωn=ωr/parenleftbigg 1−qr 2+1 2Qn+ζ 2YnQn√qr/parenrightbigg . /B4/BE/BC/B5/BY /D3/D6 /CP/D2 /D3/D4 /CT/D2/BB /D0/D3/D7/CT/CS /DD/D0/CX/D2/CS/CT/D6 ωn= (2n−1)πc/2L /B8 /D8/CW/CT/D6/CT/D7/D9/D0/D8 /CX/D7 (krL)min≃(2n−1)π 2/parenleftbigg 1+qr 2−1 2Qn−ζ 2YnQn√qr/parenrightbigg ./B4/BE/BD/B5/CC /DD/D4/CX /CP/D0 /DA /CP/D0/D9/CT/D7Yn= 1/25 /B8ζ= 0.4 /B8 /CP/D2/CSqr= 0.4 /D0/CT/CP/CS /D8/D3/CP/D2 /CX/D2 /D6/CT/CP/D7/CT /CX/D2γmin /D6/CT/D0/CP/D8/CX/DA /CT /D8/D3γ0 /D3/CU /CP/CQ /D3/D9/D88% /B8 /D3/D2/AS/D6/D1/CX/D2/CV/D8/CW/CT /D4/D6/CT/D4 /D3/D2/CS/CT/D6/CP/D2 /D8 /CT/AR/CT /D8 /D3/CU /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1/D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /CW/D3 /DB /D3/D9/D4/D0/CX/D2/CV /CP /D3/D9/D7/D8/CX /CP/D0 /CP/D2/CS/D1/CT /CW/CP/D2/CX /CP/D0 /D6/CT/D7/D3/D2/CP/D2 /CT/D7 /D3/D9/D0/CS /D6/CT/CS/D9 /CT /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/D3 /DA /CT/D6 /CP /DB/CX/CS/CT/D6 /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /D6/CP/D2/CV/CT/B8 /D8/CW/CT /D2/CT/CX/CV/CW /CQ /D3/D6/CW/D3 /D3 /CS /D3/CU /D8/CW/CT/D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /D1/CT/D2 /D8/CX/D3/D2/CT/CS /D1/CX/D2/CX/D1 /D9/D1 /CW/CP/D7 /CQ /CT/CT/D2 /D7/D8/D9/CS/CX/CT/CS/BA /CD/D7/CX/D2/CV/CP/CV/CP/CX/D2 /CP /D7/CX/D2/CV/D0/CT /CP /D3/D9/D7/D8/CX /CP/D0 /D1/D3 /CS/CT /CU/D3/D6 /D8/CW/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2/B8 /CS/CT/D6/CX/DA /CP/B9/D8/CX/D3/D2 /D3/CU /CP /D4/CP/D6/CP/CQ /D3/D0/CX /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /DB /CP/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT/BA /CF /D6/CX/D8/CX/D2/CV γ=γ/D1/CX/D2(1+ε2) /B8θ2=θ2/D1/CX/D2+δ /CP/D2/CSωn= (ωn)/D1/CX/D2(1+ν) /B8/DB/CX/D8/CWε /B8δ /B8 /CP/D2/CSν /D7/D1/CP/D0/D0 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/B8 /D8/CW/CT /CC /CP /DD/D0/D3/D6 /CT/DC/D4/CP/D2/D7/CX/D3/D2/D3/CU /BX/D5/D7/BA /B4/BD/BI /B5 /CP/D2/CS /B4/BD/BJ /B5 /DB/CX/D8/CW /D6/CT/D7/D4 /CT /D8 /D8/D3 /D8/CW/CT/D7/CT /DA /CP/D0/D9/CT/D7 /D0/CT/CP/CS/D7/D8/D3 /D8/CW/CT /D2/CT/DC/D8 /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/D7/BM ε2∼δ2/(2q2 r), /B4/BE/BE/B5 2q2 rYnQn(δ−2ν) =−δζ√qr. /B4/BE/BF/B5/BY/CX/D2/CP/D0/D0/DD /B8 /D2/CT/CP/D6 /D8/CW/CT /D1/CX/D2/CX/D1 /D9/D1 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CS/CT/D4 /CT/D2/B9/CS/CT/D2 /CT /D3/D2 /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD γ=γ/D1/CX/D2 1+2qr/parenleftbigg q3/2 r+ζ 2YnQn/parenrightbigg2/parenleftbiggkrL−krL/D1/CX/D2 krL/D1/CX/D2/parenrightbigg2 ./B4/BE/BG/B5/C1/D2 /CP /AS/D6/D7/D8 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B8 /D8/CW/CT /CP/D4 /CT/D6/D8/D9/D6/CT /D3/CU /D8/CW/CT /CP/D4/D4/D6/D3 /DC/CX/B9/D1/CP/D8/CT/CS /D4/CP/D6/CP/CQ /D3/D0/CP/B8 /D8/CW /D9/D7 /D8/CW/CT /DB/CX/CS/D8/CW /D3/CU /D8/CW/CT /D6/CP/D2/CV/CT /CU/D3/D6 /DB/CW/CX /CW/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CX/D7 /D0/D3 /DB /CT/D6/CT/CS/B8 /CX/D7 /D1/CP/CX/D2/D0/DD /D3/D2 /D8/D6/D3/D0/D0/CT/CS /CQ /DD /D8/CW/CT /D1 /D9/D7/CX /CX/CP/D2 /CT/D1 /CQ /D3/D9 /CW /D9/D6/CT/B8 /CX/BA/CT/BA/B8 /CQ /DD /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /CP/D2/CS /D0/CX/D4/D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS/BA /CC/CW/CX/D7 /D1/CT/CP/D2/D7 /D8/CW/CP/D8/B8 /D8/CW/CP/D2/CZ/D7 /D8/D3 /CX/D8/D7 /CT/D1/B9/CQ /D3/D9 /CW /D9/D6/CT/B8 /D8/CW/CT /D4/D0/CP /DD /CT/D6 /CP/D2 /CT/DC/D4 /CT /D8 /CP/D2 /CT/CP/D7/CX/CT/D6 /D4/D6/D3 /CS/D9 /D8/CX/D3/D2 /D3/CU/D8/D3/D2/CT/D7 /CU/D3/D6 /CT/D6/D8/CP/CX/D2 /D2/D3/D8/CT/D7/BA/BY /D3/D6 /CP /D0/D3/D7/D7/DD /DD/D0/CX/D2/CS/D6/CX /CP/D0 /D3/D4 /CT/D2/BB /D0/D3/D7/CT/CS /CQ /D3/D6/CT/B8 /D1/D3 /CS/CP/D0 /CT/DC/B9/D4/CP/D2/D7/CX/D3/D2 /D3/CU /CX/D2/D4/D9/D8 /CX/D1/D4 /CT/CS/CP/D2 /CT /CV/CX/DA /CT/D7YnQn=ωnL/2c= (2n−1)π/4 /D7/D3 /D8/CW/CP/D8 /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /CS/D3 /D2/D3/D8 /D7/CT/CT/D1 /D8/D3 /CW/CP /DA /CT /CP/CV/D6/CT/CP/D8 /CX/D2/AT/D9/CT/D2 /CT /D3/D2 /D4/D0/CP /DD/CX/D2/CV /CU/CP /CX/D0/CX/D8 /DD /B8 /CP/D8 /D0/CT/CP/D7/D8 /DB/CW/CT/D2 /D3/D2/D7/CX/CS/B9/CT/D6/CX/D2/CV /D1/CX/D2/CX/D1/CP/D0 /CQ/D0/D3 /DB/CX/D2/CV /D4/D6/CT/D7/D7/D9/D6/CT γmin /B4Qn /CS/D3 /CT/D7 /D2/D3/D8 /CP/D4/D4 /CT/CP/D6/CP/D0/D3/D2/CT /CX/D2 /AS/D6/D7/D8 /D3/D6/CS/CT/D6 /CP/D0 /D9/D0/CP/D8/CX/D3/D2/B5/BA /C7/D2 /D8/CW/CT /D3/D2 /D8/D6/CP/D6/DD /B8 /D8/CW/CT/DD/CP/D6/CT /CT/D7/D7/CT/D2 /D8/CX/CP/D0 /CU/D3/D6 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D8/CW/CT /CT/DC/D8/CX/D2 /D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/D4/CW/CT/D2/D3/D1/CT/D2/D3/D2 /BF/BH/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2 /D8/CW/CT /D6/CT/CT/CS /CX/D7 /CW/CT/D0/CS /D1/D3/D8/CX/D3/D2/D0/CT/D7/D7/D0/DD/CP/CV/CP/CX/D2/D7/D8 /D8/CW/CT /D0/CP /DD /BA/C1 /C1 /C1/BA /C5/C7/BW/BX/C4 /C1/C5/C8/CA/C7 /CE/BX/C5/BX/C6/CC/CB/C4/CP/D7/D8 /CU/D3/D9/D6 /CS/CT /CP/CS/CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /CU/D6/D9/CX/D8/CU/D9/D0 /CX/D2 /D4/CW /DD/D7/CX /CP/D0 /D1/D3 /CS/B9/CT/D0/CX/D2/CV /D3/CU /D1 /D9/D7/CX /CP/D0 /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7/B8 /CT/D7/D4 /CT /CX/CP/D0/D0/DD /CU/D3/D6 /D7/CX/D2/CV/D0/CT /D6/CT/CT/CS/CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7/BA /C8/CX/D4 /CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /D8/CW/CT /CU/D3 /D9/D7 /D3/CU /CP /CV/D6/CT/CP/D8 /D2 /D9/D1/B9/CQ /CT/D6 /D3/CU /D7/D8/D9/CS/CX/CT/D7 /D7/CX/D2 /CT /BU/CT/D2/CP/CS/CT /BF/BI/B8 /CP/D7 /DB /CT/D0/D0 /CP/D7 /D8/CW/CT /CS/CT/D7 /D6/CX/D4/D8/CX/D3/D2 /D3/CU/D4 /CT /D9/D0/CX/CP/D6/CX/D8/CX/CT/D7 /D3/CU /D8/CW/CT /AT/D3 /DB /B4/BU/CP /CZ/D9/D7 /BF/B8 /C0/CX/D6/D7 /CW /CQ /CT/D6/CV /BE/BI/B8 /CP/D2/CS /BW/CP/D0/B9/D1/D3/D2 /D8 /CT/D8 /CP/D0/BA /BE/BJ/B5/BA /CC/CW/CT /CP/CX/D1 /CW/CT/D6/CT /CX/D7 /D8/D3 /D8/D6/DD /D8/D3 /D6/CT/CS/D9 /CT /CS/CX/D7 /D6/CT/D4/CP/D2/B9 /CX/CT/D7 /CQ /CT/D8 /DB /CT/CT/D2 /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CP/D2/CS /D8/CW/CT/D3/D6/DD /B8 /CQ/CP/D7/CT/CS /D3/D2 /D7/D3/D1/CT/D3/CU /D8/CW/D3/D7/CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2/D7 /DB/CW/CX /CW /D0/D3 /D3/CZ /D6/CT/D0/CT/DA /CP/D2 /D8 /D8/D3 /D8/CW/CT /D7/D8/D9/CS/DD /D3/CU/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /C1/D2 /CP /D6/CT/CP/D0 /D0/CP/D6/CX/D2/CT/D8/B9/D4/D0/CP /DD /CT/D6 /D7/DD/D7/D8/CT/D1/B8 /D8/CW/CT/DA /D3 /CP/D0 /D8/D6/CP /D8 /D1/CP /DD /CW/CP /DA /CT /CP/D2 /CT/AR/CT /D8 /D3/D2 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BF/BJ /B5/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /D9/D7/CT /D3/CU /D4 /D3/D6/D3/D9/D7 /D1/CP/D8/CT/D6/CX/CP/D0/CX/D2 /D8/CW/CT /CF/BU /CP/D4/D4/CP/D6/CP/D8/D9/D7 /D7/D9/CV/CV/CT/D7/D8/D7 /D8/CW/CP/D8 /CP /D3/D9/D7/D8/CX /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2/CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CW/CP/D1 /CQ /CT/D6 /CP/D2/CS /D8/CW/CT /D1/D3/D9/D8/CW/D4/CX/CT /CT /CW/CP/D7 /CQ /CT/CT/D2 /CP/D2/B9 /CT/D0/CT/CS /D3/D6 /CP/D8 /D0/CT/CP/D7/D8 /D7/D8/D6/D3/D2/CV/D0/DD /D6/CT/CS/D9 /CT/CS/BA /CC/CW/CT/D6/CT/CU/D3/D6/CT /DB /CT /CS/D3 /D2/D3/D8/CS/CX/D7 /D9/D7/D7 /D8/CW/CT /CT/AR/CT /D8 /D3/CU /D8/CW/CT /D9/D4/D7/D8/D6/CT/CP/D1 /D4/CP/D6/D8/BA/BT/BA /CE/CX/D7 /D3/D8/CW/CT/D6/D1/CP/D0 /D0/D3/D7/D7/CT/D7 /D1/D3 /CS/CT/D0 /CP/D2/CS /DA/CT/D2/CP /D3/D2/D8/D6/CP /D8/CP/C1/D8 /D7/CW/D3/D9/D0/CS /CQ /CT /D2/D3/D8/CX /CT/CS /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP /CV/CP/D4 /CX/D2 /D4/D6/CT/D7/D7/D9/D6/CT/D8/CW/D6/CT/D7/CW/D3/D0/CS /DA /CP/D0/D9/CT/D7 /CQ /CT/D8 /DB /CT/CT/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /CP/D2/CS /D8/CW/CT/D3/D6/DD /CX/D2 /CF/BU/CP/D6/D8/CX /D0/CT/BA /CC/CW/CX/D7 /D3 /D9/D6/D7 /CT/DA /CT/D2 /CU/D3/D6 /D0/D3/D2/CV /CQ /D3/D6/CT/D7 /DB/CW/CT/D2 /D6/CT/CT/CS /CS/DD/B9/D2/CP/D1/CX /CP/D0 /CQ /CT/CW/CP /DA/CX/D3/D6 /D7/CW/D3/D9/D0/CS /D2/D3/D8 /CS/CT/DA/CX/CP/D8/CT /CU/D6/D3/D1 /D8/CW/CT /CX/CS/CT/CP/D0 /D7/D4/D6/CX/D2/CV/D1/D3 /CS/CT/D0 /B4/CQ /CT /CP/D9/D7/CT /D3/CU /CP/D2 /CT/D1/CT/D6/CV/CX/D2/CV /CU/D6/CT/D5/D9/CT/D2 /DD /D1 /D9 /CW /D7/D1/CP/D0/D0/CT/D6/D8/CW/CP/D2ωr /B5/BA /BY /D3/D6 /D8/CW/CP/D8 /CP/D7/CT/B8 /C3/CT/D6/CV/D3/D1/CP/D6/CS /CT/D8 /CP/D0/BA /BE/BD/D4/D6/D3 /DA/CX/CS/CT/CS/CP/D2 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /CU/D3/D6/D1 /D9/D0/CP /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /CQ /D3/D8/CW /D6/CT/CT/CS/CS/DD/D2/CP/D1/CX /D7 /CP/D2/CS /CQ /D3/D6/CT /D0/D3/D7/D7/CT/D7 /D8/CW/CP/D8 /CP/D2 /CQ /CT /CT/DC/D8/CT/D2/CS/CT/CS /D8/D3 γ≃1−θ2 n 3−θ2n+2 /CA/CT(Ye(θn)) 3√ 3ζ, θn= (2n−1)π 2krL. /B4/BE/BH/B5 θn /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CTn /D8/CW /D6/CT/D7/D3/D2/CP/D2 /CT /CU/D6/CT/D5/D9/CT/D2 /DD /D3/CU /D8/CW/CT/CQ /D3/D6/CT/BA /C1/D2 /D3/D1/D4/CP/D6/CX/D7/D3/D2 /DB/CX/D8/CW /CX/CS/CT/CP/D0 /D1/D3 /CS/CT/D0 /B4/D0/D3/D7/D7/D0/CT/D7/D7 /CQ /D3/D6/CT /CP/D2/CS/D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS/B8 /CX/BA/CT/BA/B8γ= 1/3 /B5/B8 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D3/D6/D6/CT /D8/CX/DA /CT /D8/CT/D6/D1/D7/CP/D6/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CX/D2 /BX/D5/BA /B4/BE/BH /B5/B8 /D3/D2/CT /D0/D3 /DB /CT/D6/CX/D2/CV /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS /CS/D9/CT /D8/D3 /D8/CW/CT /D3/D0/D0/CP/CQ /D3/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT 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/C8/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/D7/DB /D3/D9/D0/CS /CQ /CT /CX/D2/CP /D9/D6/CP/D8/CT /DB/CW/CT/D2 /CW/CX/CV/CW/CT/D6/B9/D3/D6/CS/CT/D6 /D1/D3 /CS/CT/D7 /D3/CU /CQ /D3/D6/CT/D3/D7 /CX/D0/D0/CP/D8/CT /AS/D6/D7/D8/BA/CC/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS /CU/D3/D6/D1 /D9/D0/CP /CU/D3/D6 /D8/CW/CT /CX/D2/D4/D9/D8 /CX/D1/D4 /CT/CS/CP/D2 /CT /DB/CX/D0/D0 /CQ /CT/CW/CT/D2 /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS/BM Ze(ω) =jZctan(kL) /DB/CX/D8/CWjk=jω c+η/radicalbigg jω c /B4/BE/BI/B5/DB/CW/CT/D6/CTη /CX/D7 /CP /D3 /CTꜶ /CX/CT/D2 /D8 /D5/D9/CP/D2 /D8/CX/CU/DD/CX/D2/CV /D8/CW/CT /DA/CX/D7 /D3/D8/CW/CT/D6/D1/CP/D0/CQ /D3/D9/D2/CS/CP/D6/DD /D0/CP /DD /CT/D6/D7/B8 /CT/D5/D9/CP/D0/D7 /D8/D30.0421 /CX/D2mks /D9/D2/CX/D8/D7 /CU/D3/D6 /CP 7mm /D6/CP/CS/CX/D9/D7 /DD/D0/CX/D2/CS/CT/D6/BA /CC/CW/CX/D7 /D1/D3 /CS/CT/D0 /CX/D2 /D8/D6/D3 /CS/D9 /CT/D7 /CS/CX/D7/D7/CX/D4/CP/B9/D8/CX/D3/D2 /CP/D2/CS /CS/CX/D7/D4 /CT/D6/D7/CX/D3/D2 /CP/D2/CS /D0/CT/CP/CS/D7 /D8/D3 /CP /DE/CT/D6/D3 /DA /CP/D0/D9/CT/CS 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/D3/D2/D7/CT/D5/D9/CT/D2 /CT /CX/D7 /D8/CW/CP/D8 /D8/CW/CT/D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CX/D2 /D6/CT/CP/D7/CT/D7 /CP/D7 /D8/CW/CT /D6/CT/D7/D3/D2/CP/D8/D3/D6 /D0/CT/D2/CV/D8/CW/CT/D2/D7/BA/BT/D2/D3/D8/CW/CT/D6 /CT/AR/CT /D8 /D1/CP /DD /D3 /D9/D6 /CP/D2/CS /D1/D3 /CS/CX/CU/DD /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS/BA /C0/CX/D6/D7 /CW /CQ /CT/D6/CV /BE/BI/CP/D2/CS /C0/CX/D6/D7 /CW /CQ /CT/D6/CV /CT/D8 /CP/D0/BA /BF/BK/CQ/D6/D3/D9/CV/CW /D8 /D8/D3 /CP/D8/B9/D8/CT/D2 /D8/CX/D3/D2 /D3/D2 /DA /CT/D2/CP /D3/D2 /D8/D6/CP /D8/CP /D4/CW/CT/D2/D3/D1/CT/D2/D3/D2/BM /CS/D9/CT /D8/D3 /D7/CW/CP/D6/D4/D2/CT/D7/D7/D3/CU /CT/CS/CV/CT/D7/B8 /AT/D3 /DB /D7/CT/D4/CP/D6/CP/D8/CX/D3/D2 /D1/CP /DD /D6/CT/D7/D9/D0/D8 /CX/D2 /D8/CW/CT /CU/D3/D6/D1/CP/D8/CX/D3/D2 /D3/CU /CP/CU/D6/CT/CT /CY/CT/D8 /CX/D2 /D6/CT/CT/CS /CW/CP/D2/D2/CT/D0/B8 /D8/CW/CX/D7 /D3/D2 /D8/D6/CP /D8/CX/D3/D2 /CT/AR/CT /D8 /D6/CT/D7/D9/D0/D8/CX/D2/CV/CX/D2 /CP /CY/CT/D8 /D6/D3/D7/D7/B9/D7/CT 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/CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D6/CT/CP/D0/CX/D7/D8/CX /D0/D3/D7/D7/CT/D7 /CP/D2/CS /DA /CT/D2/CP /D3/D2 /D8/D6/CP /D8/CP/D4/CW/CT/D2/D3/D1/CT/D2/D3/D2 /D6/CT/CS/D9 /CT/D7 /CS/CX/D7 /D6/CT/D4/CP/D2 /CX/CT/D7 /CX/D2 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS/B8 /CT/D7/D4 /CT /CX/CP/D0/D0/DD /CU/D3/D6 /CW/CX/CV/CW /DA /CP/D0/D9/CT/D7 /D3/CU /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /CP/D2/CS /CU/D3/D6/CP /D7/D8/D6/D3/D2/CV/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/B8 /CX/BA/CT/BA/B8 /DB/CW/CT/D2 /D6/CT/CT/CS /CS/DD/D2/CP/D1/CX /D7 /CW/CP/D7/CP /D7/D1/CP/D0/D0 /CX/D2/AT/D9/CT/D2 /CT /D3/D2 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /C6/CT/DA /CT/D6/D8/CW/CT/D0/CT/D7/D7/D4/D6/CT/D7/D7/D9/D6/CT /DA /CP/D0/D9/CT/D7 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D0/DD /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /CF/BU /D7/D8/CX/D0/D0 /D6/CT/B9/D1/CP/CX/D2 /D5/D9/CX/D8/CT /CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D3/D2/CT/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /D8/CW/CT/D1/D3 /CS/CX/AS/CT/CS /D1/D3 /CS/CT/D0/B8 /DB/CW/CX/D0/CT /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CP/D6/CT /D9/D2/CP/D0/D8/CT/D6/CT/CS/BA/BT/D2 /CP/D8/D8/CT/D1/D4/D8 /D8/D3 /CT/DC/D4/D0/CP/CX/D2 /D8/CW/CT /CS/CX/D7 /D6/CT/D4/CP/D2 /CX/CT/D7 /CX/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /DB/CX/D0/D0 /CQ /CT /D4/D6/D3 /DA/CX/CS/CT/CS /CX/D2 /CB/CT /BA /C1/CE/BA /BY/C1/BZ/BA /BF/BA /CA/CT/D7/D9/D0/D8/D7 /DB/CX/D8/CW /D8/CW/CT /D1/D3 /CS/CT/D0 /D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /DA /CT/D2/CP /D3/D2 /D8/D6/CP /D8/CP/D4/CW/CT/D2/D3/D1/CT/D2/D3/D2 /B4/D8/CW/CX/D2 /D0/CX/D2/CT/B5 /CP/D2/CS /DA/CX/D7 /D3/D8/CW/CT/D6/D1/CP/D0 /D0/D3/D7/D7/CT/D7 /B4/D8/CW/CX /CZ /D0/CX/D2/CT/B5/B4/D7/CP/D1/CT /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D7 /CX/D2 /BY/CX/CV/BA /BD /B5/BA /CF/BU /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /CP/D0/D7/D3 /D6/CT /CP/D0/D0/CT/CS/B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/B5/BA/BU/BA /CA/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB/CC/CW/CT /CX/D2/AT/D9/CT/D2 /CT /D3/CU /D8/CW/CT /D6/CT/CT/CS /CX/D7 /D2/D3/D8 /D0/CX/D1/CX/D8/CT/CS /D8/D3 /CX/D8/D7 /D6/CT/D7/D3/D2/CP/D2 /CT/BA/CB/CX/D2 /CT /C6/CT/CS/CT/D6/DA /CT/CT/D2 /BG/BD/CP/D2/CS /CC/CW/D3/D1/D4/D7/D3/D2 /BG/BE/B8 /CX/D8 /CX/D7 /D4/D6/D3 /DA /CT/CS /D8/CW/CP/D8 /D8/CW/CT/AT/D3 /DB /CT/D2 /D8/CT/D6/CX/D2/CV /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /D6/CT/CT/CS /D3/D4 /CT/D2/CX/D2/CV /CX/D7 /CS/CX/DA/CX/CS/CT/CS /CX/D2 /D3/D2/CT/D4/CP/D6/D8 /CT/DC /CX/D8/CX/D2/CV /D8/CW/CT /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CP/D2/CS /CP/D2/D3/D8/CW/CT/D6 /D4/CP/D6/D8 /CX/D2/CS/D9 /CT/CS /CQ /DD/D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2/BA /C1/D2 /CU/CP /D8/B8 /D8/CW/CT /DA/CX/CQ/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D9/D6/CU/CP /CT /D3/CU/D8/CW/CT /D6/CT/CT/CS /D4/D6/D3 /CS/D9 /CT/D7 /CP/D2 /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /D3/D7 /CX/D0/D0/CP/D8/CX/D2/CV /AT/D3 /DB/BA /CC/CW /D9/D7/D8/CW/CT /CT/D2 /D8/CT/D6/CX/D2/CV /AT/D3 /DBU /CP/D2 /CQ /CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7 U(ω) =Ye(ω)P(ω)+Sr(jωY(ω)), /B4/BE/BJ/B5/DB/CW/CT/D6/CTSr /CX/D7 /D8/CW/CT /CT/AR/CT /D8/CX/DA /CT /CP/D6/CT/CP /D3/CU /D8/CW/CT /DA/CX/CQ/D6/CP/D8/CX/D2/CV /D6/CT/CT/CS /D6/CT/D0/CP/D8/CT/CS/D8/D3 /D8/CW/CT /D8/CX/D4 /CS/CX/D7/D4/D0/CP /CT/D1/CT/D2 /D8 y(t) /BA /BT/D0/D8/CT/D6/D2/CP/D8/CT/D0/DD /B8 /CP /D0/CT/D2/CV/D8/CW∆l /CP/D2 /CQ /CT /CP/D7/D7/D3 /CX/CP/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /AS /D8/CX/D8/CX/D3/D9/D7 /DA /D3/D0/D9/D1/CT /DB/CW/CT/D6/CT /D8/CW/CT/D6/CT/CT/CS /D7/DB/CX/D2/CV/D7/BA /BW/CP/D0/D1/D3/D2 /D8 /CT/D8 /CP/D0/BA /BD/BF/D6/CT/D4 /D3/D6/D8/CT/CS /D8 /DD/D4/CX /CP/D0 /DA /CP/D0/D9/CT/D7/D3/CU10mm /CU/D3/D6 /CP /D0/CP/D6/CX/D2/CT/D8/BA /C6/CT/CS/CT/D6/DA /CT/CT/D2 /BG/BD/D0/CX/D2/CZ /CT/CS∆l /D8/D3 /D6/CT/CT/CS/D7/D8/D6/CT/D2/CV/D8/CW /B4/D3/D6 /CW/CP/D6 /CS/D2/CT/D7/D7 /B5/BM∆l /D1/CP /DD /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/D0/DD /DA /CP/D6/DD /CU/D6/D3/D1 6mm /B4/D7/D8/D6/D3/D2/CV /D6/CT/CT/CS/D7/B5 /D8/D39mm /B4/D7/D3/CU/D8/CT/D6 /D6/CT/CT/CS/D7/B5/BA /CC/CW/CT/D7/CT /DA /CP/D0/D9/CT/D7/CQ /CT/CX/D2/CV /D7/D1/CP/D0/D0 /D3/D1/D4/CP/D6/CT/CS /D8/D3 /D0/CP/D6/CX/D2/CT/D8 /CS/CX/D1/CT/D2/D7/CX/D3/D2/D7/B8 /D8/CW/CT /D6/CT/CT/CS/D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /CP/D2 /CQ /CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /D8/CW/D6/D3/D9/CV/CW /CP /D1/CT/D6/CT/D0/CT/D2/CV/D8/CW /D3/D6/D6/CT /D8/CX/D3/D2 /CX/D2 /D3/D1/D1/D3/D2 /DB /D3/D6/CZ/B8 /CQ/D9/D8 /CX/D8/D7 /CX/D2/AT/D9/CT/D2 /CT /D3/D2/D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CP /D3/D9/D7/D8/CX /D6/CT/D7/D3/D2/CP/D8/D3/D6 /CP/D2/CS /D6/CT/CT/CS /CX/D7/D2/D3/D8 /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /D3/D2 /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CU/D6/CT/D5/D9/CT/D2 /DD /B8 /CP/D7 /CX/D8 /CX/D7 /D7/D8/D9/CS/CX/CT/CS/D2/D3 /DB/BA/C1/D2 /CP /AS/D6/D7/D8 /D7/D8/CT/D4/B8 /D8/CW/CX/D7 /CT/AR/CT /D8 /CX/D7 /D3/D2/D7/CX/CS/CT/D6/CT/CS /D7/CT/D4/CP/D6/CP/D8/CT/D0/DD /B8 /CP/D0/D0/D0/D3/D7/D7/CT/D7 /CQ /CT/CX/D2/CV /CX/CV/D2/D3/D6/CT/CS /B4qr= 0 /CP/D2/CSη= 0 /B5/BA /BX/D5/D9/CP/D8/CX/D3/D2 /B4/BE/BJ /B5 /D3/D9/D4/D0/CT/CS /D8/D3 /BX/D5/BA /B4/BJ /B5 /D0/CT/CP/CS/D7 /D8/D3 /D8/CW/CT /CU/D3/D0/D0/D3 /DB/CX/D2/CV /D7/DD/D7/D8/CT/D1/BM − /C1/D1(Ye) =kr∆lθ 1−θ2, /B4/BE/BK/B5 1 1−θ2−1−γ 2γ= 0⇔γ=1−θ2 3−θ2. /B4/BE/BL/B5/BT/D7 /D7/CT/CT/D2 /D4/D6/CT/DA/CX/D3/D9/D7/D0/DD /B8 /D7/CT/DA /CT/D6/CP/D0 /CU/D6/CT/D5/D9/CT/D2 /DD /D7/D3/D0/D9/D8/CX/D3/D2/D7 θ /CT/DC/CX/D7/D8 /CU/D3/D6/CS/CX/AR/CT/D6/CT/D2 /D8 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/D7 γ /BA /BX/DC/CP/D1/CX/D2/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BE/BL /B5/CP/D7 /CP /CU/D9/D2 /D8/CX/D3/D2 γ=f(θ) /B4/CU/D3/D6θ <1 /B5 /D6/CT/DA /CT/CP/D0/D7 /D8/CW/CP/D8 /D8/CW/CT /D7/D3/B9/D0/D9/D8/CX/D3/D2 /CW/CP /DA/CX/D2/CV /D8/CW/CT /D0/D3 /DB /CT/D7/D8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CX/D7 /D8/CW/CT /D3/D2/CT /CQ /CT/CX/D2/CV /D8/CW/CT /D0/D3/D7/CT/D7/D8 /D8/D3 /D8/CW/CT /D6/CT/CT/CS /CU/D6/CT/D5/D9/CT/D2 /DD /BA/BT/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/D7 /CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS /CX/D2 /D7/D3/D1/CT /D7/CX/D8/D9/CP/D8/CX/D3/D2/D7/BA/CF/CW/CT/D2 /D4/D0/CP /DD/CX/D2/CV /D0/D3/D7/CT /D8/D3 /CP /CQ /D3/D6/CT /D6/CT/D7/D3/D2/CP/D2 /CT /CU/D6/CT/D5/D9/CT/D2 /DD θn≪/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BI/BY/C1/BZ/BA /BG/BA /BV/D3/D1/D4/CP/D6/CX/D2/CV /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /CT/AR/CT /D8 /CP/D2/CS/D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /CT/AR/CT /D8 /B4∆l= 12mm /CP/D2/CS /D3/D8/CW/CT/D6 /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D7/CX/D2 /BY/CX/CV/BA /BD/B5/BM /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /D3/D2/D0/DD /B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/D7/B5/B8/D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2/D0/DD /B4/CS/CP/D7/CW/B9/CS/D3/D8/D8/CT/CS /D0/CX/D2/CT/D7/B5/B8 /CP/D2/CS /CQ /D3/D8/CW /CT/AR/CT /D8/D7 /B4/D4/D0/CP/CX/D2/D0/CX/D2/CT/D7/B5/BA /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT /CP/D0/D0/CT/CS /B4/D7/D5/D9/CP/D6/CT/D7/B5/BA 1 /B8 /D8/CW/CT /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D8/CT/D6/D1 /CX/D2 /BX/D5/BA /B4/BE/BK /B5 /CX/D7 /D7/D1/CP/D0/D0/B8 /D7/D3 /D8/CW/CP/D8/B8 /DB/CX/D8/CW Ye=−jcot(θkrL) /B8 /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /CP /D8/D7/D1/CT/D6/CT/D0/DD /CP/D7 /CP /D0/CT/D2/CV/D8/CW /D3/D6/D6/CT /D8/CX/D3/D2/BM ∆l1 1−θ2n /DB/CW/CT/D6/CTθn=(2n−1)π 2krL. /B4/BF/BC/B5/CC/CW/CX/D7 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2 /CX/D7 /DA /CP/D0/CX/CS /DB/CW/CT/D2 /D8/CW/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS /CQ /D3/D6/CT/CU/D6/CT/D5/D9/CT/D2 /DD θn /D6/CT/D1/CP/CX/D2/D7 /D7/D1/CP/D0/D0/CT/D6 /D8/CW/CP/D2 /D9/D2/CX/D8 /DD /BA /BT/D2 /CT/D5/D9/CX/DA /CP/D0/CT/D2 /D8/CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CT/CS /D0/CT/D2/CV/D8/CW /D3/D6/D6/CT /D8/CX/D3/D2 /CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS /CU/D3/D6 /D8/CW/CT/CT/AR/CT /D8 /D3/CU /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2 /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CU/D6/D3/D1 /BX/D5/BA /B4/BK/B5/BM ∆lq≃ζqr√ 3kr. /B4/BF/BD/B5/BT /D8/D6/CX/CP/D0 /CP/D2/CS /CT/D6/D6/D3/D6 /D4/D6/D3 /CT/CS/D9/D6/CT /CW/CP/D7 /CQ /CT/CT/D2 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /D8/D3 /CP/CS/CY/D9/D7/D8/D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /BA /CE /CT/D6/DD /CU/CT/DB /CX/D8/CT/D6/CP/D8/CX/D3/D2/D7 /DB /CT/D6/CT /D2/CT/CT/CS/CT/CS/D8/D3 /CT/DC/CW/CX/CQ/CX/D8 /CP /DA /CP/D0/D9/CT /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /B4∆l≃ 12mm /B5 /CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /B4∆lq≃2mm /B5/CX/D2 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2/D7 /D3/CU /BY/CX/CV/D7/BA /BD /CP/D2/CS /BG/BA /CF/CW/CT/D2 /CP /D3/D9/D7/D8/CX /CP/D0 /CP/D2/CS/D1/CT /CW/CP/D2/CX /CP/D0 /D6/CT/D7/D3/D2/CP/D2 /CT/D7 /CP/D6/CT /DA /CT/D6/DD /D0/D3/D7/CT /B4θ= 1−ε /CP/D2/CSθn= 1−εn /B5/B8 /CP /D7/CT /D3/D2/CS/B9/D3/D6/CS/CT/D6 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CP/D2 /CQ /CT /CS/CT/CS/D9 /CT/CS/BM ε=εn 2/parenleftigg 1+/radicaligg 1+2∆l Lε2n/parenrightigg , /B4/BF/BE/B5/D8/CW/CT /CP/D4/D4/CP/D6/CX/D8/CX/D3/D2 /D3/CU /CP /D7/D5/D9/CP/D6/CT /D6/D3 /D3/D8 /CQ /CT/CX/D2/CV /D8 /DD/D4/CX /CP/D0 /D3/CU /D1/D3 /CS/CT /D3/D9/B9/D4/D0/CX/D2/CV/B8 /D1/CP/CZ/CX/D2/CV /D8/CW/CT /CP /CW/CX/CT/DA /CT/D1/CT/D2 /D8 /D3/CU /CP/D2/CP/D0/DD/D8/CX /CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7/CS/CXꜶ /D9/D0/D8/BA /CC/CW/CT/D2/B8 /DB/CW/CT/D2 /D8/CW/CT /CQ /D3/D6/CT /D0/CT/D2/CV/D8/CW /CS/CT /D6/CT/CP/D7/CT/D7 /CT/D2/D3/D9/CV/CW/D7/D3 /D8/CW/CP/D8 /D3/D2/CT /D3/CU /CX/D8/D7 /D6/CT/D7/D3/D2/CP/D2 /CT/D7 /CX/D2 /D6/CT/CP/D7/CT/D7 /CP/CQ /D3 /DA /CT /D8/CW/CT /D6/CT/CT/CS /D3/D2/CT/B4θn>1 /B5/B8 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /CP/D4/D4/D6/D3/CP /CW/CT/D7 /D8/CW/CT /D6/CT/CT/CS/D3/D2/CT /D9/D2 /D8/CX/D0 /D8/CW/CT /CX/D2 /D8/CT/D6/D7/CT /D8/CX/D3/D2 /D4 /D3/CX/D2 /D8 /CS/CX/D7/CP/D4/D4 /CT/CP/D6/D7 /CU/D3/D6krL=nπ/B4/D7/CT/CT /BY/CX/CV/BA /BH /B5/BA /C6/CT/CP/D6 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT/B8 /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /CP/D4/D4/D6/D3 /DC/B9/CX/D1/CP/D8/CX/D3/D2/D7 /CP/D2 /CQ /CT /CS/CT/D6/CX/DA /CT/CS/BM θ≃1−1 2kr∆ltan(krL), /B4/BF/BF/B5 γ≃1 2kr∆ltan(krL). /B4/BF/BG/B51rL k2/ π=θ1rL k2/ π2 =θ13rL k2/ π3 =θ θ0r)2θ−1 (/ l ∆ k θ r)L k θ(t o c/BY/C1/BZ/BA /BH/BA /BZ/D6/CP/D4/CW/CX /CP/D0 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /BX/D5/BA /B4/BE/BK /B5 /CV/CX/DA/CX/D2/CV /D3/D7 /CX/D0/D0/CP/B9/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /CP/D8 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BM /D0/CT/CU/D8/B9/CW/CP/D2/CS /D8/CT/D6/D1− /C1/D1(Y) /B4/D8/CW/CX /CZ/D0/CX/D2/CT/D7/B5/B8 /D6/CX/CV/CW /D8/B9/CW/CP/D2/CS /D8/CT/D6/D1kr∆lθ/(1−θ2) /B4/D8/CW/CX/D2 /D0/CX/D2/CT/B5/B8 /CP/D2/CS /D7/D3/D0/D9/B9/D8/CX/D3/D2/D7 /B4/D1/CP/D6/CZ /CT/D6/D7/B5/BA/CC/CW/CT/D7/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2/D7 /CP/D6/CT /DA /CP/D0/CX/CS /CX/CUtan(krL)/greaterorsimilar0 /B8 /CX/BA/CT/BA/B8krL/greaterorsimilar nπ /BA /BT /D3/D6/CS/CX/D2/CV /D8/D3 /BX/D5/BA /B4/BE/BL /B5/B8 /DB/CW/CT/D2 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /CP/D4/B9/D4/D6/D3/CP /CW/CT/D7 fr /B8 /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CS/CT /D6/CT/CP/D7/CT/D7 /D8/D3 /DE/CT/D6/D3 /D3/D2/B9/D8/D6/CP/D6/DD /D8/D3 /DB/CW/CP/D8 /CW/CP/D4/D4 /CT/D2/D7 /DB/CW/CT/D2 /D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CT /D6/CT/CT/CS /CS/CP/D1/D4/B9/CX/D2/CV /CT/AR/CT /D8/BA/BY/CX/CV/D9/D6/CT/D7 /BG /CP/D2/CS /BI /D7/CW/D3 /DB /CP /D2 /D9/D1/CT/D6/CX /CP/D0 /D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D8/CW/CT/D6/CT/D7/D4 /CT /D8/CX/DA /CT /CT/AR/CT /D8/D7 /D3/CU /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /CP/D2/CS /D6/CT/CT/CS/CS/CP/D1/D4/CX/D2/CV/BA /CC/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /DB/CX/D8/CW∆l= 12mm /CX/D2 /BY/CX/CV/BA /BG /CP/D2/CS5mm /CX/D2 /BY/CX/CV/BA /BI /CP/CS/CY/D9/D7/D8/D7 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD/CS/CT/DA/CX/CP/D8/CX/D3/D2 /CU/D3/D6 /CQ /D3/D8/CW /CW/CT/CP /DA/CX/D0/DD /CP/D2/CS /D7/D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/D7/B8/CT/DA /CT/D2 /DB/CW/CT/D2 /CP/D4/D4/D6/D3/CP /CW/CX/D2/CV /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT/B8 /CP/D2/CS /CX/D7 /D4/D6/CT/D4 /D3/D2/B9/CS/CT/D6/CP/D2 /D8 /D3/D1/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /CT/AR/CT /D8/BA /BT/D2 /CT/DC/CP/D1/D4/D0/CT /D3/CU/CP /D1/CX/D7/D8/D9/D2/CT/CS /D0/CT/D2/CV/D8/CW /D3/D6/D6/CT /D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /B4∆l= 12mm /B5/CX/D7 /CT/DC/CW/CX/CQ/CX/D8/CT/CS /CX/D2 /BY/CX/CV/BA /BI /B8 /D6/CT/DA /CT/CP/D0/CX/D2/CV /CP /D1/CX/D7/D1/CP/D8 /CW /D3/CU /D3/D7 /CX/D0/D0/CP/B9/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /BA /CC/CW/CX/D7 /D0/CP/D8/D8/CT/D6 /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT /D1/CP/CX/D2/D0/DD /D3/D2 /D8/D6/D3/D0/D0/CT/CS/CQ /DD /D8/CW/CT /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /CT/AR/CT /D8 /DB/CW/CT/D2 /CP/D4/D4/D6/D3/CP /CW/B9/CX/D2/CV /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT /CU/D6/CT/D5/D9/CT/D2 /DD /BA /BV/D0/CP/D7/D7/CX /CP/D0 /CP/D4/D4/D6/D3/CP /CW/CT/D7/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BG/BD /B5 /CU/D3/D6 /D8/CW/CT /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D4/D0/CP /DD/CX/D2/CV /CU/D6/CT/D5/D9/CT/D2/B9 /CX/CT/D7/B8 /CX/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT /AT/D3 /DB /CS/D9/CT /D8/D3 /D4/D6/CT/D7/D7/D9/D6/CT /CS/D6/D3/D4/B8 /CP/D2/CS /D7/CT/CP/D6 /CW/B9/CX/D2/CV /CU/D3/D6 /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/CU /D8/CW/CT /D4/CP/D7/D7/CX/DA /CT /D7/DD/D7/D8/CT/D1 /CX/D2 /D0/D9/CS/CX/D2/CV/D8/CW/CT /CQ /D3/D6/CT /CP/D2/CS /D8/CW/CT /D6/CT/CT/CS /D3/D2/D0/DD /CP/D6/CT /D8/CW /D9/D7 /CY/D9/D7/D8/CX/AS/CT/CS/BA /C1/D8 /CP/D2 /CQ /CT/D2/D3/D8/CX /CT/CS /D8/CW/CP/D8 /BX/D5/BA /B4/BE/BK/B5 /DB /CP/D7 /CP/D0/D6/CT/CP/CS/DD /CV/CX/DA /CT/D2 /CQ /DD /CF /CT/CQ /CT/D6 /BG/BF/CX/D2/D8/CW/CT /CT/CP/D6/D0/DD /BD/BL/D8/CW /CT/D2 /D8/D9/D6/DD /B4/D7/CT/CT /D4/CP/CV/CT /BE/BD/BI/B5/B8 /CP/D7/D7/D9/D1/CX/D2/CV /CP /D6/CT/CT/CS/CP/D6/CT/CP /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT /DD/D0/CX/D2/CS/D6/CX /CP/D0 /D8/D9/CQ /CT /D7/CT /D8/CX/D3/D2/BA /CC/CW/CX/D7 /D8/CW/CT/D3/D6/DD /B8/D9/D7/CT/CS /CQ /DD /D7/CT/DA /CT/D6/CP/D0 /CP/D9/D8/CW/D3/D6/D7 /B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BF/BE /B5/B8 /DB /CP/D7 /CS/CX/D7 /D9/D7/D7/CT/CS/CQ /DD /DA /D3/D2 /C0/CT/D0/D1/CW/D3/D0/D8/DE /BF/BF/CP/D2/CS /BU/D3/D9/CP/D7/D7/CT /BG/BG/B8 /CT/D7/D4 /CT /CX/CP/D0/D0/DD /D3/D2 /CT/D6/D2/CX/D2/CV/D8/CW/CT /D0/CP /CZ /D3/CU /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2 /D3/D2 /D8/CW/CT /D4/D6/D3 /CS/D9 /D8/CX/D3/D2 /D3/CU /D7/CT/D0/CU/B9/D7/D9/D7/D8/CP/CX/D2/CT/CS/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7/BA /C7/D2 /D8/CW/CT /D3/D2 /D8/D6/CP/D6/DD /B8 /D8/CW/D6/CT/D7/CW/D3/D0/CS /D4/D6/CT/D7/D7/D9/D6/CT /D9/D6/DA /CT/D7/CT/DC/CW/CX/CQ/CX/D8 /D8/CW/CP/D8 /CQ /D3/D8/CW /CS/CP/D1/D4/CX/D2/CV /CP/D2/CS /CP/CS/CS/CX/D8/CX/D3/D2/CP/D0 /AT/D3 /DB /CW/CP /DA /CT /CX/D2/B9/AT/D9/CT/D2 /CT /D3/D2 /D8/CW/CT /D4/D6/CT/D7/D7/D9/D6/CT /D6/CT/D5/D9/CX/D6/CT/CS /CU/D3/D6 /D8/CW/CT /D6/CT/CT/CS /D8/D3 /D3/D7 /CX/D0/D0/CP/D8/CT/BA/CB/D3 /CP /D3/D1 /CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4/CW/CT/D2/D3/D1/CT/D2/CP /CW/CP/D7 /D8/D3 /CQ /CT /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3/CP /D3/D9/D2 /D8/B8 /D2/D3/D2/CT /D3/CU /D8/CW/CT/D1 /CQ /CT/CX/D2/CV /D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /CX/D2 /D8/CW/CT /D3/D2/D7/CX/CS/CT/D6/CT/CS/CS/D3/D1/CP/CX/D2/BA/C1/CE/BA /BZ/C7/C1/C6/BZ /BU/BX/CH/C7/C6/BW /C1/C6/CB/CC /BT/BU/C1/C4/C1/CC/CH /CC/C0/CA/BX/CB/C0/C7/C4/BW/BT/BA /C4/CX/D2/CT/CP /D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D2/CP/D0/DD/D7/CX/D7 /DB/CX/D8/CW /D1/D3 /CS/CP/D0 /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2/BT/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2 /CQ /CT /D4 /CT/D6/CU/D3/D6/D1/CT/CS/D9/D7/CX/D2/CV /D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2 /DD /CU/D3/D6/D1/CP/D0/CX/D7/D1/BA /BY /D3/D6 /CP /CV/CX/DA /CT/D2 /D3/D2/AS/CV/B9/D9/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D7/DD/D7/D8/CT/D1 /CQ /D3/D6 /CT/B9/D6 /CT /CT /CS/B9/D1/D9/D7/CX /CX/CP/D2 /B4L /B8S /B8/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BJ/BY/C1/BZ/BA /BI/BA /BV/D3/D1/D4/CP/D6/CX/D2/CV /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS /AT/D3 /DB /CT/AR/CT /D8 /CP/D2/CS /D6/CT/CT/CS/CS/CP/D1/D4/CX/D2/CV /CT/AR/CT /D8 /B4 /D3/D2/CS/CX/D8/CX/D3/D2/D7 /CP/D7 /CX/D2 /BY/CX/CV/BA /BE/B5/BM /D6/CT/CT/CS /D1/D3/D8/CX/D3/D2 /CX/D2/CS/D9 /CT/CS/AT/D3 /DB /D3/D2/D0/DD /B4∆l= 5mm /BM /D4/D0/CP/CX/D2 /D8/CW/CX/D2 /D0/CX/D2/CT/D7/B8∆l= 12mm /BM /CS/D3/D8/D8/CT/CS/D0/CX/D2/CT /CX/D2 /D9/D4/D4 /CT/D6 /CV/D6/CP/D4/CW /D3/D2/D0/DD/B5/B8 /D6/CT/CT/CS /CS/CP/D1/D4/CX/D2/CV /D3/D2/D0/DD /B4/CS/CP/D7/CW/CT/CS /D8/CW/CX/D2/D0/CX/D2/CT/D7/B5/B8 /CP/D2/CS /CQ /D3/D8/CW /CT/AR/CT /D8/D7 /B4/D4/D0/CP/CX/D2 /D8/CW/CX /CZ /D0/CX/D2/CT/D7/B5/BA /CF/BU /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D6/CT/D7/D9/D0/D8/D7 /CP/D6/CT /D6/CT/CP /CP/D0/D0/CT/CS /B4/D7/D5/D9/CP/D6/CT/D7/B5/BA ωr /B8qr /B8γ /B8 /CP/D2/CSζ /CQ /CT/CX/D2/CV /D7/CT/D8/B5/B8 /CX/D8/D7 /D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 sn=jωn−αn /CP/D2 /CQ /CT /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS/BA /CC/CW/CT /CX/D1/CP/CV/CX/D2/CP/D6/DD /D4/CP/D6/D8/D3/CUsn /D3/D6/D6/CT/D7/D4 /D3/D2/CS/D7 /D8/D3 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /B8 /D8/CW/CT /D6/CT/CP/D0 /D4/CP/D6/D8αn /CQ /CT/B9/CX/D2/CV /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /D3/CU /D8/CW/CX/D7 /D1/D3 /CS/CT/B8 /CU/D3/D6 /D8/CW/CT /D3/D9/D4/D0/CT/CS /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS/D7/DD/D7/D8/CT/D1 /D0/D3/D7/CT /D8/D3 /D8/CW/CT /D7/D8/CP/D8/CX /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT/BA /BV/D0/CP/D7/D7/CX /CP/D0/D0/DD /B8/DB/CW/CT/D2 /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /CQ /CT/D0/D3 /DB /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/B9/D3/D0/CS/B8 /CP/D0/D0 /CT/CX/CV/CT/D2/CS/CP/D1/D4/CX/D2/CV/D7 αn /CP/D6/CT /D4 /D3/D7/CX/D8/CX/DA /CT/B8 /D8/CW/CT /D7/D8/CP/D8/CX /D6/CT/CV/CX/D1/CT/CQ /CT/CX/D2/CV /D7/D8/CP/CQ/D0/CT/BA /BT/D2 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D1/CP /DD /CP/D4/D4 /CT/CP/D6 /DB/CW/CT/D2 /CP/D8 /D0/CT/CP/D7/D8/D3/D2/CT /D1/D3 /CS/CT /D3/CU /D8/CW/CT /DB/CW/D3/D0/CT /D7/DD/D7/D8/CT/D1 /CQ /CT /D3/D1/CT/D7 /D9/D2/D7/D8/CP/CQ/D0/CT/B8 /CX/BA/CT/BA/B8/DB/CW/CT/D2 /CP/D8 /D0/CT/CP/D7/D8 /D3/D2/CT /D3/CU /D8/CW/CTαn /CQ /CT /D3/D1/CT/D7 /D2/CT/CV/CP/D8/CX/DA /CT/BA /C4/D3 /D3/CZ/CX/D2/CV/CU/D3/D6 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2 /CQ /CT /CS/D3/D2/CT /CQ /DD /DA /CP/D6/DD/CX/D2/CV /CP /CQ/CX/CU/D9/D6 /CP/B9/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 /B4/CT/CX/D8/CW/CT/D6L /B8γ /B8 /D3/D6ζ /B5 /CP/D2/CS /CT/DC/CP/D1/CX/D2/CX/D2/CV /D8/CW/CT /D6/CT/CP/D0/D4/CP/D6/D8 /D3/CU /D3/D1/D4/D9/D8/CT/CS /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7/BA /CC /DB /D3 /CT/DC/CP/D1/D4/D0/CT/D7 /CP/D6/CT/D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/D7/BA /BJ /CP/D2/CS /BK /BA /C1/D8 /CX/D7 /D2/D3/D8/CX /CT/CP/CQ/D0/CT /D8/CW/CP/D8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2/B9 /CX/CT/D7 /CT/DA /D3/D0/DA /CT /D3/D2/D0/DD /D7/D0/CX/CV/CW /D8/D0/DD /DB/CX/D8/CW /D8/CW/CT /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /D4/CP/D6/CP/D1/CT/D8/CT/D6 γ/CP/D2/CS /CP/D6/CT /D0/D3/D7/CT /D8/D3 /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/CU /CT/CX/D8/CW/CT/D6 /D8/CW/CT /CQ /D3/D6/CT /D3/D6 /D8/CW/CT/D6/CT/CT/CS /B4 /C1/D1(jω/ωr)≃1 /B5/BA /BY /D3/D6 /D8/CW/CT /AS/D6/D7/D8 /CT/DC/CP/D1/D4/D0/CT/B8 /D8/CW/CT /D7/D8/CP/D8/CX /D6/CT/CV/CX/D1/CT /CQ /CT /D3/D1/CT/D7 /D9/D2/D7/D8/CP/CQ/D0/CT /CU/D3/D6γ≃0.28 /CP/D2/CS /CP /CU/D6/CT/D5/D9/CT/D2 /DD/D2/CT/CP/D6 /D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D2/CP/D2 /CT /D3/CU /D8/CW/CT /CQ /D3/D6/CT /B4/CS/D3/D8/B9/CS/CP/D7/CW/CT/CS /D9/D6/DA /CT/D7/B5/BA/C7/D8/CW/CT/D6 /CP /D3/D9/D7/D8/CX /D6/CT/D7/D3/D2/CP/D2 /CT/D7 /B4/D8/CW/CT/CX/D6 /CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CQ /CT/CX/D2/CV /D3 /CS/CS/D1 /D9/D0/D8/CX/D4/D0/CT/D7 /D3/CU /D8/CW/CT /AS/D6/D7/D8 /D3/D2/CT/B5 /CP/D2/CS /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT /CW/CP /DA /CT /CW/CX/CV/CW/CT/D6/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2/CS /D6/CT/D1/CP/CX/D2 /CS/CP/D1/D4 /CT/CS /CU/D3/D6 /D8/CW/CX/D7 /D3/D2/AS/CV/D9/B9/D6/CP/D8/CX/D3/D2/BA /BY /D3/D6 /CP /D0/D3/D2/CV/CT/D6 /D8/D9/CQ /CT /B4/BY/CX/CV/BA /BK/B5/B8 /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CP/D4/D4 /CT/CP/D6/D7 /CU/D3/D6 γ≃0.3 /CP/D8 /CP /CU/D6/CT/D5/D9/CT/D2 /DD /D0/D3 /CP/D8/CT/CS /D2/CT/CP/D6 /D8/CW/CT /D8/CW/CX/D6/CS /D6/CT/D7/D3/D2/CP/D2 /CT/D3/CU /D8/CW/CT /CQ /D3/D6/CT /CJ/CS/D3/D8/D8/CT/CS /D0/CX/D2/CT /CP/D8 /C1/D1(jω/ωr)≃0.8≃3×0.27 ℄/B8/D8/CW/CT /AS/D6/D7/D8 /D6/CT/D7/D3/D2/CP/D2 /CT /CQ /CT /D3/D1/CX/D2/CV /D9/D2/D7/D8/CP/CQ/D0/CT /CU/D3/D6 /CP /D0/CP/D6/CV/CT/D6 /D1/D3/D9/D8/CW/D4/D6/CT/D7/D7/D9/D6/CT/BA /CC/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /D0/D3/D7/CT /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT /B4/D7/D3/D0/CX/CS /D9/D6/DA /CT/D7/B5 /D7/D8/CX/D0/D0 /D6/CT/D1/CP/CX/D2/D7 /CS/CP/D1/D4 /CT/CS/BA/BV/CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /D1/CP /DD /CQ /CT /D7/CX/D1/D4/D0/CX/AS/CT/CS /CP/D2/CS /CP /CT/D0/CT/D6/CP/D8/CT/CS /CQ /DD /D9/D7/B9/CX/D2/CV /CP /D1/D3 /CS/CP/D0 /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CQ /D3/D6/CT /CX/D1/D4 /CT/CS/CP/D2 /CT Ze(ω) /BM/D8/CW/CX/D7 /CP/D0/D0/D3 /DB/D7 /CU/D3/D6 /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2 /D8/D3 /CQ /CT /DB/D6/CX/D8/D8/CT/D2/CP/D7 /CP /D4 /D3/D0/DD/D2/D3/D1/CX/CP/D0 /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CUjω /B8 /CP/D2/CS /D3/D4/D8/CX/D1/CX/DE/CT/CS /CP/D0/CV/D3/B9/D6/CX/D8/CW/D1/D7 /CU/D3/D6 /D4 /D3/D0/DD/D2/D3/D1/CX/CP/D0 /D6/D3 /D3/D8 /AS/D2/CS/CX/D2/CV /CP/D2 /CQ /CT /D9/D7/CT/CS/BA /C5/D3 /CS/CP/D0/CT/DC/D4/CP/D2/D7/CX/D3/D2 /D3/D2/D7/CX/CS/CT/D6/D7 /D8/CW/CTN /AS/D6/D7/D8 /CP /D3/D9/D7/D8/CX /D6/CT/D7/D3/D2/CP/D2 /CT/D7 /D3/CU /D8/CW/CT /BY/C1/BZ/BA /BJ/BA /BX/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CP/D7 /CP /CU/D9/D2 /B9/D8/CX/D3/D2 /D3/CU /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /BAr= 7mm /B8ωr= 2π×1000rad/s /B8 qr= 0.3 /B8L= 16cm /B4θ1= 0.53 /CP/D2/CSkrL= 2.96 /B5/B8 /CP/D2/CSζ= 0.2 /BA /BY/C1/BZ/BA /BK/BA /BX/DA /D3/D0/D9/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /CP/D7 /CP /CU/D9/D2 /B9/D8/CX/D3/D2 /D3/CUγ /BAr= 7mm /B8ωr= 2π×1000rad/s /B8qr= 0.3 /B8 L= 32cm /B4θ1= 0.26 /CP/D2/CSkrL= 5.91 /B5/B8 /CP/D2/CSζ= 0.2 /BA/CQ /D3/D6/CT/BM Ze(ω) Zc=jtan/parenleftbiggωL c−jα(ω)L/parenrightbigg/B4/BF/BH/B5 Ze(ω) Zc≃2c LN/summationdisplay n=1jω ω2n+jqnωωn+(jω)2, /B4/BF/BI/B5/DB/CW/CT/D6/CT /D1/D3 /CS/CP/D0 /D3 /CTꜶ /CX/CT/D2 /D8/D7 ωn /CP/D2/CSqn /CP/D2 /CQ /CT /CS/CT/CS/D9 /CT/CS /CT/CX/B9/D8/CW/CT/D6 /CU/D6/D3/D1 /D1/CT/CP/D7/D9/D6/CT/CS /CX/D2/D4/D9/D8 /CX/D1/D4 /CT/CS/CP/D2 /CT /D3/D6 /CU/D6/D3/D1 /CP/D2/CP/D0/DD/D8/CX /CP/D0/CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CJ/BX/D5/BA /B4/BF/BH /B5℄/B8 /CP/D7/D7/D9/D1/CX/D2/CV α(ω) /D8/D3 /CQ /CT /CP /D7/D0/D3 /DB/D0/DD /DA /CP/D6/DD/B9/CX/D2/CV /CU/D9/D2 /D8/CX/D3/D2 /D3/CU /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD /BA/BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /CS/CX/D6/CT /D8 /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D8/CW/D6/CT/D7/CW/D3/D0/CS γ/D8/CW /D9/D7/CX/D2/CV /CF/BU /D1/CT/D8/CW/D3 /CS /CP/D2/CS /CT/D7/D8/CX/D1/CP/D8/CX/D3/D2 /D9/D7/CX/D2/CV/D1/D3 /CS/CP/D0 /CS/CT /D3/D1/D4 /D3/D7/CX/D8/CX/D3/D2 /CP/D2/CS /D3/D1/D4/D0/CT/DC /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/D1/B9/D4/D9/D8/CX/D2/CV /CW/CP/D7 /CQ /CT/CT/D2 /CS/D3/D2/CT/BA /CF/CW/CT/D6/CT/CP/D7 /D8/CW/CT /CS/CXꜶ /D9/D0/D8/CX/CT/D7 /CU/D3/D6 /D8/CW/CT/AS/D6/D7/D8 /D1/CT/D8/CW/D3 /CS /CP/D6/CX/D7/CT /CS/D9/CT /D8/D3 /D8/CW/CT /D8/D6/CP/D2/D7 /CT/D2/CS/CT/D2 /D8/CP/D0 /CW/CP/D6/CP /D8/CT/D6/CX/D7/B9/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2/B8 /D8/CW/CT /D7/CT /D3/D2/CS /D3/D2/CT /D6/CT/D5/D9/CX/D6/CT/D7 /CP/D0 /D9/D0/CP/D8/CX/D3/D2 /D3/CU /CT/CX/CV/CT/D2/B9/DA /CP/D0/D9/CT/D7 /CU/D3/D6 /DA /CP/D6/CX/D3/D9/D7 /DA /CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT γ /B8 /D9/D7/CX/D2/CV/CP/D2 /CX/D8/CT/D6/CP/D8/CX/DA /CT /D7/CT/CP/D6 /CW /D3/CU /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /CC/CW/CT /D2 /D9/D1/B9/CQ /CT/D6 /D3/CU /D1/D3 /CS/CT/D7 /D8/CP/CZ /CT/D2 /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /CW/CP/D7 /CQ /CT/CT/D2 /CW/D3/D7/CT/D2 /D7/D9 /CW/D8/CW/CP/D8 /D6/CT/D7/D3/D0/D9/D8/CX/D3/D2 /D3/CUγ/D8/CW /CX/D7 /D0/CT/D7/D7 /D8/CW/CP/D20.01 /B8 /DB/CW/CX /CW /CX/D7 /CP/D0/D7/D3 /D8/CW/CT/D8/D3/D0/CT/D6/CP/D2 /CT /D9/D7/CT/CS /CU/D3/D6 /D8/CW/CT /CX/D8/CT/D6/CP/D8/CX/DA /CT /D7/CT/CP/D6 /CW/BA /BT/D2 /CT/DC/CP/D1/D4/D0/CT /CX/D7/CV/CX/DA /CT/D2 /CX/D2 /CC /CP/CQ/D0/CT /C1 /CU/D3/D6 /CP /CW/CT/CP /DA/CX/D0/DD /CS/CP/D1/D4 /CT/CS /CP/D2/CS /D7/D8/D6/D3/D2/CV /D6/CT/CT/CS/BA/C1/D8 /D7/CW/D3 /DB/D7 /DA /CT/D6/DD /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 /D1/CT/D8/CW/D3 /CS/D7/CU/D3/D6 /CQ /D3/D8/CWγ /CP/D2/CSθ /BA /CC/CW/CX/D7 /DA /CP/D0/CX/CS/CP/D8/CX/D3/D2 /CP/D0/D0/D3 /DB/D7 /D8/CW/CT /D9/D7/CT /D3/CU /D8/CW/CT /D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2 /DD /CP/D4/D4/D6/D3/CP /CW/B8 /DB/CW/CX /CW /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /CP/D2 /CTꜶ /CX/CT/D2 /D8/CP/D0/CV/D3/D6/CX/D8/CW/D1 /D8/CW/CP/D8 /CP/D2 /CQ /CT /CT/CP/D7/CX/D0/DD /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D1/D3/D6/CT /D3/D1/D4/D0/CT/DC /D6/CT/D7/B9/D3/D2/CP/D8/D3/D6/D7 /DB/CW/CT/D2/CT/DA /CT/D6 /CP /D1/D3 /CS/CP/D0 /CS/CT/D7 /D6/CX/D4/D8/CX/D3/D2 /CX/D7 /CP /DA /CP/CX/D0/CP/CQ/D0/CT/BA/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BKkrLθCFθWB∆θγCFγWB∆γ/BK/BA/BH /BC/BA/BD/BK/BH /BC/BA/BD/BK/BI /BC/BA/BH/B1 /BC/BA/BG/BF /BC/BA/BG/BF /BC/BA/BD/B1/BE /BC/BA/BJ/BG/BJ /BC/BA/BJ/BG/BJ /BC/BA/BD/B1 /BC/BA/BF/BC /BC/BA/BF/BC /BC/BA/BI/B1/BD /BD/BA/BC/BE/BE /BD/BA/BC/BE/BG /BC/BA/BE/B1 /BF/BA/BK/BI /BF/BA/BK/BE /BD/BA/BC/B1/BC/BA/BK/BD /BD/BA/BC/BF/BF /BD/BA/BC/BF/BG /BC/BA/BD/B1 /BK/BA/BJ/BJ /BK/BA/BJ/BD /BC/BA/BJ/B1/CC /BT/BU/C4/BX /C1/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /CP/D2/CS /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/CU/D6/CT/D5/D9/CT/D2 /DD /CP/D0 /D9/D0/CP/D8/CT/CS /D9/D7/CX/D2/CV /D3/D1/D4/D0/CT/DC /CU/D6/CT/D5/D9/CT/D2 /DD /CU/D3/D6/D1/CP/D0/CX/D7/D1 /B4/CX/D2/B9/CS/CT/DC/CT/CS /CQ /DDCF /B5 /CP/D2/CS /CF/CX/D0/D7/D3/D2 /B2 /BU/CT/CP /DA /CT/D6/D7 /D1/CT/D8/CW/D3 /CS /B4WB /B5 /CU/D3/D6 r= 7mm /B8ωr= 2π×750rad/s /B8qr= 0.4 /CP/D2/CSζ= 0.13 /BA/CC/CW/CT /DB/D6/CX/D8/CX/D2/CV /D3/CU /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2 /CU/D3/D6 /CP /D7/CX/D2/B9/CV/D0/CT /CP /D3/D9/D7/D8/CX /D1/D3 /CS/CT /CT/DC/CW/CX/CQ/CX/D8/D7 /D8/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /D3/D9/D4/D0/CT/CS/D3/D7 /CX/D0/D0/CP/D8/D3/D6/D7/BM /bracketleftbigg ω2 n+jω/parenleftbigg qnωn+c Lζ1−γ√γ/parenrightbigg −ω2/bracketrightbigg ×/bracketleftbig ω2 r+jqrωωr−ω2/bracketrightbig =jω2c Lω2 r/parenleftbigg ζ√γ+jω∆l c/parenrightbigg ./B4/BF/BJ/B5/BV/D3/D9/D4/D0/CX/D2/CV /D6/CT/CP/D0/CX/DE/CT/CS /CQ /DD /D8/CW/CT /AT/D3 /DB /CX/D2 /D8/CW/CT /D6/CT/CT/CS /CW/CP/D2/D2/CT/D0 /D1/D3 /CS/B9/CX/AS/CT/D7 /D8/CW/CT /CS/CP/D1/D4/CX/D2/CV /D3/CU /D8/CW/CT /CP /D3/D9/D7/D8/CX /D1/D3 /CS/CT/BM /CX/D2 /CP/CS/CS/CX/D8/CX/D3/D2 /D8/D3/D8/CW/CT /D9/D7/D9/CP/D0 /D8/CT/D6/D1 /B4 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /DA/CX/D7 /D3/D8/CW/CT/D6/D1/CP/D0 /D0/D3/D7/D7/CT/D7 /CP/D2/CS/CT/DA /CT/D2 /D8/D9/CP/D0/D0/DD /D6/CP/CS/CX/CP/D8/CX/D3/D2/B5/B8 /CS/CP/D1/D4/CX/D2/CV /CX/D7 /CX/D2 /D6/CT/CP/D7/CT/CS /CQ /DD /CP /D5/D9/CP/D2 /D8/CX/D8 /DD/D6/CT/D0/CP/D8/CT/CS /D8/D3 /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CP/D2/CS /D7/D8/D6/CT/D7/D7 /D3/D2 /D8/CW/CT /D6/CT/CT/CS/BA /CC/CW/CX/D7/D1/CP /DD /CQ /CT /D6/CT/CV/CP/D6/CS/CT/CS /D8/D3 /CP/D7 /CP /D6/CT/D7/CX/D7/D8/CX/DA /CT /CP /D3/D9/D7/D8/CX /CQ /CT/CW/CP /DA/CX/D3/D6 /CP/D8 /D8/CW/CT/CQ /D3/D6/CT /CT/D2 /D8/D6/CP/D2 /CT/BA/BT/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8 /CP /D0/CX/D2/CT/CP/D6/CX/DE/CT/CS /D1/D3 /CS/CT/D0 /CX/D7 /D7/D8/CX/D0/D0 /D6/CT/D0/CT/DA /CP/D2 /D8 /CS/D9/D6/B9/CX/D2/CV /D8/CW/CT /CV/D6/D3 /DB/D8/CW /D3/CU /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /B4/CQ /CT/CU/D3/D6/CT /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT /D7/CP/D8/B9/D9/D6/CP/D8/CX/D3/D2 /D1/CT /CW/CP/D2/CX/D7/D1 /CP/D4/D4 /CT/CP/D6/D7/B5/B8 /D8/CW/CX/D7 /CP/D4/D4/D6/D3/CP /CW /CP/D2 /CQ /CT /CT/DC/B9/D8/CT/D2/CS/CT/CS /D8/D3 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT /D8/CW/CT /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /D6/CT/D7/D4 /D3/D2/D7/CT /D3/CU /D8/CW/CT /D3/D9/B9/D4/D0/CT/CS /D7/DD/D7/D8/CT/D1/BA /BV/CW/CP/D6/CP /D8/CT/D6/CX/DE/CX/D2/CV /D8/CW/CT /CS/CT/CV/D6/CT/CT /D3/CU /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD/D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CQ /DDσ= min nαn /B8 /D8/CW/CT /D7/D0/D3/D4 /CT /D3/CU /D8/CW/CT /D9/D6/DA /CT σ=f(γ) /CV/CX/DA /CT/D7 /CP/D2 /CX/D2/CU/D3/D6/D1/CP/D8/CX/D3/D2 /CP/CQ /D3/D9/D8 /D8/CW/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /CS/CT/B9/CV/D6/CT/CT /D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1 /DB/CW/CT/D2 /D8/CW/CT /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT /CX/D7 /D7/D0/CX/CV/CW /D8/D0/DD/CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /BT /CV/D6/CT/CP/D8 /D7/D0/D3/D4 /CT/DB /D3/D9/D0/CS /D3/D6/D6/CT/D7/D4 /D3/D2/CS /D8/D3 /CP /DA /CT/D6/DD /D9/D2/D7/D8/CP/CQ/D0/CT /D3/D2/AS/CV/D9/D6/CP/D8/CX/D3/D2 /CP/D2/CS/CP /D5/D9/CX /CZ /CV/D6/D3 /DB/D8/CW /D3/CU /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/B8 /DB/CW/CT/D6/CT/CP/D7 /D2/CT/CP/D6/D0/DD /D3/D2/D7/D8/CP/D2 /D8 /D9/D6/DA /CT /DB /D3/D9/D0/CS /D0/CT/CP/CS /D8/D3 /CP /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/AS /CP/D8/CX/D3/D2 /D3 /CTꜶ /CX/CT/D2 /D8 /CP/D2/CS/D7/D0/D3 /DB/D0/DD /D6/CX/D7/CX/D2/CV /DA/CX/CQ/D6/CP/D8/CX/D3/D2/D7 /CP/D2/CS /D8/CW/CT/D2 /D8/D3 /D0/D3/D2/CV/CT/D6 /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /CP/D8/B9/D8/CP /CZ /CQ /CT/CU/D3/D6/CT /D7/D8/CP/CQ/CX/D0/CX/DE/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CP/CV/D2/CX/D8/D9/CS/CT /D3/CU /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7/BA/C4/CX/D2/CZ/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D3/D1/D4/D9/D8/CT/CS /CT/CX/CV/CT/D2/CU/D6/CT/D5/D9/CT/D2 /CX/CT/D7 /D3/CU /D8/CW/CT /D3/D9/B9/D4/D0/CT/CS /D7/DD/D7/D8/CT/D1 /D4/D6/CT/D7/CT/D2 /D8/CT/CS /CW/CT/D6/CT /CP/D2/CS /D8/CW/CT /D8/D6/CP/D2/D7/CX/CT/D2 /D8 /CQ /CT/CW/CP /DA/CX/D3/D6/CW/CP /DA /CT /D7/D8/CX/D0/D0 /D8/D3 /CQ /CT /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS/BA/BU/BA /C5/D3/D9/D8/CW /D4 /D6/CT/D7/D7/D9/D6/CT /D6/CT/D5/D9/CX/D6/CT/CS /D8/D3 /D3/CQ/D8/CP/CX/D2 /CP /CV/CX/DA/CT/D2 /B4/D7/D1/CP/D0/D0/B5/CP/D1/D4/D0/CX/D8/D9/CS/CT/CC/CW/CT /D4/D6/CT/DA/CX/D3/D9/D7 /D7/CT /D8/CX/D3/D2/D7 /D3/CU /D8/CW/CT /D4/CP/D4 /CT/D6 /CS/CT/CP/D0 /DB/CX/D8/CW /D8/CW/CT /D7/D8/CP/CQ/CX/D0/B9/CX/D8 /DD /D3/CU /D8/CW/CT /D7/D8/CP/D8/CX /D6/CT/CV/CX/D1/CT/B8 /D0/D3 /D3/CZ/CX/D2/CV /CU/D3/D6 /D8/CW/CT /D3/D2/CS/CX/D8/CX/D3/D2 /D8/D3 /D1/CP/CZ /CT/CP /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /D4 /D3/D7/D7/CX/CQ/D0/CT/BA /CB/D3/D1/CT /CS/CT/DA /CT/D0/D3/D4/D1/CT/D2 /D8/D7 /D3/D2 /CT/D6/D2/CX/D2/CV/D8/CW/CT /CT/DC/CX/D7/D8/CT/D2 /CT /D3/CU /D3/D7 /CX/D0/D0/CP/D8/CX/D2/CV /D6/CT/CV/CX/D1/CT /CP/CQ /D3 /DA /CT /D8/CW/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS/CP/D6/CT /CS/CT/D6/CX/DA /CT/CS /D2/D3 /DB/BA /C6/CT/CX/D8/CW/CT/D6 /D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D3/CU /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D2/D3/D6 /D8/D3/D2/CT/CS/CT/DA/CX/CP/D8/CX/D3/D2 /CX/D7/D7/D9/CT /DB/CX/D0/D0 /CQ /CT /CS/CX/D7 /D9/D7/D7/CT/CS /CW/CT/D6/CT/BA/BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA /BE/BC/D7/D9/CV/CV/CT/D7/D8/CT/CS /D8/CW/CT /CX/D2 /D8/D6/D3 /CS/D9 /D8/CX/D3/D2 /D3/CU /D8/CW/CT /D0/CX/D1/B9/CX/D8/CT/CS /BY /D3/D9/D6/CX/CT/D6 /D7/CT/D6/CX/CT/D7 /D3/CU /D4/D6/CT/D7/D7/D9/D6/CT /CX/D2 /D8/CW/CT /D1/CP/D7/D7/D0/CT/D7/D7 /D6/CT/CT/CS /CP/D7/CT/BA /CC/CW/CT /D8/CT /CW/D2/CX/D5/D9/CT /CX/D7 /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX /CQ/CP/D0/CP/D2 /CT /CP/D4/D4/D0/CX/CT/CS /D8/D3 /D3/D7 /CX/D0/B9/D0/CP/D8/CX/D3/D2/D7 /D3/CU /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT/D7/BA /BV/CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/D3/D2/CT /CW/CT/D6/CT/B9/CP/CU/D8/CT/D6 /CQ /DD /D8/CP/CZ/CX/D2/CV /CX/D2 /D8/D3 /CP /D3/D9/D2 /D8 /D8/CW/CT /D6/CT/CT/CS /CS/DD/D2/CP/D1/CX /D7 /CX/D2 /D8/CW/CT/DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/B8 /DB/CW/CX /CW /CS/D3 /CT/D7 /D2/D3/D8 /CP/D4/D4 /CT/CP/D6 /CX/D2 /D8/CW/CT/D1/CT/D2 /D8/CX/D3/D2/CT/CS /D4/CP/D4 /CT/D6/BA /BY /D3/D9/D6/CX/CT/D6 /D7/CT/D6/CX/CT/D7 /D3/CU /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /CS/CT/B9/D4 /CT/D2/CS/D7 /D3/D2 /BY /D3/D9/D6/CX/CT/D6 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /D3/CU /D7/CX/CV/D2/CP/D0Pm−p(t) /CP/D2/CS y(t) /BA /BT/D7/D7/D9/D1/CX/D2/CV /D7/D8/CT/CP/CS/DD /D7/D8/CP/D8/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW /CP/D2/CV/D9/D0/CP/D6 /CU/D6/CT/B9/D5/D9/CT/D2 /DDω /B8 /D8/CW/CT /CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /D7/CX/CV/D2/CP/D0/D7 /CP/D6/CT /DB/D6/CX/D8/D8/CT/D2 /CP/D7 p(t) =/summationdisplay n/negationslash=0pnenjωt, u(t) =u0+/summationdisplay n/negationslash=0Ynpnenjωt, /B4/BF/BK/B5 y(t) y0= (1−γ)+/summationdisplay n/negationslash=0Dnpnenjωt, /B4/BF/BL/B5/DB/CW/CT/D6/CTYn=Ye(nω) /CP/D2/CSDn=D(nω) /CP/D6/CT /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU/CS/CX/D1/CT/D2/D7/CX/D3/D2/D0/CT/D7/D7 /CQ /D3/D6/CT /CP/CS/D1/CX/D8/D8/CP/D2 /CT /CP/D2/CS /D6/CT/CT/CS /D8/D6/CP/D2/D7/CU/CT/D6 /CU/D9/D2 /D8/CX/D3/D2/CU/D3/D6 /CP/D2/CV/D9/D0/CP/D6 /CU/D6/CT/D5/D9/CT/D2 /DD nω /BA /CC/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/CX/D4/CX/D7 /D6/CT/DB/D6/CX/D8/D8/CT/D2 /CP/D7 u2(t) =ζ2(y(t)/y0)2(γ−p(t)). /B4/BG/BC/B5/CB/D9/D7/D8/CP/CX/D2/CT/CS /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /D3/CU /DA /CT/D6/DD /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT /CP/D6/CT /D7/D8/D9/CS/B9/CX/CT/CS/B8 /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8p1 /CX/D7 /CP /D2/D3/D2 /DA /CP/D2/CX/D7/CW/CX/D2/CV /D3 /CTꜶ /CX/CT/D2 /D8 /D3/D2/B9/D7/CX/CS/CT/D6/CT/CS /CP/D7 /CP /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /D5/D9/CP/D2 /D8/CX/D8 /DD /BA /C6/D3/D8/CP/D8/CX/D3/D2/D7 CEn /CP/D2/CS F(n m) /CP/D6/CT /CX/D2 /D8/D6/D3 /CS/D9 /CT/CS/BM CEn=Yn/(ζ√γ)+1−γ 2γ−Dn, /B4/BG/BD/B5 F(m n)=F(n m)=DnDm−1−γ γ(Dn+Dm)−YnYm ζ2γ/B4/BG/BE/B5/BV/CP/D2 /CT/D0/D0/CP/D8/CX/D3/D2 /D3/CUCEn /CU/D3/D6 /CV/CX/DA /CT/D2ω /CP/D2/CSγ /D1/CT/CP/D2/D7 /D8/CW/CP/D8 /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2 /B4/BJ/B5 /CX/D7 /D7/D3/D0/DA /CT/CS /CU/D3/D6nω /CP/D2/CSγ /BA /BX/DC/B9/D4/CP/D2/CS/CX/D2/CV /BX/D5/BA /B4/BG/BC /B5 /D0/CT/CP/CS/D7 /D8/D3 0 =/bracketleftbiggu2 0 ζ2γ−(1−γ)2/bracketrightbigg +2(1−γ)/summationdisplay n/negationslash=0/bracketleftbiggu0Yn ζ2γ(1−γ)−Dn+1−γ 2γ/bracketrightbigg pnenjωt −/summationdisplay n,m/negationslash=0F(n m)pnpme(n+m)jωt +1 γ/summationdisplay n,m,q/negationslash=0DnDmpnpmpqe(n+m+q)jωt./B4/BG/BF/B5/C1/D8 /CX/D7 /CW/CT/D6/CT /CP/D7/D7/D9/D1/CT/CS /D8/CW/CP/D8pn /CX/D7 /D3/CU /D3/D6/CS/CT/D6|n| /B4/DB/CX/D8/CWp−n=p∗ n /B5/B4/D7/CT/CT/B8 /CT/BA/CV/BA/B8 /CA/CT/CU/BA /BE/BC /B5/BA /CC/CW/CT /D3/D2 /D8/CX/D2 /D9/D3/D9/D7 /D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU /D8/CW/CT/DA /D3/D0/D9/D1/CT /AT/D3 /DB /CX/D7 /CP/D0 /D9/D0/CP/D8/CT/CS /D9/D4 /D8/D3 /D3/D6/CS/CT/D6 /BE/BM u2 0 ζ2γ=(1−γ)2+/summationdisplay n/negationslash=0F“+n −n”|pn|2 −1 γ/summationdisplay n,m,n+m/negationslash=0DnDmpnpmp∗ n+m ≃(1−γ)2+2F“+1 −1”|p1|2+o(p2 1) /B4/BG/BG/B5/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BL/D7/D3 /D8/CW/CP/D8/B8 /D3/D2/D7/CX/CS/CT/D6/CX/D2/CV u0 /D8/D3 /CQ /CT /D6/CT/CP/D0 u0≃ζ√γ(1−γ) 1+|p1|2F“+1 −1” (1−γ)2 +o(p2 1). /B4/BG/BH/B5/BY /D6/D3/D1 /BX/D5/BA /B4/BG/BF/B5/B8 /D8/CW/CT /CU/D6/CT/D5/D9/CT/D2 /DD (Nω) /CX/D7 /CT/DC/D8/D6/CP /D8/CT/CS /CU/D3/D6 N≥1 /BM 0 = 2(1−γ)/bracketleftbigg DN−1−γ 2γ−u0Yn ζ2γ(1−γ)/bracketrightbigg pN +/summationdisplay n/negationslash=0F(n N−n)pnpN−n −1 γ/summationdisplay n,m/negationslash=0DnDmpnpmpN−n−m. /B4/BG/BI/B5/BY /D3/D6N≥2 /B8 /CC /CP /DD/D0/D3/D6 /D7/CT/D6/CX/CT/D7 /CT/DC/D4/CP/D2/D7/CX/D3/D2 /D9/D4 /D8/D3 /D3/D6/CS/CT/D6N /CX/D7/CP/D4/D4/D0/CX/CT/CS/BM /CX/D2 /D8/CW/CT /AS/D6/D7/D8 /D7/D9/D1/B8 /D3/D2/D0/DD /D8/CT/D6/D1/D7 /D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 0≤n≤N /D3/D2 /D8/D6/CX/CQ/D9/D8/CT /CP/D8 /D3/D6/CS/CT/D6N /B8 /DB/CW/CX/D0/CT/B8 /CX/D2 /D8/CW/CT /D7/CT /D3/D2/CS/D3/D2/CT/B8 /D8/CW/CT /D8/CT/D6/D1/D7 /D8/D3 /D3/D2/D7/CX/CS/CT/D6 /CP/D6/CT /D8/CW/CT /D3/D2/CT/D7 /CU/D3/D6 /DB/CW/CX /CW0< n < N /CP/D2/CS0< m < N −n /BA /CC/CW/CT /D3/D1/D4 /D3/D2/CT/D2 /D8 pN /CP/D2 /CQ /CT/CS/CT/CS/D9 /CT/CS /CU/D6/D3/D1 /D8/CW/CT /D7/CT/D5/D9/CT/D2 /CT (pn)0<n<N /BM pN=1 2(1−γ)CEN/bracketleftigg/summationdisplay 0<n<NF(n N−n)pnpN−n −1 γ/summationdisplay 0<n<N 0<m<N −nDnDmpnpmpN−n−m/bracketrightigg +o(pN 1). /B4/BG/BJ/B5/BT/D7 /CP/D2 /CT/DC/CP/D1/D4/D0/CT/B8 /CU/D3/D6N= 2 /B8 /D8/CW/CT /D7/CT /D3/D2/CS /D7/D9/D1 /CQ /CT/CX/D2/CV /CT/D1/D4/D8 /DD/BM p2≃F(1 1)p2 1 2(1−γ)CE2+o(p2 1). /B4/BG/BK/B5/BT/D7 /CT/DC/D4 /CT /D8/CT/CS /CP/D2/CS /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D7/D3/B9 /CP/D0/D0/CT/CS /AG/CF /D3/D6/B9/D1/CP/D2 /D6/D9/D0/CT/AH /BD/BK/B8 /CW/CX/CV/CW/CT/D6 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /CP/D4/D4 /CT/CP/D6 /D8/D3 /CQ /CT /CW/CX/CV/CW/CT/D6 /D3/D6/B9/CS/CT/D6 /D5/D9/CP/D2 /D8/CX/D8/CX/CT/D7/BM /D3/D6/CS/CT/D6 /BE /CU/D3/D6p2 /B8 /BF /CU/D3/D6p3 /B8 /BG /CU/D3/D6p4 /B8 /CT/D8 /BA/BY /D3 /D9/D7 /CX/D7 /D2/D3 /DB /CV/CX/DA /CT/D2 /D8/D3 /CU/D9/D2/CS/CP/D1/CT/D2 /D8/CP/D0 /CU/D6/CT/D5/D9/CT/D2 /DD /BA /BV/CP/D0 /D9/B9/D0/CP/D8/CX/D3/D2/D7 /CP/D6/CT /CS/D3/D2/CT /D9/D4 /D8/D3 /D3/D6/CS/CT/D6 /BF/BM 0 =2(1−γ)/bracketleftbigg D1−1−γ 2γ−u0Y1 ζ2γ(1−γ)/bracketrightbigg p1 +/summationdisplay n/negationslash=0,1F(n 1−n)pnp1−n −1 γ/summationdisplay 1−n−m,n,m/negationslash=0DnDmpnpmp1−n−m ≃2(1−γ)/bracketleftbigg D1−1−γ 2γ−u0Y1 ζ2γ(1−γ)/bracketrightbigg p1 +2F(2 −1)p2p∗ 1−1 γD1p1|p1|2(D1+2D∗ 1) /B4/BG/BL/B5/CA/CT/D4/D0/CP /CX/D2/CV u0 /CP/D2/CSp2 /CV/CX/DA /CT/D7 |p1|2≃2(1−γ)2CE1 F(1 1)F“+2 −1” CE2−2Y1 ζ√γF“+1 −1”−1−γ γ(D2 1+2|D1|2)./B4/BH/BC/B5 /CC/CW/CT /D0/CX/D1/CX/D8/CP/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /AS/D6/D7/D8 /CW/CP/D6/D1/D3/D2/CX /D1/CT/D8/CW/D3 /CS /CX/CV/D2/D3/D6/CX/D2/CV /D8/CW/CT/CX/D2/AT/D9/CT/D2 /CT /D3/CUpn≥2 /D3/D2 /D8/CW/CT /CP/D1/D4/D0/CX/D8/D9/CS/CT |p1| /D0/CT/CP/CS/D7 /D8/D3 |p1|2≃2(1−γ)2CE1 −2Y1 ζ√γF“+1 −1”−1−γ γ(D2 1+2|D1|2). /B4/BH/BD/B5/C1/D2 /D8/CW/CT/CX/D6 /D4/CP/D4 /CT/D6/B8 /BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA /BE/BC/D7/D8/CP/D8/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D4/D6/D3 /CT/CS/D9/D6/CT/D9/D7/CT/CS /CX/D2 /CP /AS/D6/D7/D8 /D7/CX/D1/D4/D0/CX/AS/CT/CS /CP/D7/CT /CP/D2 /CQ /CT /CP/D4/D4/D0/CX/CT/CS /D8/D3 /CP /D1/D3 /CS/CT/D0/CX/D2 /D0/D9/CS/CX/D2/CV /D6/CT/CT/CS /CS/DD/D2/CP/D1/CX /D7 /D4/D6/D3 /DA/CX/CS/CT/CS /D8/CW/CP/D8Ze(ω) /CX/D7 /D6/CT/D4/D0/CP /CT/CS/CQ /DDZe(ω)D(ω) /BA /BV/D3/D2 /D0/D9/D7/CX/D3/D2 /CX/D7 /D2/D3/D8 /D7/D3 /D7/D8/D6/CP/CX/CV/CW /D8/CU/D3/D6/DB /CP/D6/CS /CP/D7/D6/CT/CT/CS /CS/DD/D2/CP/D1/CX /D7 /CP/D0/D7/D3 /CX/D2 /D8/CT/D6/CU/CT/D6/CT/D7 /DB/CX/D8/CW /D8/CW/CT /DA /D3/D0/D9/D1/CT /AT/D3 /DB /D6/CT/D0/CP/B9/D8/CX/D3/D2/D7/CW/CX/D4 /D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D2/CV /D8/D3 /CP /D1/D3/D6/CT /D3/D1/D4/D0/CT/DC /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /CU/D3/D6/D8/CW/CT /AS/D6/D7/D8 /D3/D1/D4 /D3/D2/CT/D2 /D8 /CP/D1/D4/D0/CX/D8/D9/CS/CT/BA /BT /D3/D6/CS/CX/D2/CV /D8/D3 /D8/CW/CT/CX/D6 /DB /D3/D6/CZ/B8 Dn /D8/CT/D6/D1/D7 /DB /D3/D9/D0/CS /D3/D2/D0/DD /D3 /D9/D6 /DB/CX/D8/CWYn /CX/D2 /D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /D3/CU |p1|2/DB/CW/CX /CW /CX/D7 /D2/D3/D8 /D8/CW/CT /CP/D7/CT/BA /BY/C1/BZ/BA /BL/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU|p1| /D3/D1/D4/D9/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX /CQ/CP/D0/B9/CP/D2 /CT /CU/D3/D6 /D7/D1/CP/D0/D0 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /DA /CP/D6/CX/D3/D9/D7 /DA /CP/D0/D9/CT/D7 /D3/CUkrL /BM /CU/D6/D3/D1/D8/CW/CX/D2 /D8/D3 /D8/CW/CX /CZ /D0/CX/D2/CT/D7/B8krL= 9,5,4,3,2,1.8,1.5,1.25 /B4∆l= 0 /B8 qr= 0.4 /B8 /CP/D2/CSfr= 1050Hz /B5/BA/C1/D2 /BY/CX/CV/BA /BL /CP/D6/CT /D7/CW/D3 /DB/D2 /D8/CW/CT /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /CS/CX/CP/CV/D6/CP/D1/D7 /CU/D3/D6 /CP /CW/CT/CP /DA/B9/CX/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS/BA /BY /D6/D3/D1 /D8/CW/CT /AT/CP/D8/D8/CT/D7/D8 /D8/D3/D2/CT /B4krL= 9.00 /B5 /D8/D3/D8/CW/CT /D7/CW/CP/D6/D4 /CT/D7/D8 /D3/D1/D4/D9/D8/CT/CS /B4krL= 1.25 /B5/B8 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /DB/CX/D8/CW/DA /CT/D6/DD /D7/D1/CP/D0/D0 /CP/D1/D4/D0/CX/D8/D9/CS/CT/D7 /CP/D6/CT /D4 /D3/D7/D7/CX/CQ/D0/CT /CP/D7 /D8/CW/CT /CS/CX/CP/CV/D6/CP/D1/D7 /D7/CW/D3 /DB/C0/D3/D4/CU /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2/D7 /D8/CW/CP/D8 /CP/D6/CT /D7/D9/D4 /CT/D6 /D6/CX/D8/CX /CP/D0/B8 /CX/BA/CT/BA/B8 /CS/CX/D6/CT /D8/B8 /CU/D3/D6 /CP/D0/D0/D8/CW/CT /D3/D1/D4/D9/D8/CT/CS /CP/D7/CT/D7/BA /C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /CU/D3/D6 /CP/D7/CT/D7 /DB/CW/CT/D6/CT/D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /CX/D7 /D0/D3/D7/CT/CS /D8/D3 /D8/CW/CT /D6/CT/CT/CS /D3/D2/CT /B4/BY/CX/CV/BA /BD/BC /B5/B8/CP /D7/D9/CQ /D6/CX/D8/CX /CP/D0 /C0/D3/D4/CU /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /D3 /D9/D6/D7 /CU/D3/D6 /CP /D6/CP/D2/CV/CT /D3/CU /CQ /D3/D6/CT/D0/CT/D2/CV/D8/CW/B8 /DB/CX/D8/CW /CP /DA /CT/D6/DD /D7/D1/CP/D0/D0 /D4/D6/CT/D7/D7/D9/D6/CT /D8/CW/D6/CT/D7/CW/D3/D0/CS /D0/CX/D2/CZ /CT/CS /D8/D3/D8/CW/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /D3/CU /D6/CT/CT/CS /D6/CT/D7/D3/D2/CP/D2 /CT /DB/CX/D8/CW /D8/CW/CT /D7/CT /D3/D2/CS /CQ /D3/D6/CT/D6/CT/D7/D3/D2/CP/D2 /CT/BM /D8/CW/CT /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /CX/D7 /D8/CW/CT/D6/CT /CX/D2 /DA /CT/D6/D7/CT/BA/C1/D2 /CP /D3/D6/CS/CP/D2 /CT /DB/CX/D8/CW /BZ/D6/CP/D2/CS /CT/D8 /CP/D0/BA /BE/BC/B8 /CX/D8 /CP/D4/D4 /CT/CP/D6/D7 /D8/CW/CP/D8 /D8/CW/CT/CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /CX/D7 /D2/D3/D8 /CP/D0/DB /CP /DD/D7 /CS/CX/D6/CT /D8/BA /CC/CW/CT/D6/CT /CT/DC/CX/D7/D8 /D3/D2/AS/CV/D9/D6/CP/B9/D8/CX/D3/D2/D7 /CU/D3/D6 /DB/CW/CX /CW /D3/D1/D4/D9/D8/CT/CS /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /CS/CX/CP/CV/D6/CP/D1 /D7/CW/D3 /DB/D7 /CP/D7/D9/CQ /D6/CX/D8/CX /CP/D0 /D4/CX/D8 /CW/CU/D3/D6/CZ/B8 /CX/BA/CT/BA/B8 /D7/D1/CP/D0/D0 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /DA /CP/D0/D9/CT/D7 /D3/CU/D4/D6/CT/D7/D7/D9/D6/CT /CQ /CT/D0/D3 /DB /D7/D8/CP/D8/CX /D6/CT/CV/CX/D1/CT /CX/D2/D7/D8/CP/CQ/CX/D0/CX/D8 /DD /D8/CW/D6/CT/D7/CW/D3/D0/CS/BA /CC/CW/CT/CA/CT/CT/CS/B9/CQ /D3/D6/CT /CX/D2 /D8/CT/D6/CP /D8/CX/D3/D2 /CX/D2 /DB /D3 /D3 /CS/DB/CX/D2/CS /CX/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /BD/BC/BY/C1/BZ/BA /BD/BC/BA /BV/D3/D1/D4/CP/D6/CX/D7/D3/D2 /D3/CU|p1| /D3/D1/D4/D9/D8/CT/CS /DB/CX/D8/CW /D8/CW/CT /CW/CP/D6/D1/D3/D2/CX /CQ/CP/D0/CP/D2 /CT /CU/D3/D6 /D7/D1/CP/D0/D0 /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2/D7 /CU/D3/D6 /BD/BG /DA /CP/D0/D9/CT/D7 /D3/CUkrL /D6/CT/CV/D9/D0/CP/D6/D0/DD/D7/D4/CP /CT/CS /CQ /CT/D8 /DB /CT/CT/D2 3.6 /CP/D2/CS4.20 /DB/CX/D8/CW /CP /D0/CX/CV/CW /D8/D0/DD /CS/CP/D1/D4 /CT/CS /D6/CT/CT/CS /B4∆l= 0 /B8qr= 0.01 /B8 /CP/D2/CSfr= 1050Hz /B5/BA/CQ /D3/D9/D2/CS/CP/D6/DD /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /D8 /DB /D3 /CP/D7/CT/D7 /CX/D7 /D2/D3/D8 /D8/D6/CX/DA/CX/CP/D0 /D8/D3 /CT/DC/B9/D4/D0/D3/D6/CT /CP/D2/CP/D0/DD/D8/CX /CP/D0/D0/DD /B8 /D6/CT/D5/D9/CX/D6/CX/D2/CV /D8/CW/CT /D1/CP/D8/CW/CT/D1/CP/D8/CX /CP/D0 /D7/D8/D9/CS/DD /D3/CU/BX/D5/BA /B4/BH/BC/B5/BA /BT/D0/D8/CT/D6/D2/CP/D8/CT/D0/DD /D2 /D9/D1/CT/D6/CX /CP/D0 /CT/DC/D4/D0/D3/D6/CP/D8/CX/D3/D2 /D3/CU /D4/CP/D6/CP/D1/B9/CT/D8/CT/D6 /D7/D4/CP /CT /CP/D2 /D0/CT/CP/CS /D8/D3 /D7/D3/D1/CT /D4/CP/D6/D8/CX/CP/D0 /D3/CQ/D7/CT/D6/DA /CP/D8/CX/D3/D2/D7/BA /C7/D2/CT/D3/CU /D8/CW/CT/D1 /CX/D7 /D8/CW/CP/D8 /D8/CW/CT /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT /CS/CX/D6/CT /D8 /DB/CW/CT/D2/D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2 /DD /CX/D7 /D0/D3/D7/CT /D8/D3 /D3/D2/CT /D3/CU /D8/CW/CT /CQ /D3/D6/CT /D6/CT/D7/D3/B9/D2/CP/D2 /CT/D7/BA/CC/CW/CT /D0/CX/D1/CX/D8 /D3/CU /CP/D0 /D9/D0/CP/D8/CX/D3/D2/D7 /CS/CT/D6/CX/DA /CT/CS /CW/CT/D6/CT /D2/CT/CT/CS/D7 /D8/D3 /CQ /CT /CT/D1/B9/D4/CW/CP/D7/CX/DE/CT/CS/BA /CF/CW/CT/D2|p1| /D8/CT/D2/CS/D7 /D8/D3 /DE/CT/D6/D3/B8 /D8/CW/CT /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /CU/D6/CT/B9/D5/D9/CT/D2 /DD /CX/D7 /D8/CW/CT /D3/D2/CT /CU/D3/D6 /DB/CW/CX /CW /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2/CX/D7 /D7/D3/D0/DA /CT/CS /B4CE1= 0 /B5/BA /C0/CX/CV/CW/CT/D6 /D3/D1/D4 /D3/D2/CT/D2 /D8/D7 /B4/D8/CW/CT /D7/CT /D3/D2/CS/D3/D2/CT /CW/CT/D6/CT /CP/D8 /D8/CW/CT /AS/D6/D7/D8/B9/D3/D6/CS/CT/D6 /CP/D4/D4/D6/D3 /DC/CX/D1/CP/D8/CX/D3/D2/B5 /CW/CP /DA /CT /CP /D2/D3/D2/B9/D2/CT/CV/D0/CX/CV/CX/CQ/D0/CT /CX/D2/AT/D9/CT/D2 /CT /D3/D2/D0/DD /CX/CUCE2 /CQ /CT /D3/D1/CT/D7 /D7/D1/CP/D0/D0 /D8/D3 /D3 /DB/CW/CT/D2/CP/D4/D4/D6/D3/CP /CW/CX/D2/CV /D3/D7 /CX/D0/D0/CP/D8/CX/D3/D2 /D8/CW/D6/CT/D7/CW/D3/D0/CS/B8 /CX/BA/CT/BA/B8 /CX/CU /D8/CW/CT /CW/CP/D6/CP /D8/CT/D6/B9/CX/D7/D8/CX /CT/D5/D9/CP/D8/CX/D3/D2 /CP /CT/D4/D8/D7 /D8/CW/CT /D7/D3/D0/D9/D8/CX/D3/D2/D7 ω /CP/D2/CS2ω /CU/D3/D6 /D8 /DB /D3 /D0/D3/D7/CT /DA /CP/D0/D9/CT/D7 /D3/CU /D1/D3/D9/D8/CW /D4/D6/CT/D7/D7/D9/D6/CT/BM CE1 /CP/D2/CSCE2 /CP/D6/CT /D7/D1/CP/D0/D0/D7/CX/D1 /D9/D0/D8/CP/D2/CT/D3/D9/D7/D0/DD /BA /C1/D2 /DA /CT/D6/D7/CT /CQ/CX/CU/D9/D6 /CP/D8/CX/D3/D2 /D1/CP /DD /D3 /D9/D6 /CX/D2 /D8/CW/CT 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2008-10-16

Sound emergence in clarinetlike instruments is investigated in terms of instability of the static regime. Various models of reed-bore coupling are considered, from the pioneering work of Wilson and Beavers ["Operating modes of the clarinet", J. Acoust. Soc. Am. 56, 653--658 (1974)] to more recent modeling including viscothermal bore losses and vena contracta at the reed inlet. The pressure threshold above which these models may oscillate as well as the frequency of oscillation at threshold are calculated. In addition to Wilson and Beavers' previous conclusions concerning the role of the reed damping in the selection of the register the instrument will play on, the influence of the reed motion induced flow is also emphasized, particularly its effect on playing frequencies, contributing to reduce discrepancies between Wilson and Beavers' experimental results and theory, despite discrepancies still remain concerning the pressure threshold. Finally, analytical approximations of the oscillating solution based on Fourier series expansion are obtained in the vicinity of the threshold of oscillation. This allows to emphasize the conditions which determine the nature of the bifurcation (direct or inverse) through which the note may emerge, with therefore important consequences on the musical playing performances.

Interaction of reed and acoustic resonator in clarinetlike systems

0810.2870v1

arXiv:0810.3893v1 [quant-ph] 21 Oct 2008On Wigner functions and a damped star product in dissipative phase-space quantum mechanics B. Belchev, M.A. Walton Department of Physics, University of Lethbridge Lethbridge, Alberta, Canada T1K 3M4 borislav.belchev@uleth.ca, walton@uleth.ca March 1, 2022 Abstract Dito and Turrubiates recently introduced an interesting mo del of the dissipative quantum mechanics of a damped harmonic oscilla tor in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter . We com- pare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products , and extend it to incorporate Wigner functions. The deformed (or damped ) star prod- uct is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find th at with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quad ratic Hamilto- nians share this property. 11 Introduction The quantum mechanics of dissipative systems has been studied inte nsively; for reviews, see [1, 2, 3]. Work started soon after the birth of quantu m mechanics and continues today. Perhaps the most fundamental approachis to consider the syste m of interest as interacting with an appropriate reservoir. Then the well-known q uantization methods that are valid for non-dissipative, closed systems can be a pplied to the system plus reservoir as a whole. Effective equations of motion for t he system can then be found by integrating out the reservoir degrees of fre edom, while making appropriate physical assumptions. Another technique is to work backwards, and derive the effective e quations by adapting quantization procedures to non-dissipative systems. The adapted quantization procedures should, in the end, agree with more funda mental treat- ments. Provided they do, they would be helpful, as shortcuts for t he more fundamental derivations, and possibly more. Canonical (operator) quantization has been adapted to dissipativ e systems either by using time-dependent Hamiltonians, complex Hamiltonians, o r by modifying the canonical Poisson brackets, and the corresponding operator com- mutators.1 We concern ourselves here, however, with a different quantization method: quantum mechanics in phase space, or deformation quantization (s ee Sect. 2). Specifically, we study Dito and Turrubiates [8] recent adaptation o f deforma- tion quantization to the paradigmatic dissipative system, the dampe d harmonic oscillator. The important innovation they introduce is a non-Hermitian γ-deformation of the Moyal star product of non-dissipative deformation quantiz ation, where γdenotes the damping constant. Their damped star product is built in turn on aγ-deformation of the classical Poisson bracket, and so recalls the m odified brackets of canonical operator quantum mechanics adapted to d issipation that werejust mentioned. Perhapsthis is not surprising, since canonica lclassicalme- chanics is a key ingredient of deformation quantization; the algebra of quantum observables is described as an ¯ h-deformation of the classical Poisson algebra on phasespace. Unlike in other schemes, the one proposedby Dito and Turrubiates onlymodifies the classical bracket and thereby the corresponding qua ntum star product–itusestheundamped Hamiltonian, forexample. TheDito-T urribiates 1Some relatively recent works with the latter approach are [4 , 5, 6, 7]. 2proposal is at least economical, since in other formulations involving m odified brackets, extra structure must be input. This paper is organized as follows. The next section is a brief review of ordinary (undamped) quantum mechanics in phase space. Section 3 studies the Dito-Turrubiates scheme and extends it to include Wigner functions and their evolution. The final section is a discussion, consisting of a concise su mmary, and a list of some of the many questions that remain. 2 Phase-space quantum mechanics We now give a brief review of phase-space quantum mechanics, to es tablish our notation and provide the essential background for next sect ion’s discussion of the Dito-Turrubiates scheme. Throughout this paper our trea tment will be limited to one non-relativistic particle travelling on the real line with coo rdinate q, and conjugate momentum p. For pedagogical introductions to phase-space quantum mechanic s (or defor- mation quantization), see [9, 10], for examples; and for more advan ced reviews, see the book [11], and references therein, including the pioneering work [12]. In our context, operators will be expressible entirely in terms of th e operator position ˆqandmomentum ˆ p. Thesemoregeneraloperatorswillalsobeindicated by carets. Phase-space quantum mechanics has a direct relations hip with the more commonly used operator quantum mechanics. The Wigner tran sformW maps operators to functions and distributions on phase space: W/parenleftBig ˆf/parenrightBig =f(p,q). (1) Hereˆfis an operator, and f(p,q) is the corresponding phase-space distribution, sometimes called the symbolofˆf. The most important property of the Wigner transform is encoded in W(ˆfˆg) = (Wˆf)∗(Wˆg). (2) Consequently, operators can be Wigner-mapped to symbols whose algebra is homomorphic to the original operator algebra, as long as the symbo ls are mul- tiplied using the so-called star-product ( ∗-product) ∗= exp/bracketleftBigi¯h 2(← ∂q→ ∂p−← ∂p→ ∂q)/bracketrightBig . (3) Operators can therefore be replaced by functions/distributions on phase space, and quantum mechanics can be done in phase space, without refere nce to oper- ators. 3The Moyal star product ∗inherits associativity from the operator product, by (2). Since ( ˆfˆg)†= ˆg†ˆf†, we also have (f∗g) =g∗f . (4) Herefindicates the complex conjugate of f. We say that ∗is Hermitian. The inverse of the Wigner transform is familiar in operator quantizat ion. The so-called Weyl map W−1, satisfies W−1(f∗g) = (W−1f)(W−1g). (5) The image of a polynomial in qandpof this Weyl map is the corresponding Weyl-ordered operator. The famous Moyal star product ∗of (3) and (5) is therefore intimately related to Weyl ordering. With other operato r orderings, other∗-products arise. The operator encoding information about the quantum state is the density matrix, and its Wigner transform is the central object of phase-s pace quantum mechanics. Up to normalization, it is the celebrated Wigner function. To consider evolution, start with the Liouville Theorem for the classic al phase-space density ρc: dρc dt=∂ρc ∂t+{ρc,H}= 0. (6) The Dirac correspondence then yields the equation of motion of the density operator (matrix) ˆ ρ i¯h∂ˆρ ∂t+ [ ˆρ,ˆH] = 0. (7) The Wigner transform of this is the evolution equation for the Wigner function ρ=ρ(p,q;t) i¯h∂ρ ∂t+ [ρ, H]∗= 0, (8) where the ∗-commutator is [a, b]∗=a∗b−b∗a . (9) Equivalently, the phase-space version of the Dirac quantization ru le, {a,b} →[a, b]∗ i¯h, (10) leads directly from (6) to (8). Defining adf[∗]g:= [f , g]∗, (11) 4(8) can be rewritten as i¯h∂ρ ∂t= adH[∗]ρ=:i¯hLρ . (12) The solution is just ρ(p,q;t) = exp/braceleftbigg−it ¯hadH[∗]/bracerightbigg ρ(p,q;0) = exp {tL}ρ(p,q;0).(13) Using the associativity of ∗-multiplication, this reduces to ρ(p,q;t) =U(p,q;t)∗ρ(p,q;0)∗U(p,q;−t), (14) with U(p,q;t) =∞/summationdisplay n=01 n!/parenleftbigg−itH ¯h/parenrightbigg∗n = Exp[∗]/parenleftbigg−itH ¯h/parenrightbigg .(15) Here Exp[∗](a) :=∞/summationdisplay n=01 n!a∗n(16) denotes the so-called ∗-exponential. Note that U(p,q;t) =U(p,q;−t), and U(p,q;t)∗U(p,q;t) =U(p,q;t)∗U(p,q;t) = 1. (17) Alternatively, substituting (14) into (8) yields the dynamical equat ion i¯h∂tU(p,q;t) =H∗U(p,q;t), (18) and its complex conjugate. Its solution, the symbol of the propag ator, is (15). The spectrum ofenergies and the Wigner functions ofstationarys tates can then be found by the spectral decomposition (or Fourier-Dirichlet expa nsion) U(p,q;t) =/summationdisplay EρE(p,q)e−iEt/¯h, (19) where H∗ρE=ρE∗H=EρE. (20) For the simple harmonic oscillator, with Hamiltonian H=p2 2m+1 2mω2q2, (21) the well-known results are U(p,q;t) =1 cos(ωt/2)exp/braceleftbigg2Htan(ωt/2) i¯hω/bracerightbigg , (22) 5and ρEn= 2(−1)nLn/parenleftbigg4H ¯hω/parenrightbigg exp/parenleftbigg −2H ¯hω/parenrightbigg , (23) whereEn= ¯hω(n+1/2), andLnis then-th Laguerre polynomial. The time evolution invokes more than the pure-state, or diagonal W igner functions, however. We need the off-diagonal ρE,E′=1 2π¯hW(|E/an}bracketri}ht/an}bracketle{tE′|), (24) satisfying the ∗-eigen equations H∗ρE,E′=EρE,E′, ρE,E′∗H=E′ρE,E′; (25) so thatρE,E=ρE. Assuming that the ∗-eigen functions (25) are complete, we expand ρ(p,q;t) =/summationdisplay E,E′RE,E′(t)ρE,E′(p,q). (26) Substituting into the equation of motion then gives the time-evolved Wigner function ρ(p,q;t) =/summationdisplay E,E′RE,E′(0) exp/bracketleftbigg−i(E−E′)t ¯h/bracketrightbigg ρE,E′(p,q).(27) To close this section, we will discuss alternatives to the Moyal star p roduct (see [10] and [12], e.g.). First, consider operator-ordering ambiguit ies. Since the Moyal product is intimately related to Weyl ordering, a different operator ordering will lead to a different star product. For example, if we use t he so- called standard ordering, where every ˆ qis placed to the left of every ˆ p, we find instead the standard star product ∗S=ei¯h← ∂q→ ∂p=∗ei¯h 2(← ∂q→ ∂p+← ∂p→ ∂q). (28) Since the two orderings can be simply related, there is a nice relation b etween the two star products: TS(f∗Sg) = (TSf)∗(TSg), (29) involving the invertible transition operator TS= exp/braceleftbigg −i¯h 2∂q∂p/bracerightbigg . (30) An ordering change preserves a bi-grading in the powers of pandq. By the Heisenberg commutation relation, ¯ hhas bi-grade (1,1) and so that of TSis (0,0). 6This kind of connection between different star products can be ext ended. In general, a star product ˜∗is considered equivalent to the Moyal one ∗, if we have ˜T(f∗g) = (˜Tf)˜∗(˜Tg). (31) Here˜Tstands for an invertible transition operator. If ˜∗, like∗, is an ¯h- deformation of the point-wise product of functions, then ˜T= 1 + O(¯h). (32) Associativity of ˜∗is guaranteed by the relation (31), if ˜Tis invertible: f˜∗g˜∗h=˜T/parenleftBig (˜T−1f)∗(˜T−1g)∗(˜T−1h)/parenrightBig . (33) If˜Tisreal,then ˜∗willbeHermitian, as ∗is.∗Sisanexampleofanon-Hermitian product, since TSis not real. By the Liebniz product rule, the equivalence relation (31) can be rew ritten as ˜∗=∗˜T−1[← ∂]˜T[← ∂+→ ∂]˜T−1[→ ∂]. (34) Here∂stands for either ∂qor∂p, ˜T[← ∂+→ ∂] :=˜T| ∂→← ∂+→ ∂, (35) and similarly for ˜T−1[← ∂] and˜T−1[→ ∂]. For example, TS[← ∂+→ ∂] = exp/braceleftbigg −i¯h 2(← ∂q+→ ∂q)(← ∂p+→ ∂p)/bracerightbigg , (36) by (30). Eqn. (34) makes it clear that equivalent star products ∗and˜∗only differ by a factor that is left-right symmetric. Consequently, it can be show n that lim ¯h→0[f,g]˜∗ i¯h= lim ¯h→0[f,g]∗ i¯h={f,g}, (37) the Poisson bracket. The equivalence between ˜∗and∗defined by (31) is known as c-equivalence. It is important because the Moyal ∗-product (3) is the unique, associative de- formation of the point-wise multiplication of functions on R2(andR2n), up to c-equivalence. The c in c-equivalence stands for cohomological. As such, it is a mathe mat- ical, rather than a physical equivalence. For example, consider the Husimi star product∗H(see [13], e.g.) ∗H=∗exp/braceleftbigg¯h 2(s2← ∂q→ ∂q+1 s2← ∂p→ ∂p)/bracerightbigg , (38) 7related to ∗by the transition operator TH= exp/braceleftbigg¯h 4/parenleftbig s2∂2 q+1 s2∂2 p/parenrightbig/bracerightbigg . (39) It is also easy to see that ∗Hsatisfies the condition (37). But ∗His used with the Husimi phase space distribution ρH(p,q;t) :=THρ(p,q;t) (40) whichdoesnotencodepreciselythesamephysicsastheWignerfunc tionρ(p,q;t). Eqn. (40) can be rewritten as ρH(p,q;t) =1 π¯h/integraldisplay dp′dq′ρ(p′,q′;t) ×exp/braceleftbigg −1 ¯h/bracketleftbigg(q−q′)2 s2+s2(p−p′)2/bracketrightbigg/bracerightbigg ,(41) indicating that the Husimi distribution is a smoothed version of the Wig ner function, coarse-grained by a squeezed2Gaussian weighting in phase space. So, although ∗His c-equivalent to the Moyal product, it describes different physics . 3 The damped harmonic oscillator in phase- space quantum mechanics We start in this section with a review of the Dito and Turrubiates [8] mo del of damping in a quantum harmonic oscillator in phase space, with comme nts added. A new contribution will then be described: our incorporation of Wigner functions and their evolution. First, a na¨ ıve proposal will be examin ed, and rejected, since it leads to an unacceptable evolution equation. The equation of motion will then be modified, and explained. Finally, its consequence s are outlined. One such consequence is that the Wigner function of the d amped harmonic oscillator follows the canonical flow. This property is also co mmon to all non-dissipative systems having quadratic Hamiltonians, including t he simple (undamped) harmonic oscillator. Since the damped and undamped ha rmonic oscillator and such systems are treated similarly in some other appro aches, this seems physically reasonable. Dito and Turrubiates [8] describe the damped harmonic oscillator usin g the Hamiltonian of the undamped simple harmonic oscillator. In a sense the n, they 2More precisely, the distribution functions for general swere introduced in [14, 15], while the Husimi distribution has s= 1. We call the Gaussian weighting of (41) squeezed because it is proportional to the Wigner transform of a squeezed stat e. 8describe its dissipation as kinematics. More precisely, the dissipation is encoded in a deformation of the classical Poisson bracket of the harmonic os cillator and its consequent quantum ∗-product. Theinitialobservationisthattheclassicalequationsofmotionofth edamped harmonic oscillator can be written as ˙q={q,H}γ=p/m; ˙p={p,H}γ=−mω2q−2γp , (42) using the undamped Hamiltonian (21). The price paid is that we must use a deformed Poisson bracket {f,g}γ:={f,g} −2γm∂pf ∂pg =f/parenleftBig← ∂q→ ∂p−← ∂p→ ∂q−2γm← ∂p→ ∂p/parenrightBig g , (43) with the damping constant γas deformation parameter. Notice that the de- formed bracket is no longer skew,3and doesn’t obey the Jacobi identity. This result applies to a general class of classical systems: if we use a ny Hamiltonian of the form ˜H=p2 2m+V(q), (44) with an arbitrary potential V(q), the same damping term 2 mγ˙qis still the only modification of the equation of motion for q(t). We will restrict attention here to the simple harmonic oscillator and its damped version, however. Just as the Moyal product is related to the Poisson bracket, the D ito- Turrubiates damped ∗-product is obtained by exponentiation of the deformed classical bracket (43): ∗γ:= exp/bracketleftBigi¯h 2(← ∂q→ ∂p−← ∂p→ ∂q−2γm← ∂p→ ∂p)/bracketrightBig =∗e−i¯hγm← ∂p→ ∂p.(45) Whyexponentiate? Intheundampedcase,itisnecessaryfortheh omomorphism of the∗-algebra with the operator algebra. Also, the Moyal ∗-product is the unique associative deformation of the point-wise multiplication of fun ctions on R2n, up to isomorphism (by transition operators – see below). No similar r esult forthe dampedproductisknown; neitheristhe operatoralgebraf orthe damped case. Exponentiation is therefore an assumption. 3The deformed bracket can be obtained from the Poisson bracke t by the substitutions ← ∂q→← ∂q−2αmγ← ∂pand→ ∂q→→ ∂q+2βmγ→ ∂p, for any real αandβsuch that α+β= 1. The sign difference in front of γbetween the substitutions for the left- and right-acting de rivatives foreshadows our proposal (70). 9On the other hand, suppose that we posit ∗γ:=F/parenleftbiggi¯h 2(← ∂q→ ∂p−← ∂p→ ∂q−2γm← ∂p→ ∂p)/parenrightbigg , (46) for some function F. Since{·,·}γ→ {·,·}asγ→0, requiring that lim γ→0∗γ= ∗selects the exponential function. The possibility that the form (46 ) is too restrictive remains, however. One key result of [8] is T(f∗g) = (Tf)∗γ(Tg), (47) with T= exp/parenleftBig−i¯hmγ 2∂2 p/parenrightBig . (48) This can be provedeasily using (34). That is, the Dito-Turrubiates d amped star product is c-equivalent to the Moyal star product. As discussed in the previous section, c-equivalence does not imply physical equivalence, and so t he damped star product has the potential to describe the dissipation of a dam ped harmonic oscillator. So, what effect does the Dito-Turrubiates transition operator Thave? First, it is clear that Tdoes more than change the ordering, since it has no fixed bi-grade in the powers of pandq. Its effect is therefore more profound. An important example is T/parenleftbiggp2 2m+V(q)/parenrightbigg =p2 2m+V(q)−i¯hγ 2; (49) (see also (58)).4Ttransforms a real Hamiltonian into a complex one. For such a conversion to be possible, it is necessary, but not sufficie nt, that T/ne}ationslash=T. Another consequence of a non-real transition operator is that ∗γis not Hermitian: a∗γb=b∗−γa/ne}ationslash=b∗γa . (50) Its c-equivalence with the Moyal star product ensures that the d amped star product∗γis associative, for any value of the damping constant γ. For example, ∗−γwill be useful later because of (50), and it is also associative. Howev er, problems exist if any two different damping parameters γandγ′are used. Care must be taken with regard to the order of multiplications. Clearly, f∗γ(1∗γ′g)/ne}ationslash= (f∗γ1)∗γ′g , (51) 4This last result is also interesting in its own right, since i t recalls a different approach to damped systems that uses complex Hamiltonians (see [1, 2], a nd also [16]). 10ifγ/ne}ationslash=γ′, for example. A unique product such as a∗γb∗−γcdoes not (automat- ically) exist. Use of a na¨ ıve substitution {a,b}γ?→[a, b]∗γ i¯h(52) leads to 0 =∂ργ ∂t+1 i¯h[ργ,H]∗γ, (53) whereργindicates the Wigner function of the dampedharmonic oscillator. This equation of motion integrates to ργ(p,q;t) =Uγ(p,q;t)∗γργ(p,q;0)∗γUγ(p,q;−t). (54) Here Uγ(p,q;t) :=∞/summationdisplay n=01 n!/parenleftbigg−itH ¯h/parenrightbigg∗γn = Exp[∗γ]/parenleftbigg−itH ¯h/parenrightbigg (55) satisfies the damped analogue of (18), i.e. i¯h∂tUγ(p,q;t) =H∗γUγ(p,q;t). (56) Dito-Turrubiatessolve(56), withoutdiscussingtheevolutionofWig nerfunc- tions. They then Fourier-Dirichlet expand the solution to find the sp ectrum of eigenvalues and eigenfunctions of H∗γργ,E=Eγργ,E. (57) Using (47), one finds Uγ(p,q;t) = Exp[ ∗γ]/parenleftbigg−itH ¯h/parenrightbigg =T/parenleftBig Exp[∗]/parenleftbigg−itT−1(H) ¯h/parenrightbigg/parenrightBig =T/parenleftBig Exp[∗]/parenleftbigg−it[H+i¯hγ/2] ¯h/parenrightbigg/parenrightBig =eγt/2T/parenleftBig U(p,q;t)/parenrightBig .(58) Using (22) and (48), this gives [8] Uγ(p,q;t) =exp(γt/2) cos(ωt/2)/bracketleftbig 1+2γ ωtan(ωt/2)/bracketrightbig ×exp/braceleftBigg 2tan(ωt/2) i¯hω/parenleftBigg p2 2m[1+2γ ωtan(ωt/2)]+1 2mω2q2/parenrightBigg/bracerightBigg .(59) The expansion of Uγgives the solutions of (57) in terms of the simple harmonic oscillator analogues: ργ,n=T/parenleftBig ρEn/parenrightBig , (60) 11with Eγ,n=En+i¯hγ 2=¯h 2[(2n+1)ω+iγ]. (61) For example, for n= 0 Dito and Turrubiates [8] find ρEγ,0=2/radicalbig 1−2iγ/ω ×exp/braceleftbigg −2 ¯hω/parenleftbiggp2 2m[1−2iγ/ω]+1 2mω2q2/parenrightbigg/bracerightbigg . (62) However, there is a fundamental problem with the basic equation of motion, eqn. (53). For the damped harmonic oscillator it yields 0 =∂ργ ∂t+1 i¯h[ργ,H]∗γ=∂ργ ∂t+1 i¯h[ργ,H]∗+iγ¯h∂p∂qργ. (63) Notice that the term proportional to the damping constant γis not real. But the eigenvaluesof the density matrix arethe populations, and they must be real. Time evolution must preserve the reality of the density matrix. The im aginary damping term in (63) means that a real density matrix does not rema in real as it evolves. An equally important flaw is revealed by the limit lim ¯h→01 i¯h[H,ργ]∗γ={H,ργ} /ne}ationslash={H,ργ}γ. (64) This is a direct consequence of the c-equivalence of ∗γand∗. But this shows that there is no damping in the classical limit. The classical limit of (53) is not correct! To try to understand better the origin of the difficulties with the pur ported equation of motion (53), let us consider how it might be “derived”. No tice that (53) would result from the undamped evolution equation (8) by the s ubstitution ργ?=T(ρ), (65) ageneralizationof(60). Thissimpleidentificationisappealinginpartbe causeit is similar to (40), that defining the Husimi phase-space distribution ρH(p,q;t). The Husimi equation of motion is found directly from that for the Wign er function, by substituting (40). Additional terms arise in the Husimi equation of motion compared to that for the Wigner function, since the coar se-graining evolves in time [13]. Just as the “twisting” by THmodifies the equation of motion, so does application of the Dito-Turrubiates transition oper atorT. The physical damping is meant to be introduced that way. 12Following (41), one might hope for an interpretation of T(ρ) as the Wigner distribution coarse-grained in momentum space. That point of view is not sensible, however, since the weighting would have to be Gaussian-like with an imaginaryexponent. Thisdoespointtotheoriginofthemainproblem, however: unlike the Husimi transition operator TH, the Dito-Turrubiates Tis not real. As a consequence, eqn. (49) can hold: Ttransforms a real Hamiltonian into a complex one. That is the crucial point here, we believe. Compare to a similar situation – if a non-self-adjoint Hamiltonian is used, the equation of m otion of the density operator is modified,5to 0 =i¯h∂ˆρ ∂t+ ˆρˆH−ˆH†ˆρ . (66) By analogy, we should consider6 −i¯h∂ργ ∂t=ργ∗γH−ργ∗γH . (67) Notice that the right-hand side of this equation is purely imaginary, s o that ∂ργ ∂t=−2 ¯hIm/parenleftbig ργ∗γH/parenrightbig . (68) This implies that the reality of the Wigner function ργ=ργ (69) is preserved in evolution.7We can therefore also write −i¯h∂ργ ∂t=ργ∗γH−H∗−γργ. (70) With this prescription it is easy to show that the classical limit makes se nse for the simple harmonic oscillator Hamiltonian: lim ¯h→0ργ∗γH−H∗−γργ i¯h={ργ,H}γ− {H,ργ}−γ 2={ργ,H}γ.(71) We emphasize that the relation (65) is then notobeyed. Let us now attempt to argue for (70) from other grounds. As a st arting point, let us assume that the Liouville Theorem still holds, and try to m odify the argument that led to (8) to justify the damped equation of mot ion (70). An 5See [17], for example, where the analogous modified equation of motion for a Heisenberg operator was shown to lead to a quantum anomaly. 6This form may possibly be related to the bi-orthogonal quant um mechanics discussed by Curtright and Mezincescu [18]. For related work in phase spa ce, see [19] and [20]. 7According to (68), if ργhad a non-zero imaginary part, it would not evolve. 13important advantageofthe Dito-Turrubiatesmethod is that it only modifies the classical brackets. That advantage would be lost if we need to input something to replace the Liouville Theorem. Therefore, we assume the equatio n of motion isdργ,c dt=∂ργ,c ∂t+{ργ,c,H}γ= 0. (72) The phase-space version of the Dirac quantization rule (10) above should there- fore be deformed to {a,b}γ→a∗γb−a∗γb i¯h=a∗γb−b∗−γa i¯h(73) for real observables aandb, in order to recover the equation of motion (67). One might be tempted to write ργ(p,q;t)?=Uγ(p,q;t)∗γργ(p,q;0)∗−γUγ(p,q;t) (74) as a real solution to the equation of motion (70). But this expressio n is ambigu- ous at best, as shown by the discussion around eqn. (51). Luckily, however, a formal solution to (70) canbe written, by deforming the undamped solution of (11-13). If we define ad(γ) ∗[f]g:=f∗−γg−g∗γf , (75) then (70) is i¯h∂ργ ∂t= ad(γ) ∗[H]ργ=:i¯hLγργ. (76) The solution is just ργ(p,q;t) = exp/braceleftbigg−it ¯had(γ) ∗[H]/bracerightbigg ργ(p,q;0) = exp/braceleftbig tLγ/bracerightbig ργ(p,q;0).(77) There is no associative ambiguity in this explicit solution. A more explicit result can be given immediately for the damped harmonic oscillator, having Lγ=mω2q∂ ∂p−p m/parenleftbigg∂ ∂q−2mγ∂ ∂p/parenrightbigg . (78) The quantum evolution is (70), but that reduces to (72), with ργ,c→ργ. The quantum Wigner function satisfiesthe classical equation ofmotion. The Wigner function at time tis therefore simply ργ(p,q;t) =f(pc(−t),qc(−t)), (79) 14wherepc(t) andqc(t) characterize the classical (damped) trajectories in phase space. Thus the quantum damped harmonic oscillatorfollows the clas sicalback- ward flow of the phase-space coordinates. Perhaps this is not sur prising, since for any quadratic Hamiltonian, such as that of the undamped simple h armonic oscillator, the same results holds (see [11], e.g.). 4 Discussion We first summarize. We have shown how to describe the dynamics of W igner functions in the Dito-Turrubiates scheme [8]. The non-Hermitian da mped star product ∗γis c-equivalent to the Moyal product ∗. Therefore, if the evolution equation of the damped Wigner function only involves ∗γ-commutators, only Poisson brackets survive in the classical limit, rather than the damp ed bracket {·,·}γ. The classical limit would then be damping-free, and therefore incor rect. However, the damped transition operator Ttransforms the simple harmonic oscillator Hamiltonian into a complex one, according to (49). Consequ ently, the evolution equation does not involve ∗γ-commutation, but must instead be (70). This ensures that a real Wigner function remains real as it evolves, and the classical limit is correct. Not only is the classical limit correct, it is exac t. The damped harmonicoscillatorin this formalismthereforefollowsthe clas sicalflow, a property shared with non-dissipative systems in phase space hav ing quadratic Hamiltonians. Most helpful to us were (i) comparisons of the damped star produc t∗γ with other physical star products that are also c-equivalent to th e Moyal∗, viz. the standard product ∗S, and the Husimi product ∗H; and (ii) the Heisenberg equation of motion for an operator observable modified for a non-H ermitian Hamiltonian (see [17], e.g.). Let us now conclude with a few of the many questions that remain. We hope progress can be made toward their answers. Should the damped ∗γ-product be obtained by exponentiating i¯h{·,·}γ/2?In the case at hand, much of the structure of the γ-deformed star product is irrel- evant. Since we use the quadratic Hamiltonian (21), the only terms t hat enter are those that are up to quadratic in← ∂and→ ∂. In this sense, our main result has a certain robustness. More work on this question would be helpful, h owever. Is there a deformed structure analogous to the Heisenberg-W eyl group that is relevant to the damped case? In the undamped case, the Heisenberg-Weyl group is the structure of paramount importance. The Moyal ∗-product simply 15provides a ∗-realization of that group. As discussed around (51), associativit y can be a problem in the damped case. But if (77) is an appropriate guid e, perhaps exp/braceleftBig ad(γ) ∗[ap+bq]/bracerightBig ecp+dq=ecp+dqe−i¯h(ad−bc+2mγac)(80) can servethe purpose. Here a,b,canddareconstants, so that eap+bqandecp+dq ∗-representelements ofthe Heisenberg-Weylgroup. When γ→0 in (80), a form of the defining ∗-relations of the Heisenberg-Weyl group is recovered. Can a solution to the damped equation of motion (70) analogou s to the un- damped formula (27) be written? It is clear from (77) that the ∗-eigen equation of ad(γ) ∗[H] is relevant, so one can start there. One possible ansatz follows, although it may only have relevance for very small γ. Suppose we find the time-independent off-diagonal Wigner matrix elements ργ;E,E′=ργ;E,E′(p,q) satisfying H∗−γργ;E,E′=Eργ;E,E′, ργ;E,E′∗γH=E′ργ;E,E′,(81) where the complex eigenvalues8are E=E+iλ ,E′=E′+iλ′, (82) withE,E′∈R, andλ,λ′∈R+. If the real eigenfunctions of (81) exist and are complete at all times t, then we can expand ργ(p,q;t) =/summationdisplay E,E′RE,E′(t)ργ;E,E′(p,q). (83) The equation of motion then yields ργ(p,q;t) =/summationdisplay E,E′RE,E′(0) exp/bracketleftbigg−i(E −E′)t ¯h/bracketrightbigg ργ;E,E′(p,q) =/summationdisplay E,E′RE,E′(0) exp/bracketleftbigg−i(E−E′)t ¯h/bracketrightbigg exp/bracketleftbigg−(λ+λ′)t ¯h/bracketrightbigg ργ;E,E′(p,q).(84) If this conjecture is correct, then the physically important ∗-eigen equations are those given in (81). It would not be clear then that the phase-spac e functions of (60) are directly relevant. The ansatz of (81-84) reduces to the undamped system when ¯ h→0. It is alsodevoidofunphysicalstationarystates–the imaginarypartso feigenvalues λ 8For the rolesuch complex eigenvalues play in another descri ption ofthe damped harmonic oscillator, see [21]. Wigner functions involving such eige nvalues are considered in [22]. 16andλ′bothcontributetothedampingofthedynamics. Furthermore,thesys tem is consistent with (79) if ( E −E′)/¯his independent of ¯ h. Such eigenvalues were obtained in [8], as indicated in (61). However, the corresponding solu tions (60) are not real – see (62), e.g. Can the Dito-Turrubiates scheme be related to other quantiz ation methods for dissipative systems (see [1, 2, 3])? Certainly, hints of connections with other models are apparent. For example, Dekker’s use of a complex Hamilto nian is recalled by T(H) =H−i¯hγ/2. Bateman’s doubled system of a damped and anti-damped oscillator comes to mind from ad(γ) ∗[H]ρ=H∗γρ−ρ∗−γH; in this last expression, however, the anti-damped ( γ→ −γ) system is dual rather than extra/auxiliary. The importance of resonances has been emp hasized by [22, 21] and others, and seems relevant to the ∗γ- and∗−γ-eigen equations of the previous paragraph. Finally, if the correspondence (65) were correct, then we would be able to define a damped Wigner transform Wγ:=T−1W=WˆT−1, (85) where ˆT−1= exp/braceleftbigg −im 2¯h/parenleftBig ad[ˆx]/parenrightBig2/bracerightbigg . (86) The form of this ˆTappears to be consistent with Tarasov’s [23] proposal to include superoperators in the operator formulation of quantum me chanics in order to adapt it to dissipative systems. Can other dissipative physical systems be treated in a simil ar way? One very useful example might be spin systems, with relaxation. Instead of dissipation, might other physical effects, such a s decoherence, be describable in the Dito-Turrubiates manner? Acknowledgements We thank our colleagues Saurya Das, Arundhati Dasgupta, and So urav Sur for useful discussion. M.A.W. also thanks M. Razavy for a helpful conve rsation. ThisworkwassupportedinpartbyaDiscoveryGrantfromtheNatu ralSciences and Engineering Research Council (NSERC) of Canada. 17References [1] H. Dekker, Phys. Rep. 80 (1981) 1. [2] M. Razavy, Classical and Quantum Dissipative Systems, Imperial College Press, London, 2005. [3] U. Weiss, Quantum Dissipative Systems, 2nd ed., World Scientific, S inga- pore, 1999. [4] A.N. Kaufmann, Phys. Lett. A 100 (1984) 419. [5] M. Grmela, Phys. Lett. A 111 (1985) 36. [6] C.P. Enz, Found. Phys. 24 (1994) 1281. [7] G. Bimonte, G. Esposito, G. Marmo, C. Stornaiolo, Phys. Lett. A 318 (2003) 313. [8] G. Dito, F. Turrubiates, Phys. Lett. A 352 (2006) 309. [9] J. Hancock, M.A. Walton, B. Wynder, Eur. J. Phys. 25 (2004) 52 5. [10] A.C. Hirshfeld, P. Henselder, Am. J. Phys. 70 (2002) 537. [11] C. Zachos, D. Fairlie, T. Curtright, Quantum Mechanics in Phase Space, World Scientific, Singapore, 2005. [12] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheim er, Ann. Phys. 111 (1978) 61 and 111. [13] K. Takahashi, Prog. Theor. Phys. Suppl. 98 (1989) 109. [14] N.D. Cartwright, Physica 83 A (1976) 210. [15] A.K. Rajagopal, Phys. Rev. A 27 (1983) 558. [16] S.G. Rajeev, Ann. Phys. 322 (2007) 1541. [17] J.G. Esteve, Phys. Rev. D 66 (2002) 125013. [18] T. Curtright, L. Mezincescu, J. Math. Phys. 48 (2007) 09210 6. [19] T. Curtright, A. Veitia, J. Math. Phys. 48 (2007) 102112. [20] F.G. Scholtz, H.B. Geyer, J. Phys. A: Math. Gen. 39 (2006) 101 89. [21] D. Chruscinski, Ann. Phys. 321 (2006) 854. 18[22] D. Chruscinski, preprint arXiv:math-ph/0209008 (2002). [23] V.E. Tarasov Phys. Lett. A 288 (2001) 173. 19

2008-10-21

Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find that with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quadratic Hamiltonians share this property.

On Wigner functions and a damped star product in dissipative phase-space quantum mechanics

0810.3893v1

1 Spin Transfer Torque as a Non-Conservative Pseudo-Field Sayeef Salahuddin*, Deepanjan Datta and Supriyo Datta School of Electrical and Computer Engineering and NSF Center for Computational Nanotechnology (NCN), Purdue University, West Lafayette, IN 47906 *Present address: Electrical Engineering and Computer Science, UC Berkeley, CA-94720 Abstract: In this paper we show that the spin transfer torque can be described by a pseudo magnetic field, proportional to the magnetic moment of the itinerant electrons that enters the Landau-Lifshitz- Gilbert equation in the same way as other external or internal magnetic fields. However, unlike an ordinary magnetic field, which is always conservative in nature, the spin torque induced ‘pseudo field’ may have both conservative and non-conservative components. We further show that the magnetic moment of itinerant electrons develops an out-of-plane component only at non- equilibrium and this component is responsible for the ‘Slonczewski’ type switching that acts against the damping and is always non-conservative. On the other hand, the in-plane components of the pseudo field exist both at equilibrium and out-of-equilibrium, and are responsible for the ‘field like’ term. For tunnel based devices, this term results in lower switching current for anti- parallel (AP) to parallel (P) switching compared to P to AP, even when the torque magnitudes are completely symmetric with voltage. 2 1. Introduction Spin torque devices [1, 2] that switch the magnetization of small magnets with spin polarized currents without any external magnetic field, have stirred tremendous interest due to their potential application as non volatile memory and also as nanoscale microwave oscillators. Although the concept of spin transfer torque has been demonstrated by a number of experiments [3, 4], quantitative measurement of spin transfer torque has been achieved only very recently [5, 6, 7]. All these measurements show a significant ‘field-like’ or out-of-plane torque in addition to the original in-plane torque predicted by Slonczewski [1]. This is very different from metallic channel based devices where the field like term is minimal. Recent theoretical studies have also shown the field like term to be significant in tunnel based devices [8, 9, 10, 11]. However, the details of how this field-like torque can affect the switching behavior is yet to be understood properly [5, 6, 7, 12, 13]. In this paper we first show that spin torque can be described by a pseudo magnetic field proportional to the net magnetic moment of the itinerant electrons, (normalized to the Bohr magneton ) providing a natural relationship between Slonczewski and field like terms: (1) Eqn. (1) is the central result of this paper and is derived in Section 2, starting from the Gilbert form of the LLG equation and introducing the spin-torque in terms of obtained from non- equilibrium Green function (NEGF) formalism for the conduction electrons. Note that so that the pseudo field is in the same direction as and enters Eqn. (1) just like other 3 magnetic fields included in. This may seem surprising; since it is well-known that spin- torque leads to phenomena like coherent precession that do not arise from ordinary magnetic fields. We show in section 3 that such phenomena can also be understood in terms of Eqn. (1) once we note that the pseudo-field representing the spin-torque has both a conservative component like the conventional magnetic fields included in and also a non-conservative component that makes the curl of overall to be non-zero: ; (Note that ) . We show that the out-of-plane component of is responsible for the Slonczewski term and is always non-conservative. On the other hand, the in plane components give the field like term and can introduce asymmetry in switching currents for opposite polarity in the voltage bias. Specifically, we shall show that for tunnel based devices, this field like torque can result in a lower switching voltage for AP to P switching compared to P to AP, even when the torque magnitudes are completely symmetric with voltage. This can be understood by noting that for tunneling devices, the in plane component of the pseudo field (responsible for the field like term) remains conservative even away from equilibrium, and thus acting like an ordinary magnetic field parallel to the direction of the fixed magnet that helps switching from AP to P while hindering P to AP transition. 2. Spin-Torque as a Pseudo Field A typical spin torque device is shown schematically in Fig. 1. Left contact is the fixed ferromagnet having magnetization along . Right contact is soft layer and its magnetization points along which is free to rotate in easy (z-x) plane. An insulating layer separates the 4 ferromagnetic contacts. Following Gilbert’s prescription, we write, the rate of change of the direction of the magnetization as (2) where the spin torque is obtained by integrating the divergence of the spin current carried by the conduction electrons over the volume of the magnet. Below we will use the non- equilibrium Green’s function formalism to show that (3) where is the magnetic moment of the conduction electrons (normalized to ) and is the energy splitting of the conduction electrons due to the exchange interaction with the localized spins that comprise the magnet. Combining Eqns (2) and (3) we obtain our central result stated earlier in Eqn. (1) with . Proof of Eqn. 3: We start from the expression for the (2x2) operator representing at site in a discrete representation (see Eqn. (8.6.3), page 317, [15]) for the conduction electrons . is the (2x2) correlation matrix at site and the Hamiltonian [ ] is given by , where is the spin-independent part and is the spin-dependent part arising from the exchange interaction with the magnet pointing along , with being a 2x2 identity matrix and representing the Pauli spin matrices. The divergence of the spin-current is obtained from the operator (4) and substituting for , we get (note: is the Levi-Civita antisymmetric tensor) 5 (5) so that the spin-torque is given by (6) Defining as the magnetic moment (normalized to ) of the conduction electrons, we obtain which is the same as as stated above in Eqn. (3). This completes our proof of Eqn. (1). Note that this expression for torque is consistent with previous studies [9,10,11]. 3. Relation to the standard form It is shown in the Appendix A that if the conduction electrons are in equilibrium then the spin density can be written in the form, (7) but away from equilibrium, the spin density remarkably develops an additional out-of plane component that is perpendicular to the magnetization of both magnets: (8) so that from Eqn. (3) the spin-torque comes out as . which has the same form as the standard torque equations used extensively in literature [5, 6, 7]. Our formulation leads to a simple criterion for coherent precession which is considered one of the hallmarks of spin-torque. To see this we note that one can write Eqn. (1) in the following form 6 (9) Noting that coherent precession arises when the second term is zero, we obtain (10) so that Eqn. (9) reduces to yielding as the precession frequency. Since the " " term is zero under equilibrium conditions (see Appendix A ), coherent precession is possible only under non-equilibrium conditions, as one would expect. Nature of the pseudo field: Now that we have established the relationship between our concept of pseudo field and the standard form of torque having a Slonczewski and a field like term, let us try to examine the pseudo field more deeply. The first term in Eqn. (8) is in the same direction as the magnet. Hence this does not contribute anything to the torque and may be ignored. As for the second term, we see that if the coefficient were independent of and , . This means that for the case when is independent of and , the second term acts as a conservative field. As for the third term in Eqn. (8) we show in Appendix B , that independent of the angular dependence of c, the third term always constitutes a non-conservative field. To summarize, the pseudo field that gives the spin transfer torque has two terms, one of which is in-plane with the magnets and may or may not be a conservative field. On the other hand, the second term is out- of-plane, is always non-conservative and can only appear at out-of-equilibrium. 7 4. Switching behavior in tunneling barrier based spin torque devices: Let us now consider switching in tunneling barrier based spin torque devices. Our formulation is based on the coupled NEGF-LLG methodology described above. The details of NEGF implementation of transport for tunneling barrier based spin torque devices have been discussed in [16]. Here we shall skip the details and only present results. In brief, our formulation is based on effective mass description. We sum over the transverse modes assuming that the inter-mode coupling is negligible. Also, we only take the torque at the surface of the soft magnet. We have shown [16] that this methodology gives reasonable agreement with both the current and the tunneling magneto resistance (TMR) as a function of voltage by using effective mass and barrier height as fitting parameters. In this case, we shall use similar parameters as used in [16, 17]. A typical bias and angular dependence of and and the torque components are shown in Fig. 2. Note that the bias and angular dependence of the torque components show the same qualitative dependence as in the recent ab-initio study [10]. The bias dependence of and can be approximately written as . Also, from the Fig. 2, it is evident that both and are completely independent of and for the tunneling device as we have considered here. This means that the pseudo field will have a conservative part due to ( ), where is symmetric with voltage. Fig. 3 shows the switching of magnetization with applied voltage. One would see that it takes less time to go from AP to P configuration compared to P to AP for the same magnitude of voltage. This means that it would take more voltage to switch from P to AP for a particular width of the voltage pulse. This result is surprising considering that both the torque magnitudes shown in Fig. 2 are completely symmetric with voltage. However the reason would be clear if we look at the pseudo field. As 8 mentioned above, ( ) is conservative and does not change polarity with voltage. This means that ( ) acts as if an external magnetic field was applied in the direction of irrespective of the voltage polarity. As a result, it directly changes the potential energy of the system helping the AP to P transition while acting against the P to AP switching. An important thing to note is the fact that , the equilibrium component of , would also introduce an asymmetry in switching voltage and it manifests itself as an exchange field in the equilibrium R-H loops. However, the significance of being independent of angular position is that even if we compensate for this exchange field by making the equilibrium hysteresis loop completely symmetric, for example, by applying an external magnetic field, there will still be an asymmetry in the switching current due to . Notice that this asymmetry in the switching voltage is not dependent on the symmetric nature of shown in Fig. 2. As long as is not purely anti-symmetric, the effect remains. This asymmetry is also in addition to that arising from any voltage asymmetry in the magnitude of , i.e., the in-plane torque component. It is worth mentioning, however, that two [6,7] of the three torque measurement experiments done so far have found to be anti-symmetric (making its magnitude symmetric) at least in the low voltage region in agreement with ab-initio calculation [10]. Our own calculations also support the anti-symmetric nature of . This suggests that the dominant reason for the asymmetry in switching voltages for tunnel based devices [18] may arise from field like terms. This is surprising considering the fact that the field like term was minimal and was normally ignored in the earlier devices based on metallic channels. 9 5. Conclusion: By formulating spin transfer torque as a pseudo field proportional to the spin resolved electron density, we have been able to show how the field like torque can introduce a voltage symmetric conservative torque on the magnet and thereby cause an asymmetry in the switching voltages for tunneling barrier based spin torque devices. It will be interesting to explore if this effect can be utilized to reduce the switching voltage by appropriate device design. Our results also suggest that that one should consider maximizing the electron density while exploring novel device designs [19, 20, 21] involving spin transfer torque. Furthermore, the ability to change the potential energy of a system (by virtue of a voltage induced conservative field [22]) may also have important implications for voltage induced energy conversion and phase transition. 10 Appendix A: Proof of Eqn(s) (7), (8) Let us assume that the fixed magnet and the soft magnet are both in the plane (see Fig. 1.) so that the Hamiltonian (see Eqn. (6)) completely real, assuming the vector potential to be zero. It can then be shown that the Green’s function is symmetric (Chapter 3, [15]): . We shall use this symmetry property of the Green’s function to understand the form of which is defined in terms of correlation function , with given by [14] (A1) where, and are the partial spectral functions due to contact 1 and 2 respectively . Now, both are Hermitian, but not symmetric, since , so that . However, the total spectral function can be written as and hence symmetric: . This means that is purely real and can be expressed as (A2) While At equilibrium, only the term in Eqn. (A1) is non-zero, so that the magnetization can be written as stated in Eqn. (7) while under non-equilibrium condition, it has the more general form stated in Eqn. (8): 11 Appendix B: Non-Conservative Nature of Pseudo-field In Appendix A we showed that the pseudo-field lies entirely in-plane at equilibrium, but can have an out-of-plane component away from equilibrium. We will now show that at equilibrium it is conservative, but away from equilibrium, the out-of-plane component makes it non- conservative. Assume that the fixed magnet points along (Fig.1 ) and the soft magnet points along where defined in a spherical co-ordinate system. The other unit vectors can be written as and . We can write the curl of the pseudo-field as (B.1) where we have dropped terms involving , since we assume , to be fixed and only consider changes in the direction of the magnetization of the soft magnet relative to the fixed magnet ( ). We write the pseudo-field as so that we obtain (with ), (B.2a) (B.2b) (B.2c) Now, if we change the of the soft magnet, its angle with the fixed magnet changes and in response the pseudo field could in general change arbitrarily making both terms in Eqn.(B.1) non-zero. But the component contributes nothing to the actual torque, and we could arbitrarily define it to be a constant so that the only non-zero curl arises from the first term: 12 (B.4) This means the curl is non-zero unless and it does not change as the of the soft magnet is rotated. This can happen only if is identically zero, which is exactly what happens under equilibrium conditions (see Appendix A ): the pseudo-field only has in-plane components, which means that , making . Hence the pseudo-field is in- plane and conservative in equilibrium, but away from equilibrium it can have an out-of-plane component that will make it non-conservative. 13 References: 1. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2. L. Berger, Phys. Rev. B 54, 9353 (1996). 3. I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman, Science, 307, 228 (2005). 4. G. D. Fuchs, J. A. Katine, S. I. Kiselev, D. Mauri, K. S. Wooley, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 96, 186603 (2006). 5. Kubota et. al., Nature Phys., 4, 37 (2008). 6. Sankey J. K. et. al., Nature Phys, 4, 67 (2008). 7. Deac, A. M. et. al., Nature Phys., 4, 803 (2008). 8. Slonczewski J. C. and Sun, J. Z., J. Magn. Magn. Mater. 310, 169 (2007). 9. Theodonis, I et. al., Phys. Rev. Lett, 97, 237205 (2006). 10. Heiliger, C. and Stiles, M.D., Phys Rev. Lett., 100, 186805 (2008). 11. P.M Haney et. al., J. Magn. Magn. Mater. 320, 1300 (2008). 12. Sun, J.Z. and Ralph, D.C., J. Magn. Magn. Mater. 320, 1227 (2008). 13. Ito, K. et al., Appl. Phys Lett., 89, 252509 (2006). 14. S. Datta, Quantum Transport: Atom to transistor , Cambridge University Press (2005). 15. S. Datta. Electronic Transport in Mesoscopic Systems , Cambridge University Press (1995). 16. S. Salahuddin et. al., Technical Digest of IEDM, 121 (2007). 17. Recent theoretical studies have successfully achieved quantitative agreement for torque with experimental measurement [10] based on ab-initio band structure. However, since we are only interested in the qualitative nature of the torque, we believe that our effective mass treatment should suffice. 18. M. Hosomi et al., Technical Digest of IEDM, 473, (2005). 19. Huai et. al., Appl. Phys. Lett. 87, 222510, (2005). 20. Meng et al., Appl. Phys. Lett. 88,082504, (2006). 14 21. Fuchs G. D. et. al ., Appl. Phys. Lett. 86, 152509 (2005). 22. Di Ventra M. et. al ., Phys. Rev. Lett, 92, 176803 (2004). 15 Figure Captions: Fig. 1. Schematic of tri-layer device. The left contact is the pinned ferromagnet having magnetization along the z-axis. The right contact is the free layer and the channel material is an oxide. is the easy axis and is the easy plane. Transport occurs in y-direction. The device region is modeled using appropriate Hamiltonian, , and electrostatic potential and the contacts are taken into account by self energy matrices and , whose anti-Hermitian components are broadening matrices due to contacts 1 and 2 respectively [14]. Fig. 2. (a) Typical variation of and as a function of voltage for tunnel based spin torque devices. shows symmetric and shows anti-symmetric voltage dependence. (b) Bias dependence of in-plane and out-of-plane components of Torque for tunnel based spin torque devices. (c), (d) The variation of and as a function of at a fixed and as a function of at a fixed respectively at a fixed voltage for a tunnel based spin torque device. We see that and at a fixed voltage are independent of both and . (e) Typical variation of differential torque (w.r.t. voltage) as a function of the relative angle between the magnetizations of the ferromagnetic electrodes. Fig. 3. The switching dynamics with same voltages with opposite polarity: positive voltage for AP to P and negative voltage for P to AP. For clarity, we have only marked the z component with bold blue color. The dashed curve shows AP to P and the solid curve shows P to AP transitions. (a) For the same voltage amplitude, the AP to P transition is faster than P to AP. Note the dashed line where the AP to P transition is almost complete while the P-to-AP transition is just around its half-way mark. (b) To get a symmetric switching time, it takes almost 30% more voltage (V-) for P-to AP compared to the AP-to-P transition. No external magnetic field has been assumed. 16 Fig. 1 17 Fig. 2 (a) Fig. 2 (b) Fig. 2 (e) Fig. 2 (c) Fig. 2 (d) 18 Fig. 3 (a) Fig. 3 (b)

2008-11-21

In this paper we show that the spin transfer torque can be described by a pseudo magnetic field, proportional to the magnetic moment of the itinerant electrons that enters the Landau-Lifshitz-Gilbert equation in the same way as other external or internal magnetic fields. However, unlike an ordinary magnetic field, which is always conservative in nature, the spin torque induced pseudo field may have both conservative and non-conservative components. We further show that the magnetic moment of itinerant electrons develops an out-of-plane component only at non-equilibrium and this component is responsible for the Slonczewski type switching that acts against the damping and is always non-conservative. On the other hand, the in-plane components of the pseudo field exist both at equilibrium and out-of-equilibrium, and are responsible for the field like term. For tunnel based devices, this term results in lower switching current for anti-parallel (AP) to parallel (P) switching compared to P to AP, even when the torque magnitudes are completely symmetric with voltage.

Spin Transfer Torque as a Non-Conservative Pseudo-Field

0811.3472v1

arXiv:0812.2209v1 [quant-ph] 11 Dec 2008Frequency-dependent Drude damping in Casimir force calculations R. Esquivel-Sirvent1,∗ 1Instituto de F´ ısica, Universidad Nacional Aut’onoma de M´ exico, Apdo. Postal 20-364, M´ exico D.F. 01000, M´ exico Abstract The Casimir force is calculated between Au thin films that are described by a Drude model with a frequency dependent damping function. The model para meters are obtained from available experimental data for Au thin films. Two cases are considered ; annealed and nonannealed films that have a different damping function. Compared with the calc ulations using a Drude model with a constant damping parameter, we observe changes in the Casi mir force of a few percent. This behavior is only observed in films of no more than 300 ˚Athick. ∗Corresponding author. Email:raul@fisica.unam.mx 1I. INTRODUCTION The advent of precise and systematic Casimir force experiments sin ce the late 90 [1, 2, 3, 4, 5, 6, 7, 8] has prompted an intense research on the role of th e dielectric properties of the involved materials. Although, the Lifshitz theory [9] explicitly req uires the dielectric function of the materials, an important issue is which one is the corre ct dielectric function that is consistent in describing the optical properties of the mater ials and the measurements of the Casimir force. The first approachis to assume a plasma model for the dielectric fun ction[10] or the more realistic Drude model, that has been extensively used when extrapo lating to low frequencies tabulated data. Although it may be thought that the problem of usin g a dielectric function is straight forward, controversial results have been reported, in particular in relation to the use of the Drude model in finite temperature calculations of the Cas imir force . The use of of the Drude model in Lifshitz theory seems to violate Ner nst heat theorem, while the plasma model presents no problem at all, but is not realistic in t he representation of the dielectric properties of metals. The plethora of papers and c omments shows that the issue is far from settled [11, 12, 13, 14, 15, 16, 17]. Even without considering finite-temperature effects, the choice o f the dielectric function can change the calculations of the Casimir force. For example, for A u samples the Drude parameters extracted from tabulated data vary depending on th e sample. The variations on the Drude parameters have important implications in the Casimir forc e calculations since difference of up to 5% are obtained[18]. A similar result was obtained by Svetovoy et al. where [19] different Au samples were prepared under similar condition s with thicknesses ranging from 120 nm to 400 nm. From measured ellipsometry data it wa s verified that the plasma frequency varies from 6.8 eV to 8.4 eV for this set of particula r samples, changing the Casimir force a few percent. The conclusions of these works sh ow that there is not a standard plasma frequency or damping parameter for Au, it is samp le dependent and in situ measurements are needed. Experimentally, the effect of thin films o n the Casimir force was shown experimentally by Iannuzzi [21, 22] who demonstrated tha t the Casimir attraction between a metallic plateanda metal coatedsphere depended onthe thickness of thecoating. The reduction of size can significantly change the physical paramet ers of a system. In the caseofthinfilms, asthethickness ofthefilmapproachesthemeanf reepath, theconductivity 2show a sharp decrease in its values. This was shown experimentally by Kastle [23] with Au films whose thickness varied from 2 nmto 70nm. Indeed, a conductor-insulator transition is observed as a function of film thickness in Au [24]. As a function of film thickness, the Casimir force decreases with decreasing film thickness until a critica l thickness is reached after which the Casimir force increases even with decreasing film thic kness[25]. To further the discussion about the possible factors that influenc e Casimir force calcula- tions, in this paper we introduce a frequency dependent damping γ(ω) in the Drude model. This model describes the dielectric properties of thin films and chang es if the film has been annealed. II. FREQUENCY DEPENDENT DAMPING The classical Drude model the local dielectric function is given by ǫ(ω) = 1−ω2 p ω(ω+iγ), (1) whereωis the frequency, ωpthe plasma frequency and γthe damping parameter that is constant for a fixed temperature. Measurements of the dielectric properties of Au thin films by M. L. Th eye [26] from reflectance and transmittance data showed a deviation from the b ulk Drude behavior of Au. The deviations from the Drude model were explained by introducing a frequency dependent relaxation time due to electron-phonon and electron-ion interactio ns of the form: γ=γ0+Aω2. (2) However, a more precise correction to the Drude model was introd uced by Nagel [27] to include the frequency dependence of the damping parameter. It w as observed that sample preparationwas relevant intheoptical behavior of thematerial, sin ce themeasured datawas different for annealed and nonannealed samples. An explanation of t he frequency dependent relaxation time with the sample and the role that annealing plays in the o ptical properties was given by Nagel [27] using a classical two carrier model. The model assumes that in a thin film sample there are two regions labeled aandb. One where the electrons see a perfect lattice, inside crystallitesandasecondhighlydisorderedregionbetw een thecrystallites. The electrons respond differently in each of these regions and have a diff erent damping rate, say 3γaandγb, and a different plasma frequency ωpaandωpb. Ignoring local field corrections in the determination of the optical response, the two carrier model yields an effective damping parameter given by γeff=γa/bracketleftBigg 1+ωpb ωpa/parenleftBiggω2+γ2 a ω2+γ2 b/parenrightBigg/bracketrightBigg−1 +γb/bracketleftBigg 1+ωpa ωpb/parenleftBiggω2+γ2 b ω2+γ2 a/parenrightBigg/bracketrightBigg−1 , (3) whereωpa,b= 4πNa,be2/m∗a,b. This last expression is general and the behavior observed by They´ e Eq. (2) is obtained if ωτa>>1 andωτb<<1. Equation (3) assumes that the effective masses in both regions are the same, thus Nb/Na=ωpb/ωpa. Thus the thin film can be modeled by a Drude dielectric function with an effective damping constant. Inthispaper wewill usetheparameters considered by Nagel [27] t hat fittheexperimental data of They´ e [26] for an annealed and a nonanneald Au film. The par ameters are shown in Table 1. TABLE I: Parameters for the two films used in our calculations , after [27]. film Nb/Naγa×1014s−1γb×1014s−1 annealed 0.0077 0.93 25 nonannealed 0.058 1.18 25 In Figure 1, we plot the effective damping γeffEq. (3) as a function of frequency for the samples considered by Nagel. As expected, annealing the film will r educe the number of impurities and the damping constant should be smaller. To study the effect that a frequency dependent damping has on th e Casimir force, we consider two plates separated a distance Lwith a thin film deposited on their surface. The films are described by the parameters given in Table. 1 We calculate th e reduction factor define as the Casimir force calculated using Lifshitz formula Fdivided by the Casimir force between perfect conductors F0; this isη=F/F0. This is, η=120L4 π4/integraldisplay∞ 0QdQ/integraldisplay q>0dkk2 q(Gs+Gp), (4) whereGs= (r−1 1sr−1 2sexp(2kL)−1)−1andGp= (r−1 1pr−1 2pexp(2kL)−1)−1. In these expres- sions, the factors rp,sare the reflectivities for porspolarized light , Qis the wavevector component along the plates, q=ω/candk=√q2+Q2. 4In Figure (2) we show the reduction factor for Au samples describe d by a classical Drude (ηc) model where the damping is constant, and the Drude model with a f requency dependent damping parameter for an annealed ( ηa) and nonannealed samples ( ηn). For the classical Drude model the parameters are ωp= 9eVandγ= 0.02eV. The difference between the annealed and nonannealed sample is small. The reduction factor incre ases as compared to the annealed and nonannealed samples. The difference in reduction f actor can better be seen by computing the percent difference ∆ = 100 |ηa,n−ηc|/ηcas a function of the separation between the plates. At large separations, the percent difference is at most of ∼2.2% for the nonannealed sample and ∼1.6% for the annealed film. III. CONCLUSIONS The goal of high-precission measurements at small separations of the Casimir force, re- quires that the dielectric function of the materials are well known. I n this paper we have considered the effects of the frequency dependent damping when Au films of a few hundreds of Angstroms are considered. Besides, the effect of sample prepa ration such as annealing the film also changes the Casimir force. Although the changes are of less than 3% at large separations, we wish to point out that this is an example where using a simple Drude model, for example, to extrapolate to low frequencies tabulated data, is n ot granted unless thick films are used. This is important if high precision measurements are se ek. Furthermore, preparing a sample in ultra high vacuum can change the optical prope rties of the Au films [28], showing the need for in-situ determination of the dielectric prop erties of the samples used in Casimir experiments. Finally, the relevance of the frequency dependent damping parame ter is in the finite- temperature effects, since now the damping will also depend on the M atsubara frequencies. This will be consider in a future paper. Acknowledgements: Partial support from DGAPA-UNAM project N o. 113208 . 50 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.060.070.080.090.10.110.120.130.140.150.16 ω(eV)γeff(eV) annealed nonannealed FIG. 1: Effective damping as a function of frequency using Eq. ( 3) for an annealed sample (dashed line) and a nonannealed sample (solid line). 100 200 300 400 500 600 7000.20.30.40.50.60.70.8 Separation (nm)η DRUDE nonannealed annealed FIG. 2: Reduction factor as a function of separation for the c lassical Drude model with a constant damping parameter (solid line), the annealed Au (*) sample a nd the nonannealed Au sample (+). 6100 200 300 400 500 600 7000.811.21.41.61.822.22.42.6 Separation (nm)∆(%) annealed nonannealed FIG. 3: Percent difference ∆ between the reduction factors cal culated in Fig.(2). References [1] Lamoreaux S K 1997 Phys. Rev. Lett. 785; 1998Phys. Rev. Lett. 815475 [2] Mohideen U and Roy A 1998 Phys. Rev. Lett. 814549 ; Roy A , Lin C Y and Mohideen U 1999Phys. Rev. D60, 111101(R) ; Harris B W , Chen F and Mohideen U 2000 Phys. Rev. A 62, 052109 [3] Ederth T (2000) Phys. Rev. A62062104 [4] Chan H B, Aksyuk V A , Kleiman R N, Bishop D J and F. Capasso 20 01Science2911941; 2001Phys. Rev. Lett. 87211801 [5] Bressi G, Carugno G , Onofrio R, and Ruoso G, Phys. Rev. Lett. 88041804 (2002) [6] Decca R S, L´ opez D, Fischbach E, and Krause D E 2003 Phys. Rev. Lett. 91050402 ; Decca R S, Fischbach E, Klimchitskaya G L, Krause D E, L´ opez D, and M ostepanenko V M 2003 Phys. Rev. D68116003 [7] van Zwol P J, Palasantzas G, De Hosson J Th M 2008 Phys. Rev. B77075412 [8] Chan H B, Bao Y, Zou J Cirelli R A Klemens F Mansfield W M and Pa i S C 2008 Phys. Rev. Lett101030401 [9] E. M. Lifshitz 1956 Zh. Eksp. Teor. Fiz. 2994 [1956 Sov. Phys. JETP 273 ]. 7[10] Lambrecht A, Reynaud S 2000 Eur. Phys. J. D8309 [11] J. S Hoye, I. Brevik, S. E. Ellingsen and J. B. Aarseth 200 7 Phys. Rev. E 75, 051127 [12] Bimonte G 2007 New J. Phys. 9281 [13] Ellingsen S A 2008 Phys. Rev. E78021120 [14] Mostepanenko V M and Geyer B 2008 J. Phys. A: Math. Thor. 4116014 [15] Lamoreaux S K 2008 Possible resolution of the Casimir fo rce finite temperature correction ”controversies” Preprint arXiv:0801.1283v1 [quant-ph] [16] Decca R S, Fischbach E, Geyer B, Klimchitskaya G L, Kraus e D E, Lopez D, Mohideen U, Mostepanenko V M 2008 Comment on ” Possible resolution of t he Casimir force finite temperature correction ”controversies”” Preprint arXiv:0803.4247v1 [quant-ph] [17] Ellingsen A., Brevik I, Hoye J S , Milton K A (2008) Low tem perature Casimir-Lifshitz free energy and entropy: the case of poor conductors Preprint arXiv:0809.0763v1 [quant-ph] [18] Pirozhenko I, Lambrecht A and Svetovoy V B 2006 New Journal of Physics 8238 [19] Svetovoy V B, van Zwol P J, Palasantzas G, De Hosson J Th M 2 008Phys. Rev. B77035439 [20] Pirozhenko I and Lambrecht A 2008 Phys. Rev. A77, 013811 [21] Iannuzzi D, Lisanti M, Munday J N and Capasso F 2006 J. Phys. A: Math. Gen. 396445 [22] Lisanti M, Iannuzzi D and Capasso F 2005 Proc. Nat. Acad. Sci. 10211989 [23] Kastle G, Boyen H G , Schroder A , Plettl A and Ziemann P 200 470, 165414 [24] Walther M , Cooke D G , Sherstan C, Hajar M, Freeman M R and H egmann F A 2007 Phys. Rev.B76125408 [25] Esquivel-Sirvent R. 2008 Phys. Rev A77042107 [26] Theye M L 1970 Phys. Rev. B23060 [27] Nagel S R and Schnatterly E 1974 Phys. Rev. B91299 [28] Bennett H E, Bennett J M 1966 Optical Properties and Electronic Structure of Metals and Alloysed F Abel´ es and L Erwin (Amsterdam: North Holland) p 175 8

2008-12-11

The Casimir force is calculated between Au thin films that are described by a Drude model with a frequency dependent damping function. The model parameters are obtained from available experimental data for Au thin films. Two cases are considered; annealed and nonannealed films that have a different damping function. Compared with the calculations using a Drude model with a constant damping parameter, we observe changes in the Casimir force of a few percent. This behavior is only observed in films of no more than 300 $\AA$ thick.

Frequency-dependent Drude damping in Casimir force calculations

0812.2209v1

arXiv:0812.2570v1 [cond-mat.mtrl-sci] 13 Dec 2008Non-Adiabatic Spin Transfer Torque in Real Materials Ion Garate1, K. Gilmore2,3, M. D. Stiles2, and A.H. MacDonald1 1Department of Physics, The University of Texas at Austin, Au stin, TX 78712 2Center for Nanoscale Science and Technology, National Inst itute of Standards and Technology, Gaithersburg, MD 20899-8412 a nd 3Maryland NanoCenter, University of Maryland, College Park , MD, 20742 (Dated: October 30, 2018) The motion of simple domain walls and of more complex magneti c textures in the presence of a transport current is described by the Landau-Lifshitz-Slo nczewski (LLS) equations. Predictions of the LLS equations depend sensitively on the ratio between th e dimensionless material parameter βwhich characterizes non-adiabatic spin-transfer torques and the Gilbert damping parameter α. This ratio has been variously estimated to be close to 0, clos e to 1, and large compared to 1. By identifying βas the influence of a transport current on α, we derive a concise, explicit and relatively simple expression which relates βto the band structure and Bloch state lifetimes of a magnetic metal. Using this expression we demonstrate that intrinsic spin-orbit interactions lead to intra- band contributions to βwhich are often dominant and can be (i) estimated with some co nfidence and (ii) interpreted using the “breathing Fermi surface” mo del. PACS numbers: I. INTRODUCTION An electric current can influence the magnetic state of a ferromagnet by exerting a spin transfer torque (STT) on the magnetization.1,2,3This effect occurs whenever currents travel through non-collinear magnetic systems and is therefore promising for magnetoelectronic appli- cations. Indeed, STT’s have already been exploited in a number of technological devices.4Partly for this reason and partly because the quantitative description of order parameter manipulation by out-of-equilibrium quasipar- ticles poses great theoretical challenges, the study of the STT effect has developed into a major research subfield of spintronics. Spin transfer torques are important in both magnetic multilayers, where the magnetization changes abruptly,5 and in magnetic nanowires, where the magnetization changes smoothly.6Theories of the STT in systems with smooth magnetic textures identify two different types of spintransfer. Ononehand,theadiabaticorSlonczewski3 torque results when quasiparticle spins follow the under- lying magnetic landscape adiabatically. It can be math- ematically expressed as ( vs· ∇)s0, wheres0stands for the magnetization and vsis the “spin velocity”, which is proportional to the charge drift velocity, and hence to the current and the applied electric field. The micro- scopic physics of the Slonczewski spin-torque is thought to be well understood5,6,7, at least8in systems with weak spin-orbitcoupling. Asimpleangularmomentumconser- vation argument argues that in the absence of spin-orbit coupling vs=σsE/es0, wheres0isthe magnetization, σs is the spin conductivity and Eis the electric field. How- ever, spin-orbit coupling plays an essential role in real magnetic materials and hence the validity of this sim- ple expression for vsneeds to be tested by more rigorous calculations. The second spin transfer torque in continuous media,βs0×(vs·∇)s0, acts in the perpendicular direction and is frequently referred to as the non-adiabatic torque.9 Unfortunately, the name is a misnomer in the present context. There are two contributions that have the pre- ceeding form. The first is truly non-adiabatic and occurs in systems in which the magnetization varies too rapidly in space for the spins of the transport electrons to fol- low the local magnetization direction as they traverse the magnetization texture. For wide domain walls, these effects are expected to be small.10The contribution of interest in this paper is a dissipative contribution that occurs in the adiabatic limit. The adiabatic torque dis- cussed aboveis the reactive contribution in this limit. As we discuss below, processes that contribute to magnetic damping, whether they derive from spin-orbit coupling or spin-dependent scattering, also give a spin-transfer torque parameterized by βas above. In this paper, we follow the common convention and refer to this torque as non-adiabatic. However, it should be understood that it is a dissipative spin transfer torque that is present in the adiabatic limit. The non-adiabatic torque plays a key role in current- driven domain wall dynamics, where the ratio between βand the Gilbert parameter αcan determine the veloc- ity of domain walls under the influence of a transport current. There is no consensus on its magnitude of the parameter β.6,11Although there have a few theoretical studies12,13,14of the STT in toy models, the relationship between toy model STT’s and STT’s in either transition metal ferromagnets or ferromagnetic semiconductors is far from clear. As we will discuss the toy models most often studied neglect spin-orbit interactions in the band- structure of the perfect crystal, intrinsic spin-orbit inter- actions, which can alter STT physics qualitatively. The main objectives of this paper are (i) to shed new light on the physical meaning of the non-adiabatic STT by relating it to the change in magnetization damping due to a transport current, (ii) to derive a concise for-2 mula that can be used to evaluate βin real materials from first principles and (iii) to demonstrate that αand βhave the same qualitative dependence on disorder (or temperature), even though their ratio depends on the details of the band structure. As a byproduct of our the- oretical study, we find that the expression for vsin terms of the spin conductivity may not always be accurate in materials with strong spin-orbit coupling. We begin in Section II by reviewing and expanding on microscopic theories of α,βandvs. In short, our microscopic approach quantifies how the micromagnetic energy of an inhomogeneous ferromagnet is altered in response to external rf fields and dc transport currents which drive the magnetization direction away from lo- cal equilibrium. These effects are captured by the spin transfertorques,dampingtorques,andeffectivemagnetic fields that appear in the LLS equation. By relating mag- netization dynamics to effective magnetic fields, we de- rive explicit expressions for α,βandvsin terms of mi- croscopic parameters. Important contributions to these materials parameters can be understood in clear physical terms using the breathing Fermi surface model.15Read- ers mainly interested in a qualitative explanation for our findings may skip directly to Section VIII where we dis- cuss of our main results in that framework. Regardless of the approach, the non-adiabatic STT can be under- stood as the change in the Gilbert damping contribution to magnetization dynamics when the Fermi sea quasi- particle distribution function is altered by the transport electric field. The outcome of this insight is a concise an- alytical formula for βwhich is simple enough that it can be conveniently combined with first-principles electronic structure calculations to predict β-values in particular materials.16 In Sections III, IV and V we apply our expressionfor β to model ferromagnets. In Section III we perform a nec- essary reality check by applying our theory of βto the parabolic band Stoner ferromagnet, the only model for which more rigorous fully microscopic calculations13,14 ofβhave been completed. Section IV is devoted to the study of a two-dimensional electron-gas ferromag- net with Rashba spin-orbit interactions. Studies of this model provide a qualitative indication of the influence of intrinsic spin-orbit interactions on the non-adiabatic STT. We find that, as in the microscopic theory17,18 forα, spin-orbit interactions induce intra-band contri- butions to βwhich are proportional to the quasiparticle lifetimes. These considerations carry over to the more sophisticated 4-band spherical model that we analyze in SectionV;thereourcalculationistailoredto(Ga,Mn)As. We show that intra-band (conductivity-like) contribu- tions are prominent in the 4-band model for experimen- tally relevant scattering rates. SectionVIdiscussesthephenomenologicallyimportant α/βratio for real materials. Using our analytical results derived in Section II (or Section VIII) we are able to re- produce and extrapolate trends expected from toy mod- els which indicate that α/βshould vary across materialsin approximately the same way as the ratio between the itinerant spin density and the total spin density. We also suggest that αandβmay have the opposite signs in sys- tems with both hole-like and electron-like carriers. We present concrete results for (Ga,Mn)As, where we obtain α/β≃0.1. This is reasonable in view of the weak spin polarizationand the strong spin-orbit coupling of valence band holes in this material. Section VII describes the generalization of the torque- correlation formula employed in ab-initio calculations of the Gilbert damping to the case of the non-adiabatic spin-transfer torque. The torque correlation formula in- corporates scattering of quasiparticles simply by intro- ducing a phenomenological lifetime for the Bloch states and assumes that the most important electronic transi- tions occur between states near the Fermi surface in the same band. Our ability to make quantitative predictions based on this formula is limited mainly by an incomplete understanding of Bloch state scattering processes in real ferromagnetic materials. These simplifications give rise to ambiguitiesandinaccuraciesthat wedissect in Section VII. Our assessment indicates that the torque correlation formula for βis most accurate at low disorder and rela- tively weak spin-orbit interactions. Section VIII restates and complements the effective field calculation explained in Section II. Within the adi- abatic approximation, the instantaneous energy of a fer- romagnet may be written in terms of the instantaneous occupation factors of quasiparticle states. We determine the effect ofthe external perturbationson the occupation factors by combining the relaxation time approximation and the master equation. In this way we recover the re- sultsofSectionII andareableto interprettheintra-band contributions to βin terms of a generalized breathing Fermi surface picture. Section IX contains a brief summary which concludes this work. II. MICROSCOPIC THEORY OF α,βANDvs The Gilbert damping parameter α, the non-adiabatic spin transfer torque coefficient βand the “spin velocity” vsappear in the generalized Landau-Lifshitz-Gilbert ex- pression for collective magnetization dynamics of a fer- romagnet under the influence of an electric current: (∂t+vs·∇)ˆΩ−ˆΩ×Heff=−αˆΩ×∂tˆΩ−βˆΩ×(vs·∇)ˆΩ. (1) In Eq. (1) Heffis an effective magnetic field which we elaborate on below and ˆΩ =s0/s0≃(Ωx,Ωy,1−(Ω2 x+ Ω2 y)/2)isthedirectionofthemagnetization.19Eq.(1)de- scribes the slow dynamics of smooth magnetization tex- turesinthepresenceofaweakelectricfieldwhichinduces transport currents. It explicitly neglects the dynamics of themagnetizationmagnitudewhichisimplicitlyassumed to be negligible. For small deviations from the easy di-3 rection (which we take to be the ˆ z-direction), it reads Heff,x= (∂t+vs·∇)Ωy+(α∂t+βvs·∇)Ωx Heff,y=−(∂t+vs·∇)Ωx+(α∂t+βvs·∇)Ωy(2) The gyromagnetic ratio has been absorbed into the units of the field Heffso that this quantity has inverse time units. We set /planckover2pi1= 1 throughout. In this section we relate the α,βandvsparameters to microscopic features of the ferromagnet by consider- ing the transverse total spin response function. For a technically more accessible (yet less rigorous) theory ofαandβwe refer to Section VIII. The transverse spin re- sponse function studied here describes the change in the micromagneticenergyduetothedepartureofthemagne- tization away from its equilibrium direction, where equi- librium is characterized by the absence of currents and external rf fields. This change in energy defines an ef- fective magnetic field which may then be identified with Eq. (2), thereby allowingus to microscopicallydetermine α,βandvs. To first order in frequency ω, wave vector q and electric field, the transverse spin response function is given by S0ˆΩa=/summationdisplay bχa,bHext,b≃/summationdisplay b/bracketleftBig χ(0) a,b+ωχ(1) a,b+(vs·q)χ(2) a,b/bracketrightBig Hext,b (3) wherea,b∈ {x,y},Hextis the external magnetic field with frequency ωand wave vector q,S0=s0Vis the total spin of the ferromagnet ( Vis the sample volume), and χis the transverse spin-spin response function in the presence of a uniform time-independent electric field: χa,b(q,ω;vs) =i/integraldisplay∞ 0dt/integraldisplay drexp(iωt−iq·r)∝an}b∇acketle{t/bracketleftbig Sa(r,t),Sb(0,0)/bracketrightbig ∝an}b∇acket∇i}ht. (4) In Eq. (3), χ(0)=χ(q=0,ω= 0;E=0) de- scribes the spin response to a constant, uniform ex- ternal magnetic field in absence of a current, χ(1)= limω→0χ(q=0,ω;E=0)/ωcharacterizes the spin re- sponse to a time-dependent, uniform external mag- netic field in absence of a current, and χ(2)= limq,vs→0χ(q,ω= 0;E)/q·vsrepresents the spin re- sponsetoaconstant, non-uniformexternalmagneticfield combined with a constant, uniform electric field E. Note that first order terms in qare allowed by symmetry in presence of an electric field. In addition, ∝an}b∇acketle{t∝an}b∇acket∇i}htis a ther- mal and quantum mechanical average over states that describe a uniformly magnetized, current carrying ferro- magnet. The approach underlying Eq. (3) comprises a linear response theory with respect to an inhomogeneous mag- netic field followed by a linear response theory with re- spect toan electricfield. Alternatively, onemaytreatthe electric and magnetic perturbations on an equal footing without predetermined ordering; for further considera- tions on this matter we refer to Appendix A. In the following we emulate and appropriately gen- eralize a procedure outlined elsewhere.17First, we rec- ognize that in the static limit and in absence of a cur- rent the transverse magnetization responds to the exter- nal magnetic field by adjusting its orientation to min- imize the total energy including the internal energy Eintand the energy due to coupling with the exter- nal magnetic field, Eext=−S0ˆΩ· Hext. It follows thatχ(0) a,b=S2 0[∂2Eint/∂ˆΩa∂ˆΩb]−1and thus Hint,a=−(1/S0)∂Eint/∂ˆΩa=−S0[χ(0)]−1 a,bˆΩb, whereHintis the internal energy contribution to the effective magnetic field. Multiplying Eq. (3) on the left by [ χ(0)]−1and usingHeff=Hint+Hextwe obtain a formal equation for Heff: Heff,a=/summationdisplay b/bracketleftBig L(1) a,b∂t+L(2) a,b(vs·∇)/bracketrightBig ˆΩb,(5) where L(1)=−iS0[χ(0)]−1χ(1)[χ(0)]−1 L(2)=iS0[χ(0)]−1χ(2)[χ(0)]−1. (6) Identifying of Eqs. (5) and (2) results in concise micro- scopic expressions for αandβandvs: α=L(1) x,x=L(1) y,y β=L(2) x,x=L(2) y,y 1 =L(2) x,y=⇒vs·q=iS0/bracketleftBig (χ(0))−1χ(χ(0))−1/bracketrightBig x,y.(7) In the third line of Eq. (7) we have combined the second line of Eq. (6) with χ(2)=χ/(vs·q). When applying Eq. (7) to realistic conducting fer- romagnets, one must invariably adopt a self-consistent mean-field (Stoner) theory description of the magnetic state derived within a spin-density-functional theory (SDFT) framework.20,21In SDFT the transverse spin response function is expressed in terms of Kohn-Sham4 quasiparticleresponsetoboth externalandinduced mag- netic fields; this allows us to transform17Eq. (7) into α=1 S0lim ω→0Im[˜χQP +,−(q= 0,ω,E= 0)] ω β=−1 S0lim vs,q→0Im[˜χQP +,−(q,ω= 0,E)] q·vs vs·q=−1 S0Re[˜χQP +,−(q,ω= 0,E)], (8) where we have used22χ(0) a,b=δa,bS0/¯∆ and ˜χQP +,−(q,ω;E) =1 2/summationdisplay i,jfj−fi ǫi−ǫj−ω−iη ∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht∝an}b∇acketle{ti|S−∆0(r)e−iq·r|j∝an}b∇acket∇i}ht (9) is the quasiparticle response to changes in the direction of the exchange-correlation effective magnetic field.23To estimate βthis response function should be evaluated in the presence of an electric current. In the derivation of Eq. (8) we have made use of the fact that χ(1) x,xand χ(2) x,xare purely imaginary, whereas χ(2) x,yis purely real; this can be verified mathematically through S±=Sx± iSy. Physically, “Im” and “Re” indicate that the Gilbert dampingandthenon-adiabaticSTTaredissipativewhile the adiabatic STT is reactive. Furthermore, in the third line it is implicit that we expand Re[˜ χQP] to first order inqandE. In Eq. (9), S±is the spin-rising/loweringoperator, |i∝an}b∇acket∇i}ht, ǫiandfiare the Kohn-Sham eigenstates, eigenenergies and Fermi factors in presence of spin-dependent disorder, and ∆ 0(r) is the difference in the magnetic ground state between the majority spin and minority spin exchange- correlation potential - the spin-splitting potential. This quantity is alwaysspatially inhomogeneous at the atomic scale and is typically larger in atomic regions than in interstitial regions. Although the spatial dependence of ∆0(r) plays a crucial role in realistic ferromagnets, we replace it by a phenomenological constant ∆0in the toy models we discuss below. Our expression of vsin terms of the transverse spin response function may be unfamiliar to readers familiar with the argument given in the introduction of this pa- per in which vsis determined by the divergence in spincurrent. This argument is based on the assumption that the (transverse) angular momentum lost by spin polar- ized electrons traversing an inhomogeneous ferromagnet is transferred to the magnetization. However, this as- sumption fails when spin angular momentum is not con- served as it is not in the presence of spin-orbit coupling. In general, part of the transverse spin polarization lost by the current carrying quasiparticles is transferred to the lattice rather than to collective magnetic degrees of freedom8when spin-orbit interactions are present. It is often stated that the physics of spin non-conservation is captured by the non-adiabatic STT; however, the non- adiabatic STT per seis limited to dissipative processes and cannot describe the changes in the reactive spin torque due to spin-flip events. Our expression in terms of the transverse spin response function does not rely on spin conservation, and while it agrees with the conven- tional picture24in simplest cases (see below), it departs from it when e.g.intrinsic spin-orbit interactions are strong. In this paper weincorporatethe influence ofan electric field by simply shifting the Kohn-Sham orbital occupa- tion factors to account for the energy deviation of the distribution function in a drifting Fermi sea: fi≃f(0)(ǫi+Vi)≃f(0)(ǫi)+Vi∂f(0)/∂ǫi(10) whereViis the effective energy shift for the i-th eigenen- ergy due to acceleration between scattering events by an electric field and f(0)is the equilibrium Fermi factor. This approximation to the steady-state induced by an external electric field is known to be reasonably accurate in many circumstances, for example in theories of electri- cal transport properties, and it can be used24to provide a microscopic derivation of the adiabatic spin-transfer torque. As we discuss below, this ansatzprovidesa result forβwhich is sufficiently simple that it can be combined with realistic ab initio electronic structure calculations to estimate βvalues in particular magnetic metals. We support this ansatzby demonstrating that it agrees with full non-linear response calculations in the case of toy models for which results are available. Using the Cauchy identity, 1 /(x−iη) = 1/x+iπδ(x), and∂f(0)/∂ǫ≃ −δ(ǫ) we obtain Im[˜χQP +,−]≃π 2/summationdisplay i,j[ω−Vj,i]|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2δ(ǫi−ǫF)δ(ǫj−ǫF) Re[˜χQP +,−]≃ −1 2/summationdisplay i,j|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2Vjδ(ǫj−ǫF)−Viδ(ǫi−ǫF) ǫi−ǫj(11) where we have defined the difference in transport devia- tion energ ies by Vj,i≡Vj−Vi. (12)5 In the first line of Eq. (11), the two terms within the square brackets correspond to the energy of particle- hole excitations induced by radio frequency magnetic and static electric fields, respectively. The imaginarypart selects scattering processes that relax the spin of the particle-hole pairs mediated either by phonons or by magnetic impurities.25Substituting Eq. (11) into Eq. (8) we can readily extract α,βandvs: α=π 2S0/summationdisplay i,j|∝an}b∇acketle{tj|S+∆0(r)|i∝an}b∇acket∇i}ht|2δ(ǫi−ǫF)δ(ǫj−ǫF) β= lim q,vs→0π 2S0q·vs/summationdisplay i,j|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2Vj,iδ(ǫi−ǫF)δ(ǫj−ǫF) vs·q=1 2S0/summationdisplay i,j|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2Vjδ(ǫj−ǫF)−Viδ(ǫi−ǫF) ǫi−ǫj(13) where we have assumed a uniform precession mode for the Gilbert damping. Eq. (13) and Eq. (11) identify the non-adiabatic STT as acorrection to the Gilbert damping in the presence of an electric current; in other words, the magnetiza- tion damping at finite current is given by the sum of the Gilbert damping and the non-adiabatic STT. We feel that this simple interpretation of the non-adiabatic spin- transfertorquehasnot receivedsufficientemphasisin the literature. Strictly speaking the influence of a transport current on magnetization dynamics should be calculated by con- sidering non-linear response of transverse spin to both effective magnetic fields and the external electric field which drives the transport current. Our approach, in which we simply alter the occupation probabilities which appear in the transverse spin response function is admit- tedly somewhat heuristic. We demonstrate below that it gives approximately the same result as the complete calculation for the case of the very simplistic model for which that complete calculation has been carried out. In Eq. (13), the eigenstates indexed by iare not Bloch states of a periodic potential but instead the eigenstates of the Hamiltonian that includes all of the static dis- order. Although Eq. (13) provides compact expressions valid for arbitrary metallic ferromagnets, its practical- ity is hampered by the fact that the characterization of disorder is normally not precise enough to permit a reli-able solution of the Kohn-Sham equations with arbitrary impurities. An approximate yet more tractable treat- ment of disorder consists of the following steps: (i) re- place the actual eigenstates of the disordered system by Bloch eigenstates corresponding to a pure crystal, e.g. |i∝an}b∇acket∇i}ht → |k,a∝an}b∇acket∇i}ht, where kis the crystal momentum and a is the band index of the perfect crystal; (ii) switch Vito Va=τk,avk,a·eE, whereτis the Bloch state lifetime and vk,a=∂ǫk,a/∂kis the quasiparticle group velocity, (iii) substitute the δ(ǫk,a−ǫF) spectral function of a Bloch state by a broadened spectral function evaluated at the Fermi energy: δ(ǫk,a−ǫF)→Aa(ǫF,k)/(2π), where Aa(ǫF,k) =Γk,a (ǫF−ǫk,a)2+Γ2 k,a 4(14) and Γ a,k= 1/τa,kis the inverse of the quasiparticle lifetime. This minimal prescription can be augmented by introducing impurity vertex corrections in one of the spin-flip operators, which restores an exact treatment of disorder in the limit of dilute impurities. This task is for the most part beyond the scope of this paper (see next section, however). The expression for αin Eq. (13) has already been discussed in a previous paper;17hence from here on we shall concentrate on the expression for βwhich now reads β(0)= lim q,vs→01 8πs0/summationdisplay a,b/integraldisplay k|∝an}b∇acketle{tk+q,b|S+∆0(r)|k,a∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k+q)(vk+q,bτk+q,b−vk,aτk,a)·eE q·vs(15) where we have used/summationtext k→V/integraltext dDk/(2π)D≡V/integraltext kwithDas the dimensionality, Vas the volume and q·vs=1 2s0/summationdisplay a,b/integraldisplay k|∝an}b∇acketle{tk+q,b|S+∆0(r)|k,a∝an}b∇acket∇i}ht|2evk+q,bτk+q,bδ(ǫF−ǫk+q,b)−evk,aτk,aδ(ǫF−ǫk,a) ǫk,a−ǫk+q,b.(16)6 In Eq. (15) the superscript “0” is to remind of the absence of impur ity vertex corrections; . In addition, we recall that s0=S0/Vis the magnetization of the ferromagnet and |ak∝an}b∇acket∇i}htis a band eigenstate of the ferromagnet withoutdisorder. It is straightforward to show that Eq. (16) reduces to the usual expression vs=σsE/(es0) for vanishing intrinsic spin-orbit coupling. However, we find that in presence of spin-orbit interaction Eq. (16) is no longer connected to the spin conductivity. Determining the precise way in which Eq. (16) d eparts from the conventional formula in real materials is an open problem that may have fundamental and practic al repercussions. Expanding the integrand in Eq. (15) to first order in qand rearranging the result we arrive at β(0)=−1 8πs0q·vs/summationdisplay a,b/integraldisplay k/bracketleftbig |∝an}b∇acketle{ta,k|S+∆0(r)|b,k∝an}b∇acket∇i}ht|2+|∝an}b∇acketle{ta,k|S−∆0(r)|b,k∝an}b∇acket∇i}ht|2/bracketrightbig Aa(ǫF,k)A′ b(ǫF,k)(vk,a·eE)(vk,b·q)τa −1 4πs0q·vs/summationdisplay a,b/integraldisplay kRe/bracketleftbig ∝an}b∇acketle{tb,k|S−∆0(r)|a,k∝an}b∇acket∇i}ht∝an}b∇acketle{ta,k|S+∆0(r)q·∂k|b,k∝an}b∇acket∇i}ht +(S+↔S−)/bracketrightbig Aa(ǫF,k)Ab(ǫF,k)(vk,a·eE)τa (17) eVa,b eVa,ba,k;ω a,k;ωeVa,b b, ω+k+q;S− S−S+ S+ω(a) (b)b,k;ω +ωn n nnω FIG. 1: Feynman diagrams for (a) αand (b) β(q·vs), the latter with a heuristic consideration of the electric field ( for a more rigorous treatment see Appendix A). Solid lines corre- spond to Green’s functions of the band quasiparticles in the Born approximation, dashed lines standfor themagnon offre - quencyωand wavevector q,ωnis the Matsubara frequency andeVa,bis the difference in the transport deviation energies. whereA′(ǫF,k)≡2(ǫF−ǫk,a)Γa//bracketleftbig (ǫF−ǫk,a)2+Γ2 a/4/bracketrightbig2 stands for the derivative of the spectral function and we have neglected ∂Γ/∂k. Eq. (17) (or Eq. (15)) is the cen- tral result of this work and it provides a gateway to eval- uate the non-adiabatic STT in materials with complex band structures;16for a diagrammatic interpretation see Fig. (1). An alternativeformula with a similar aspiration has been proposed recently,26yet that formula ignores intrinsic spin-orbit interactions and relies on a detailed knowledge of the disorder scattering mechanisms. In the following three sections we apply Eq. (17) to three differ-ent simplified models of ferromagnets. For a simpler-to- implement approximate version of Eq. (15) or Eq. (17) we refer to Section VI. III. NON-ADIABATIC STT FOR THE PARABOLIC TWO-BAND FERROMAGNET The model described in this section bears little resem- blance to any real ferromagnet. Yet, it is the only model in which rigorous microscopic results for βare presently available, thus providing a valuable test bed for Eq. (17). The mean-field Hamiltonian for itinerant carriers in a two-band Stoner model with parabolic bands is simply H(k)=k2 2m−∆0Sz(18) where ∆ 0is the exchange field and Sz a,b=δa,bsgn(a). In this model the eigenstates have no momentum depen- dence and hence Eq. (17) simplifies to (vs·q)β(0)=−∆2 0 2πs0/summationdisplay a/integraldisplay kAa(ǫF,k)A′ −a(ǫF,k) k·q mk·eE mτk,a, (19) wherea= +(−) for majority (minority) spins, vk,±= k/m, andS±=Sx±iSywithSx a,b=δa,b. Also, from here on repeated indexes will imply a sum. Taking ∆ 0≤ EFand ∆ 0>>1/τ, the momentum integral in Eq. (19) is performed in the complex energy plane using a keyhole contour around the branch cut that stems from the 3D density of states:7 (vs·q)β(0)=−∆2 0 2πs02eE·q 3m/integraldisplay∞ 0ν(ǫ)Aa(ǫF,a−ǫ)A′ −a(ǫF,−a−ǫ)ǫτk,a ≃eE·q 6m∆0s0sgn(a)νaǫF,aτaΓ−a =eE·q 2m∆0s0(n↑τ↑γ↓−n↓τ↓γ↑) (20) whereǫF,a=ǫF+ sgn(a)∆0,νais the spin-dependent density of states at the Fermi surface, na= 2νaǫF,a/3 is the corresponding number density, and γa≡Γa/2. The factor 1 /3 on the first line of Eq. (20) comes from the angular integration. In the second line of Eq. (20) we have neglected a term that is smaller than the one retainedbyafactorof∆2 0/(12ǫ2 F); suchextraterm(which would have been absent in a two-dimensional version of the model) appears to be missing in previous work.13,14 The simplicity of this model enables a partial incorpo- ration of impurity vertex corrections. By adding to β(0)the contribution from the leading order vertex correction (β(1)), we shall recover the results obtained previously for this model by a full calculation of the transverse spin response function. As it turns out, β(1)is qualitatively important because it ensures that only spin-dependent impuritiescontributetothenon-adiabaticSTTintheab- sence of an intrinsic spin-orbit interaction. In Appendix B we derive the following result: (vs·q)β(1)=e∆2 0 4πs0/integraldisplay k,k′uiRe/bracketleftBig S+ a,bSi b,b′S− b′,a′Si a′,a/bracketrightBigAa(ǫF,k) (ǫF−ǫk′,a′)/bracketleftbiggAb(ǫF,k+q) (ǫF−ǫk′+q,b′)Vb,a+Ab′(ǫF,k′+q) (ǫF−ǫk+q,b)Vb′,a/bracketrightbigg ,(21) whereui≡niw2 i(i= 0,x,y,z),niis the density of scatterers, wiis the Fourier transform of the scattering potential and the overline denotes an average over different disorder config urations.13Also,Va,b= (τbvk+q,b−τavk,a)·eE. Expanding Eq. (21) to first order in q, we arrive at (vs·q)β(1)=−∆2 0 2πs0(u0−uz)/integraldisplay k,k′Aa(ǫF,k) ǫF−ǫk′,a/bracketleftbiggA′ −a(ǫF,k) ǫF−ǫk′,−a+A−a(ǫF,k′) (ǫF−ǫk,−a)2/bracketrightbiggk·q mk·eE mτk,a (22) In the derivation of Eq. (22) we have used S±=Sx±iSyand assumed that ux=uy≡ux,y, so that uiRe/bracketleftBig Sx a,bSi b,b′Sx b′,a′Si a′,a/bracketrightBig =/parenleftbig u0−uz/parenrightbig δa,a′δb,b′δa,−b. In addition, we have used/integraltext k,k′F(|k|,|k′|)kik′ j= 0. The first term inside the square brackets of Eq. (22) can be ignored in the we ak disorder regime because its contribution is linear in the scattering rate, as opposed to the second term, which contributes at zeroth order. Then, (vs·q)β(1)=−∆2 0 πs0(u0−uz)/integraldisplay k,k′Aa(ǫF,k)A−a(ǫF,k′) (ǫF−ǫk′,a)(ǫF−ǫk,−a)2k·q mk·eE mτk,a ≃ −∆2 0 πs0(u0−uz)2eE·q 3m/integraldisplay∞ −∞dǫdǫ′ν(ǫ)ν(ǫ′)Aa(ǫF,a−ǫ)A−a(ǫF,−a−ǫ′) (ǫF−ǫ′a)(ǫF−ǫ−a)2ǫτa ≃ −π(u0−uz)eE·q 2m∆0s0sign(a)naτaν−a (23) Combining this with Eq. (20), we get (vs·q)β≃(vs·q)β(0)+(vs·q)β(1) =eE·q 2ms0∆0/bracketleftbig n↑τ↑γ↓−n↓τ↓γ↑−π(u0−uz)(n↑τ↑ν↓−n↓τ↓ν↑)/bracketrightbig =πeE·q ms0∆0[n↑τ↑(uzν↓+ux,yν↑)−n↓τ↓(uzν↑+ux,yν↓)] (24) where we have used γa=π/bracketleftbig (u0+uz)νa+2ux,yν−a/bracketrightbig . In this model it is simple to solve Eq. (16) for vsanalyt-ically, whereupon Eq. (24) agrees with the results pub-8 lished by other authors in Refs.[ 13,14] from full non- linear response function calculations. However, we reit- erate that in order to reach such agreement we had to neglect a term of order ∆2 0/ǫ2 Fin Eq. (20). This extra term is insignificant in all but nearly half metallic ferro- magnets. IV. NON-ADIABATIC STT FOR A MAGNETIZED TWO-DIMENSIONAL ELECTRON GAS The model studied in the previous section misses the intrinsic spin-orbit interaction that is inevitably present in the band structure of actual ferromagnets. Further- more,sinceintrinsicspin-orbitinteractionisinstrumental for the Gilbert damping at low temperatures, a similarly prominent role may be expected in regards to the non- adiabaticspin transfertorque. Hence, thepresentsection is devoted to investigatethe relativelyunexplored26,27ef- fect of intrinsic spin-orbit interaction on β. The minimalmodel for this enterprise is the two-dimensional electron- gas ferromagnet with Rashba spin-orbit interaction, rep- resented by H(k)=k2 2m−b·S, (25) whereb= (λky,−λkx,∆0),λis the Rashba spin-orbit coupling strength and ∆ 0is the exchange field. The eigenspinors of this model are |+,k∝an}b∇acket∇i}ht= (cos(θ/2),−iexp(iφ)sin(θ/2)) and |−,k∝an}b∇acket∇i}ht= (sin(θ/2),iexp(iφ)cos(θ/2)), where the spinor an- gles are defined through cos θ= ∆0//radicalbig λ2k2+∆2 0 and tan φ=ky/kx. The corresponding eigenen- ergies are Ek±=k2/(2m)∓/radicalbig ∆2 0+λ2k2. Therefore, the band velocities are given by vk±=k/parenleftBig 1/m∓λ2//radicalbig λ2k2+∆2 0/parenrightBig =k/m±. Dis- regarding the vertex corrections, the non-adiabatic spin-torque of this model may be evaluated analytically starting from Eq. (17). We find that (see Appendix C): (vs·q)β(0)≃∆2 0eE·q 8πs0/bracketleftbiggm2 4m+m−/parenleftbigg 1+∆2 0 b2/parenrightbigg1 b2+1 4λ2k2 F∆2 0 b6/bracketrightbigg +∆2 0eE·q 8πs0/bracketleftbigg1 2m2 m2 +λ2k2 F b2/parenleftbigg 1−δm+ m∆2 0 b2/parenrightbigg τ2+1 2m2 m2 −λ2k2 F b2/parenleftbigg 1−δm− m∆2 0 b2/parenrightbigg τ2/bracketrightbigg (26) whereb=/radicalbig λ2k2 F+∆2 0(kF=√2mǫF), andδm±= m−m±. As we explain in the Appendix, Eq. (26) ap- plies forλkF,∆0,1/τ << ǫ F; for a more general analysis, Eq. (17) must be solved numerically (e.g. see Fig. (2)). Eq. (26) reveals that intrinsic spin-orbit interaction en- ablesintra-band contributions to β, whose signature is theO(τ2) dependence on the second line. In contrast, theinter-band contributions appear as O(τ0). Since vs itself is linear in the scattering time, it follows that β is proportional to the electrical conductivity in the clean regime and the resistivity in the disordered regime, much like the Gilbert damping α. We expect this qualitative feature to be model-independent and applicable to real ferromagnets. V. NON-ADIABATIC STT FOR (Ga,Mn)As Inthis sectionweshallapplyEq.(17) toamoresophis- ticated model which provides a reasonable description of (III,Mn)V magnetic semiconductors.28Since the orbitals at the Fermi energy are very similar to the states near the top of the valence band of the host (III,V) semicon- ductor, the electronic structure of (III,Mn)V ferromag- nets is remarkably simple. Using a p-d mean field theory modelfortheferromagneticgroundstateandafour-bandspherical model for the host semiconductor band struc- ture, Ga 1−xMnxAs may be described by H(k)=1 2m/bracketleftbigg/parenleftbigg γ1+5 2γ2/parenrightbigg k2−2γ3(k·S)2/bracketrightbigg +∆0Sz, (27) whereSisthe spinoperatorprojectedontothe J=3/2to- tal angularmomentum subspace at the top of the valence band and {γ1= 6.98,γ2=γ3= 2.5}are the Luttinger parametersforthesphericalapproximationtothevalence bands of GaAs. In addition, ∆ 0=JpdsNMn=Jpds0is theexchangefield, Jpd= 55 meVnm3isthep-dexchange coupling, s= 5/2 is the spin of Mn ions, NMn= 4x/a3 is the density of Mn ions and a= 0.565 nm is the lattice constant of GaAs. We solve Eq. (27) numerically and input the outcome in Eqs. (16), (17). The results are summarized in Fig. (3). We find that the intra-band contribution dominates as a consequence ofthestrongintrinsicspin-orbitinteraction,muchlikefor the Gilbert damping;18. Incidentally, βbarely changes regardless of whether the applied electric field is along the easy axis of the magnetization or perpendicular to it.9 0.00 0.05 0.10 0.15 0.20 1/(εFτ)0.000.100.200.300.400.50β∆0=0.5εF ; λkF=0.05εF intra−band inter−band total FIG. 2: M2DEG : inter-band contribution, intra-band con- tribution and the total non-adiabatic STT for a magnetized two-dimensional electron gas (M2DEG). In this figure the ex- change field dominates over the spin-orbit splitting. At hig her disorder the inter-bandpart (proportional toresistivity ) dom- inates, while at low disorder the inter-band part (proporti onal to conductivity) overtakes. For simplicity, the scatterin g time τis taken to be the same for all sub-bands. 0.00 0.10 0.20 0.30 1/(εFτ)0.000.100.200.300.40βintra−band inter−band totalx=0.08 ; p=0.4 nm−3 FIG. 3:GaMnAs :β(0)forEperpendicular to the easy axis of magnetization (ˆ z).xandpare the Mn fraction and the hole density, respectively. The intra-band contribution i s con- siderably larger than the inter-band contribution, due to t he strongintrinsicspin-orbitinteraction. Sincethe4-band model typically overestimates the influence of intrinsic spin-or bit in- teraction, it is likely that the dominion of intra-band con- tributions be reduced in the more accurate 6-band model. By evaluating βforE||ˆz(not shown) we infer that it does not depend significantly on the relative direction between t he magnetic easy axis and the electric field. VI.α/βIN REAL MATERIALS Theprecedingthreesectionshavebeenfocusedontest- ing and analyzing Eq. (17) for specific models of ferro-magnets. In this section we return to more general con- siderationsandsurveythephenomenologicallyimportant quantitative relationshipbetween αandβin realistic fer- romagnets, which always have intrinsic spin-orbit inter- actions. We begin by recollecting the expression for the Gilbert damping coefficient derived elsewhere:17 α=1 8πs0/summationdisplay a,b/integraldisplay k|∝an}b∇acketle{tb,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k) (28) where we have ignored disorder vertex corrections. This expression is to be compared with Eq. (15); for peda- gogical purposes we discuss intra-band and inter-band contributions separately. Starting from Eq. (15) and expanding the integrand to first order in qwe obtain βintra=1 8πs0/integraldisplay k|∝an}b∇acketle{ta,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)2 eτaqi∂kivj k,aEj q·vs(29) where we have neglected the momentum dependence of the scattering lifetime and a sum over repeated indices is implied. Remarkably, only matrix elements that are diagonal in momentum space contribute to βintra; the implicationsofthiswillbehighlightedinthenextsection. Recognizing that ∂kjvi k,a= (1/m)i,j a, where (1 /m)ais the inverse effective mass tensor corresponding to band a, Eq. (29) can be rewritten as βintra=1 8πs0/integraldisplay k|∝an}b∇acketle{ta,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)2q·vd,a q·vs, (30) where vi d,a=eτa(m−1)i,j aEj(31) is the “drift velocity” corresponding to the quasiparticles in band a. For Galilean invariant systems33vd,a=vs for any ( k,a) and consequently βintra=αintra. At first glance, it might appear that vs, which (at least in ab- sence of spin-orbit interaction) is determined by the spin current, must be different than vd,a. However, recall that vsis determined by the ratio of the spin current to the magnetization. If the same electrons contribute to the transport as to the magnetization, vs=vd,aprovided the scattering rates and the masses are the same for all states. These conditions are the conditions for an elec- tron system to be Galilean invariant.10 The interband contribution can be simplified by noting that τbvi k+q,b−τavi k,a= (τbvi k+q,b−τavi k+q,a)+(τavi k+q,a−τavi k,a). (32) The second term on the right hand side of Eq.( 32) can then be manipu lated exactly as in the intra-band case to arrive at βinter=1 8πs0/summationdisplay a,b(a/negationslash=b)/integraldisplay k|∝an}b∇acketle{tb,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k)q·vd,a q·vs+δβinter (33) where δβinter=1 8πs0/summationdisplay a,b(a/negationslash=b)/integraldisplay k|∝an}b∇acketle{ta,k−q|S+∆0|b,k∝an}b∇acket∇i}ht|2Aa(ǫF,k−q)Ab(ǫF,k)(τbvk,b−τavk,a)·E q·vs. (34) When Galilean invariance is preserved the quasiparticle velocity and scattering times are the same for all bands, which implies that δβ= 0 and hence that βinter=αinter. Although realistic materials are not Galilean invariant, δβis nevertheless probably not significant because the term between parenthesis in Eq. (34) has an oscillatory behavior prone to cancellation. The degree of such can- cellation must ultimately be determined by realistic cal- culations for particular materials. With this proviso, we estimate that β≃1 8πs0/integraldisplay k|∝an}b∇acketle{tb,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k) q·vd,a q·vs. (35) As long as δβ≃0 is justified, the simplicity of Eq. (35) in comparisonto Eq. (15) or (17) makes ofthe former the preferred starting point for electronic structure calcula- tions. Even when δβ∝ne}ationslash= 0 Eq. (35) may be an adequate platformfor ab-initio studiesonweaklydisorderedtransi- tion metal ferromagnets and strongly spin-orbit coupled ferromagnetic semiconductors,29whereβis largely de- termined by the intra-band contribution. Furthermore, a direct comparison between Eq. (28) and Eq. (35) leads to the following observations. First, for nearly parabolic bands with nearly identical curvature, where the “drift velocity” is weakly dependent on momentum or the band index, we obtain β≃(vd/vs)αand thus β/αis roughly proportional to the ratio of the total spin density to the itinerant spin density, in concordance with predictions from toy models.12Second, if α/β >0 for a system with purely electron-like carriers, then α/β >0 for the same system with purely hole-like carriers because for a fixed carrier polarization va dandvsreverse their signs under m→ −m. However, if both hole-like and electron-like carriers coexist at the Fermi energy, then the integrand in Eq. (35) is positive for some values of aand negative for others. In such situation it is conceivable that α/βbe either positive or negative. A negative value of βimplies adecrease in magnetization damping due to an applied current.0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 1/(εFτ)0.120.220.320.42 8 α β FIG. 4: Comparison of αandβin (Ga,Mn)As for x= 0.08 andp= 0.4nm−3. It follows that β/α≃8, with a weak dependence on the scattering rate off impurities. If we use th e torque correlation formula (Section VII), we obtain β/α≃10. As an illustration of the foregoing discussion, in Fig. (4) we evaluate α/βfor (Ga,Mn)As. We find βto be about an order of magnitude larger than α, which is reasonablebecause (i) the local moment magnetizationis larger than the valence band hole magnetization, and (ii) the spin-orbit coupling in the valence band decreases the transportspin polarization. Accordingly βis of the order of unity, in qualitative agreement with recent theoretical work30. VII. TORQUE-CORRELATION FORMULA FOR THE NON-ADIABATIC STT Thus far we have evaluated non-adiabatic STT us- ing the bare vertex ∝an}b∇acketle{ta,k|S+|b,k+q∝an}b∇acket∇i}ht. In this section, we shall analyze an alternative matrix element denoted ∝an}b∇acketle{ta,k|K|b,k+q∝an}b∇acket∇i}ht(see below for an explicit expression), which may be better suited to realistic electronic struc- ture calculations.16,31We begin by making the ap-11 0.00 0.05 0.10 0.15 0.20 1/(εFτ)0.000.100.200.300.400.50β S+ K∆0=0.5 εF ; λkF=0.05εF FIG. 5: M2DEG : comparing SandKmatrix element ex- pressions for the non-adiabatic STT formula in the weakly spin-orbit coupled regime. Both formulations agree in the clean limit, where the intra-band contribution is dominant . In more disordered samples inter-band contributions becom e more visible and SandKbegin to differ; the latter is known to be more accurate in the weakly spin-orbit coupled regime. proximation that the exchange splitting can be writ- ten as a constant spin-dependent shift Hex= ∆0Sz. Then, the mean-field quasiparticle Hamiltonian H(k)= H(k) kin+H(k) so+Hexcan be written as the sum of a spin- independent part H(k) kin, the exchange term, and the spin- orbit coupling H(k) so. With this approximation, we have the identity: ∝an}b∇acketle{ta,k|S+|b,k+q∝an}b∇acket∇i}ht =1 ∆0∝an}b∇acketle{ta,k|/bracketleftBig H(k),S+/bracketrightBig |b,k+q∝an}b∇acket∇i}ht −1 ∆0∝an}b∇acketle{ta,k|/bracketleftBig H(k) so,S+/bracketrightBig |b,k+q∝an}b∇acket∇i}ht.(36) The last term in the right hand side of Eq. (36) is the generalization of the torque matrix element used in ab- initiocalculations of the Gilbert damping: ∝an}b∇acketle{ta,k|K|b,k+q∝an}b∇acket∇i}ht ≡1 ∆0∝an}b∇acketle{ta,k|/bracketleftBig H(k) so,S+/bracketrightBig |b,k+q∝an}b∇acket∇i}ht(37) Eq. (36) implies that at q=0∝an}b∇acketle{tb,k|S+|a,k∝an}b∇acket∇i}ht ≃ ∝an}b∇acketle{tb,k|K|a,k∝an}b∇acket∇i}htprovided that ( Ek,a−Ek,b)<<∆0, which is trivially satisfied for intra-band transitions but less so for inter-band transitions.18Forq∝ne}ationslash=0the agreement between intra-band matrix elements is no longer obvi- ous and is affected by the momentum dependence of the band eigenstates. At any rate, Eq. (29) demon- strates that only q=0matrix elements contribute to βintra; therefore βintrahas the same value for SandK matrix elements. The disparity between the two formu- lations is restricted to βinter, and may be significant if the most prominent inter-band matrix elements connect states that are notclose in energy. When they disagree,0.00 0.05 0.10 0.15 0.20 1/(εFτ)0.000.100.200.300.40βS+ K∆0=0.5εF ; λkF=0.8εF FIG. 6: M2DEG : In the strongly spin-orbit coupled limit the intra-band contribution reigns over the inter-band con tri- bution and accordingly SandKmatrix element expressions display a good (excellent in this figure) agreement. Neverth e- less, this agreement does not guarantee quantitative relia bil- ity, because for strong spin-orbit interactions impurity v ertex corrections may play an important role. 0.00 0.10 0.20 0.30 1/(εFτ)0.00.20.40.6βS+ Kx=0.08 ; p=0.4 nm−3 FIG. 7: GaMnAs : comparison between SandKmatrix element expressions for E⊥ˆz. The disagreement between both formulations stems from inter-band transitions, whic h are less important as τincreases. Little changes when E/bardblˆz. it is generally unclear32whether SorKmatrix elements will yield a better estimate of βinter. The weak spin-orbit limit is a possible exception, in which the use of Kap- pearstoofferapracticaladvantageover S. Inthis regime Sgenerates a spurious inter-band contribution in the ab- sence of magnetic impurities (recall Section III) and it is onlyafterthe inclusion ofthe leadingordervertexcorrec- tion that such deficiency gets remedied. In contrast, K vanishes identically in absence of spin-orbit interactions, thus bypassing the pertinent problem without having to introduce vertex corrections. Figs. (5)- (7) display a quantitative comparison be- tween the non-adiabatic STT obtained from KandS,12 both for the M2DEG and (Ga,Mn)As. Fig. (5) reflects the aforementioned overestimation of Sin the inter-band dominated regime ofweakly spin-orbitcoupled ferromag- nets. In the strong spin-orbit limit, where intra-band contributions dominate in the disorder range of interest, KandSagree fairly well (Figs. (6) and (7)). Summing up, insofar as impurity vertex corrections play a minor roleandthedominantcontributionto βstemsfromintra- band transitions the torque-correlation formula will pro- vide a reliable estimate of β. VIII. CONNECTION TO THE EFFECTIVE FIELD MODEL As explained in Section II we view the non-adiabatic STT as the change in magnetization damping due to a transport current. The present section is designed to complement that understanding froma different perspec- tive based on an effective field formulation, which pro- vides asimple physicalinterpretationforboth intra-band and inter-band contributions to β. An effective field Heffmay be expressed as the varia- tion of the system energy with respect to the magnetiza- tion direction Heff i=−(1/s0)∂E/∂Ωi. Here we approxi- mate the energy with the Kohn-Sham eigenvalue sum E=/summationdisplay k,ank,aǫk,a. (38) The variation of this energy with respect to the magne- tization direction yields Heff i=−1 s0/summationdisplay k,a/bracketleftbigg nk,a∂ǫk,a ∂Ωi+∂nk,a ∂Ωiǫk,a/bracketrightbigg .(39) It has previously been shown that, in the absence of cur- rent, the first term in the sum leads to intra-bandGilbert damping15,35while the second term produces inter-band damping.34In the following, we generalize these resultsbyallowingthe flowofan electricalcurrent. αandβmay be extracted by identifying the the dissipative part of the effective field with −α∂ˆΩ/∂t−βvs·∇ˆΩ that appears in the LLS equation. Intra-band terms : We begin by recognizing that as the direction of magnetization changes in time, so does the shape of the Fermi surface, provided that there is an in- trinsic spin-orbit interaction. Consequently, empty (full) states appear below (above) the Fermi energy, giving rise to an out-of-equilibrium quasiparticle distribution. This configuration tends to relax back to equilibrium, but re- population requires a time τ. Due to the time delay, the quasiparticle distribution lags behind the dynamical configuration of the Fermi surface, effectively creating a friction (damping) force on the magnetization. From a quantitative standpoint, the preceding discussion means that the quasiparticle energies ǫk,afollow the magnetiza- tion adiabatically, whereas the occupation numbers nk,a deviate from the instantaneous equilibrium distribution fk,avia nk,a=fk,a−τk,a/parenleftbigg∂fk,a ∂t+˙ra·∂fk,a ∂r+˙k·∂fk,a ∂k/parenrightbigg , (40) where we have used the relaxation time approximation. As we explain below, the last two terms in Eq. (40) do not contribute to damping in the absence of an electric field and have thus been ignored by prior applications of the breathing Fermi surface model, which concentrate on Gilbert damping. It is customary to associate intra-band magnetization damping with the torque exerted by the part of the effective field Heff intra=−1 s0/summationdisplay k,ank,a∂ǫk,a ∂ˆΩ(41) that is lagging behind the instantaneous magnetization. Plugging Eq. (40) in Eq. (41) we obtain Heff intra,i=1 s0/summationdisplay k,a/bracketleftbigg −fk,a∂ǫk,a ∂Ωi+τa∂fk,a ∂ǫk,a∂ǫk,a ∂Ωi∂ǫk,a ∂Ωj∂Ωj ∂t+τa˙rl a∂fk,a ∂ǫk,a∂ǫk,a ∂Ωi∂ǫk,a ∂Ωj∂Ωj ∂rl+τa˙kj∂fk,a ∂ǫk,a∂ǫk,a ∂kj∂ǫk,a ∂Ωi/bracketrightbigg (42) where a sum is implied over repeated Latin indices. The first term in Eq. (42) is a contribution to the anisotropy field; it evolves in synchrony with the dynamical Fermi surfaceandisthusthereactivecomponentoftheeffective field. The remaining terms, which describe the time lag of the effective field due to a nonzero relaxation time, are responsible for intra-band damping. The last term van- ishesincrystalswith inversionsymmetrybecause ˙k=eE and∂ǫ/∂kis an odd function of momentum. Similarly,if we take ˙r=∂ǫ(k)/∂kthe second to last term ought to vanish as well. This leaves us with the first two terms in Eq. ( 42), which capture the intra-band Gilbert damping but not the non-adiabatic STT. This is not surprising as the latter involves the coupledresponse to spatial varia- tions of magnetization and a weak electric field, render- ing linear order in perturbation theory insufficient (see Appendix A). In order to account for the relevant non- linearity we use ˙r=∂ǫ(k−ev·Eτ)/∂kin Eq.( 42), where13 v=∂ǫ(k)/∂k. The dissipative part of Heff intrathen reads Heff,damp intra,i=1 s0/summationdisplay k,aτk,a∂fǫk,a ∂ǫk,a∂ǫk,a ∂Ωi∂ǫk,a ∂Ωj/bracketleftbigg∂Ωj ∂t+vl d,a∂Ωj ∂rl/bracketrightbigg , (43) wherevi d,a=eτa(m−1)i,j aEjis the “drift velocity” cor- responding to band a. Eq. (43) may now be identified with−αintra∂ˆΩ/∂t−βintravs· ∇ˆΩ that appears in the LLS equation. For an isotropic system this results in αintra=−1 s0/summationdisplay k,a,iτk,a∂fk,a ∂ǫk,a/parenleftbigg∂ǫk,a ∂Ωi/parenrightbigg2 βintra=−1 s0/summationdisplay k,a,iτk,a∂fk,a ∂ǫk,a/parenleftbigg∂ǫk,a ∂Ωi/parenrightbigg2q·vd,a q·vs.(44) Since∝an}b∇acketle{t[Sx,Hso]∝an}b∇acket∇i}ht=∂φ∝an}b∇acketle{texp(iSxφ)Hsoexp(−iSxφ)∝an}b∇acket∇i}ht= ∂ǫ/∂φfor an infinitesimal angle of rotation φaround the instantaneous magnetization, βin Eq. (44) may be rewritten as βintra=∆2 0 2s0/summationdisplay k,aτk,a∂fk,a ∂ǫk,a|∝an}b∇acketle{tk,a|K|k,a∝an}b∇acket∇i}ht|2q·vd,a q·vs(45) whereK= [S+,Hso]/∆0is the spin-torque operator in- troduced in Eq. ( 37) and we have claimed spin rota- tionalinvariancevia |∝an}b∇acketle{t[Sx,Hso]∝an}b∇acket∇i}ht|2=|∝an}b∇acketle{t[Sy,Hso]∝an}b∇acket∇i}ht|2. Using ∂f/∂ǫ≃ −δ(ǫ−ǫF) and recalling from Section VII that Ka,a=S+ a,a, Eq. (45) is equivalent to Eq. (30); note that the product of spectral functions in the latter yields a factor of 4 πτupon momentum integration. These obser- vations prove that βintradescribes the contribution from atransportcurrenttothe“breathingFermisurface”type of damping. Furthermore, Eq. (44) highlights the impor- tance of the ratio between the two characteristic veloci- ties of a current carrying ferromagnet, namely vsandvd. As explained in Section VI these two velocities coincide in models with Galilean invariance. Only in these arti- ficial models, which never apply to real materials, does α=βhold. Inter-band terms : The Kohn-Sham orbitals are effec- tive eigenstates of a mean-field Hamiltonian where the spins are aligned in the equilibrium direction. As spins precess in response to external rf fields and dc trans- port currents, the time-dependent part of the mean-field Hamiltonian drives transitions between the ground-state Kohn-Sham orbitals. These processes lead to the second term in the effective field and produce the inter-band contribution to damping.We thus concentrate on the second term in Eq. (39), Heff inter=−1 s0/summationdisplay k,a∂nk,a ∂ˆΩǫk,a. (46) Multiplying Eq. (46) with ∂ˆΩ/∂twe get Heff,damp inter·∂tˆΩ =−1 s0/summationdisplay k,aǫk,a/bracketleftBig ∂na,k/∂ˆΩ·∂ˆΩ/∂t/bracketrightBig =−1 s0/summationdisplay k,aǫk,a∂na,k/∂t. (47) The rate of change of the populations of the Kohn- Shamstatescanbeapproximatedbythefollowingmaster equation ∂na,k ∂t=−/summationdisplay b,k′Wa,b(nk,a−nk′,b),(48) where Wa,b= 2π|∝an}b∇acketle{tb,k′|∆0Sx|a,k∝an}b∇acket∇i}ht|2δk′,k+qδ(ǫb,k′−ǫa,k−ω) (49) is the spin-flip inter-band transition probability as dic- tated by Fermi’s golden rule. Eqs. (48) and (49) rely on the principle of microscopic reversibility36and are rather ad hocbecause they circumvent a rigorous analysis of the quasiparticle-magnon scattering, which would for instance require keeping track of magnon occu- pation number. Furthermore, quasiparticle-phonon and quasiparticle-impurity scattering are allowed for simply by broadening the Kohn-Sham eigenenergies (see below). The right hand side of Eq. (48) is now closely related to inter-band magnetization damping because it agrees37 with the netdecay rate of magnons into particle-hole excitations, where the particle and hole are in different bands. Combining Eq. (47) and (48) and rearranging terms we arrive at Heff inter·∂tˆΩ =1 2s0/summationdisplay k,k′,a,bWa,b(nk,a−nk′,b)(ǫk,a−ǫk′,b). (50) For the derivation of αinterit is sufficient to approximate nk,aas a Fermi distribution in Eq. (50); here we ac- countforatransportcurrentbyshiftingtheFermiseasas nk,a→nk,a−evk,a·Eτk,a∂nk,a/∂ǫk,a, which to leading order yields14 Heff inter·∂tˆΩ =−πω 2s0/summationdisplay k,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2δ(ǫb,k+q−ǫa,k−ω)∂nk,a ∂ǫk,a(−ω+eVb,a) =ω 8πs0/summationdisplay k,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2Aa(k,ǫF)Ab(k+q,ǫF)(−ω+eVb,a) (51) where we have used Sx= (S++S−)/2 and defined Vb,a=evk+q,b·Eτk+q,b−evk,a·Eτk,a. In the second line of Eq.( 51) we have assumed low temperatures, and have introduced a finite quasiparticle lifetime by broadening the spectral functions of the Bloch states into Lorentzians with the c onvention outlined in Eq. (14): δ(x)→A(x)/(2π). Identifying Eq.( 51) with ( −αinter∂tˆΩ−βinter(vs·∇)ˆΩ)·∂tˆΩ =−αinterω2+βinterω(q·vs) we arrive at αinter=1 8πs0/summationdisplay a,b/negationslash=a/summationdisplay k,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2Aa(k,ǫF)Ab(k+q,ǫF) βinter=1 8πs0q·vs/summationdisplay a,b/negationslash=a/summationdisplay k,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2Aa(k,ǫF)Ab(k+q,ǫF)Vb,a (52) in agreement with our results of Section II. IX. SUMMARY AND CONCLUSIONS Starting from the Gilbert damping αand including the influenceofanelectricfieldinthetransportorbitalssemi- classically, we have proposed a concise formula for the non-adiabatic spin transfer torque coefficient βthat can be applied to real materials with arbitrary band struc- tures. Our formula for βreproduces results obtained by more rigorous non-linear response theory calculations when applied to simple toy models. By applying this ex- pression to a two-dimensional electron-gas ferromagnet with Rashba spin-orbit interaction, we have found that it implies a conductivity-like contributionto β, related to thecorrespondingcontributiontotheGilbertdamping α, which is proportionalto scattering time rather than scat- tering rate and arises from intra-band transitions. Our subsequent calculations using a four-band model have shown that intra-band contributions dominate in ferro- magnetic semiconductors such as (Ga,Mn)As. We have then discussed the α/βratio in realistic materials and have confirmed trends expected from toy models, in ad- dition to suggesting that αandβcan have the oppo- site sign in systems where both hole-like and electron-like bands coexist at the Fermi surface. Afterwards, we have analyzed the spin-torque formalism suitable to ab-initio calculations, and have concluded that it may provide a reliable estimate of the intra-band contribution to β; for the inter-band contribution the spin-torque formula of-fers a physically sensible result in the weak spin-orbit limit but its quantitative reliability is questionable un- less the prominent inter-band transitions connect states that are close in energy. Finally, we have extended the breathing Fermi surface model for the Gilbert damping to current carrying ferromagnets and have accordingly found a complementary physical interpretation for the intra-band contribution to β; similarly, we have applied the master equation in order to offer an alternative inter- pretation for the inter-band contribution to β. Possible avenues for future research consist of carefully analyzing the importance of higher order vertex corrections in β, better understanding the disparities between the differ- ent approaches to vs, and finding real materials where α/βis negative. Acknowledgements We acknowledge informative correspondence with Rembert Duine and Hiroshi Kohno. In addition, I.G. is grateful to Paul Haney for interesting discussions and generous hospitality during his stay in the National In- stitute of Standards and Technology. This work was sup- ported in part by the Welch Foundation, by the National Science Foundation under grant DMR-0606489, and by the NIST-CNST/UMD-NanoCenter Cooperative Agree- ment. APPENDIX A: QUADRATIC SPIN RESPONSE TO AN ELECTRIC AND MAGNE TIC FIELD Consider a system that is perturbed from equilibrium by a time-depen dent perturbation V(t). The change in the expectation value of an operator O(t) under the influence of V(t) can be formally expressed as δ∝an}b∇acketle{tO(t)∝an}b∇acket∇i}ht=∝an}b∇acketle{tΨ0|U†(t)O(t)U(t)|Ψ0∝an}b∇acket∇i}ht−∝an}b∇acketle{tΨ0|O(t)|Ψ0∝an}b∇acket∇i}ht (A1)15 where|Ψ0∝an}b∇acket∇i}htis the unperturbed state of the system, U(t) =Texp/bracketleftbigg −i/integraldisplayt −∞V(t′)dt′/bracketrightbigg (A2) is the time-evolution operator in the interaction representation an dTstands for time ordering. Expanding the exponentials up to second order in Vwe arrive at δ∝an}b∇acketle{tO(t)∝an}b∇acket∇i}ht=i/integraldisplayt −∞dt′∝an}b∇acketle{t[O(t),V(t′)]∝an}b∇acket∇i}ht−1 2/integraldisplayt −∞dt′dt′′∝an}b∇acketle{t[[O(t),V(t′)],V(t′′)]∝an}b∇acket∇i}ht. (A3) For the present work, O(t)→Sa(a=x,y,z) and V(t) =−/integraldisplay drj·A(r,t)+/integraldisplay drS·Hext(r,t), (A4) whereAis the vector potential, Hextis the external magnetic field, and jis the current operator. Plugging Eq. (A4) into Eq. (A3) and neglecting O(A2),O(H2 ext) terms we obtain δSa(x) =/summationdisplay b/integraldisplay dx′χa,b S,jAb(x′)+/summationdisplay b/integraldisplay dx′χa,b S,SHb ext(x′)+/summationdisplay b,c/integraldisplay dx′dx′′χa,b,c S,S,jAb(x′)Hc ext(x′′),(A5) wherex≡(r,t) and/integraltext dx′≡/integraltext∞ −∞dt′/integraltext dr′. The linear and quadratic response functions introduced above ar e defined as χa,b S,j(x,x′) =i∝an}b∇acketle{t/bracketleftbig Sa(x),jb(x′)/bracketrightbig Θ(t−t′) χa,b S,S(x,x′) =i∝an}b∇acketle{t/bracketleftbig Sa(x),Sb(x′)/bracketrightbig Θ(t−t′) χa,b,c S,S,j(x,x′,x′′) =∝an}b∇acketle{t/bracketleftbig/bracketleftbig Sa(x),jb(x′)/bracketrightbig ,Sc(x′′)/bracketrightbig Θ(t−t′)Θ(t′−t′′) +∝an}b∇acketle{t/bracketleftbig/bracketleftbig Sa(x),Sb(x′′)/bracketrightbig ,jc(x′)/bracketrightbig Θ(t−t′′)Θ(t′′−t′) (A6) where we have used T[F(t)G(t′)] =F(t′)G(t′′)Θ(t′−t′′)+G(t′′)F(t′)Θ(t′′−t′), Θ being the step function. χS,jis the spin density induced by an electric field in a uniform ferromagnet, and it vanishes unless there is intrinsic spin-orbit interaction. χS,Sis the spin density induced by an external magnetic field. χS,S,jis the spin density induced by the combined action of an electric and magnetic field (see Fig. (8) for a dia grammatic representation); this quantity is closely related to ( vs·q)χ(2), introduced in Section II. APPENDIX B: FIRST ORDER IMPURITY VERTEX CORRECTION The aim of this Appendix is to describe the derivation of Eq. (21). We s hall begin by evaluating the leading order vertex correction to the Gilbert damping. From there, we shall obt ain the counterpart quantity for the non-adiabatic STT by shifting the Fermi occupation factors to first order in the e lectric field. The analytical expression for the transverse spin response with one vertex correction is (see Fig. ( 9)) ˜χQP,(1) +,−=−V∆2 0 2T/summationdisplay ωn/integraldisplay k,k′uiGa(iωn,k)S+ a,bGb(iωn+iω,k+q)Si a,b′Gb′(iωn+iω,k′+q)S− b′,a′Ga′(iωn,k′)Si a′,a.(B1) S+S− v.A FIG. 8: Feynman diagram for χS,S,j. The dashed lines correspond to magnons, whereas the wavy li ne represents a photon.16 S+S− FIG. 9: Feynman diagram for the first order vertex correction . The dotted line with a cross represents the particle-hole correlation mediated by impurity scattering. whereVis the volume of the system and the minus sign originates from fermion ic statistics. Using the Lehmannn representation of the Green’s functions Gand performing the Matsubara sum we get ˜χQP,(1) +,−=−V∆2 0 2/integraldisplay k,k′ui2 Re/bracketleftBig S+ a,bSi b,b′S− b′,a′Si a′,a/bracketrightBig/integraldisplay∞ −∞dǫ1dǫ′ 1dǫ2dǫ′ 2 (2π)4Aa(ǫ1,k)Aa′(ǫ′ 1,k′) ×Ab(ǫ2,k+q)Ab′(ǫ′ 2,k′+q)/bracketleftbiggf(ǫ1) (ǫ1−ǫ′ 1)(iω+ǫ1−ǫ2)(iω+ǫ1−ǫ′ 2)+/parenleftbigg ǫ1↔ǫ2,ǫ′ 1↔ǫ′ 2, ω↔ −ω/parenrightbigg/bracketrightbigg (B2) where twice the real part arose after absorbing two of the terms coming from the Matsubara sum. Next, we apply iω→ω+i0+and take the imaginary part: ˜χQP,(1) +,−=V∆2 0 22π/integraldisplay k,k′uiRe/bracketleftBig S+ a,bSi b,b′S− b′,a′Si a′,a/bracketrightBig/integraldisplay∞ −∞dǫ1dǫ′ 1dǫ2dǫ′ 2 (2π)4Aa(ǫ1,k)Aa′(ǫ′ 1,k′)Ab(ǫ2,k+q)Ab′(ǫ′ 2,k′+q) ×f(ǫ1) ǫ1−ǫ′ 1/bracketleftbiggδ(ω+ǫ1−ǫ2) ω+ǫ1−ǫ′ 2+δ(ω+ǫ1−ǫ′ 2) ω+ǫ1−ǫ2−/parenleftbigg ω→ −ω, q→ −q/parenrightbigg/bracketrightbigg (B3) where we used 1 /(x−iη) =PV(1/x) +iπδ(x), and invoked spin-rotational invariance to claim that terms with Sx a,bSi b,b′Sy b′,a′Si a′,awill vanish. Integrating the delta functions we arrive at ˜χQP,(1) +,−=V∆2 0 2/integraldisplay k,k′uiRe[...]/integraldisplay∞ −∞dǫ′ 1dǫ2dǫ′ 2 (2π)3f(ǫ2)Aa(ǫ2,k)Aa′(ǫ′ 1,k′) (ǫ2−ǫ′ 2)(ǫ2−ǫ′ 1) ×/bracketleftbig Ab(ǫ2+ω,k+q)Ab′(ǫ′ 2+ω,k′+q)+Ab(ǫ′ 2+ω,k+q)Ab′(ǫ2+ω,k′+q)/bracketrightbig −/parenleftbigg ω→ −ω, q→ −q/parenrightbigg (B4) The next step is to do the ǫ′ 1andǫ′ 2integrals, taking advantage of the fact that for weak disorder th e spectral functions are sharply peaked Lorentzians ( in fact at the present order of approximation one can take regard them as Dirac delta functions). The result reads ˜χQP,(1) +,−=V∆2 0 2/integraldisplay k,k′uiRe[...]/integraldisplay∞ −∞dǫ2 2πf(ǫ2)Aa(ǫ2,k) ǫ2−ǫk′,a′/bracketleftbiggAb(ǫ2+ω,k+q) ǫ2+ω−ǫk′+q,b′+Ab′(ǫ2+ω,k′+q) ǫ2+ω−ǫk+q,b/bracketrightbigg −(ω→ −ω,q→ −q) (B5) By making further changes of variables, this equation can be rewrit ten as ˜χQP,(1) +,−=V∆2 0 2/integraldisplay k,k′uiRe[...]/integraldisplay∞ −∞dǫ2 2π(f(ǫ2)−f(ǫ2+ω))Aa(ǫ2,k) ǫ2−ǫk′,a′/bracketleftbiggAb(ǫ2+ω,k+q) ǫ2+ω−ǫk′+q,b′+Ab′(ǫ2+ω,k′+q) ǫ2+ω−ǫk+q,b/bracketrightbigg (B6) This is the first order vertex correction for the Gilbert damping. In order to obtain an analogous correction for the non-adiabatic STT, it suffices to shift the Fermi factors in Eq. (B6) as indicated in the main text. This immediately results in Eq. (21). APPENDIX C: DERIVATION OF EQ. (26) Let us first focus on the first term of Eq. (17), namely Eiqj/integraldisplay k/bracketleftbig |∝an}b∇acketle{ta,k|S+|b,k∝an}b∇acket∇i}ht|2+|∝an}b∇acketle{ta,k|S−|b,k∝an}b∇acket∇i}ht|2/bracketrightbig AaA′ bvi k,avj k,bτk,a (C1)17 We shall start with the azimuthal integral. It is easy to showthat th e entire angle dependence comes from vivj∝kikj, from which the azimuthal integral vanishes unless i=j. Regarding the |k|integral, we assume that λkF,∆0,1/τ << ǫ F; otherwise the analytical calculation is complicated and must be tackled numerically. Such assumption allows us to use/integraltext k→N2D/integraltext∞ −∞dǫ. For inter-band transitions (a∝ne}ationslash=b),AaA′ bcontributes mainly thru the pole at ǫF,a, thus all the slowly varying factors in the integrand may be set at the Fermi energy. For intra-band transitions ( a=b),AaA′ ahas no peak at the Fermi energy; hence it is best to keep the slowly varying factors inside the integrand. The above observations lead straightforwardly to the following res ult: Eiqj/integraldisplay k/bracketleftbig |∝an}b∇acketle{ta,k|S+|b,k∝an}b∇acket∇i}ht|2+|∝an}b∇acketle{ta,k|S−|b,k∝an}b∇acket∇i}ht|2/bracketrightbig AaA′ bvi k,avj k,bτk,a ≃E·qm2 8m+m−/parenleftbigg 1+∆2 0 b2/parenrightbigg(ǫF,−τ−Γ+−ǫF,+τ+Γ−) b3 −E·q/bracketleftbiggm2 m2 +1 2λ2k2 F b2/parenleftbigg 1+∆2 0 b2/parenrightbigg τ2 ++m2 m2 −1 2λ2k2 F b2/parenleftbigg 1+∆2 0 b2/parenrightbigg τ2 −/bracketrightbigg (C2) The second and third line in Eq. (C2) come from inter-band and intra- band transitions, respectively. The latter vanishes in absence of spin-orbit interaction, leading to a 2D version of Eq. (20). Since the band-splitting is much smaller than the Fermi energy, one can further simplify the above e quation via τ+≃τ−→τ. Let us now move on the second term of Eq. (17), namely Eiqj/integraldisplay kRe/bracketleftbig ∝an}b∇acketle{tb,k|S−|a,k∝an}b∇acket∇i}ht∝an}b∇acketle{ta,k|S+∂kj|b,k∝an}b∇acket∇i}ht+(S+↔S−)/bracketrightbig AaAbvi k,aτk,a (C3) Most of the observations made above apply for this case as well. For instance, the azimuthal integral vanishes unlessi=j. This follows from a careful evaluation of the derivatives of the eige nstates with respect to momentum; ∂kjθ= sin(θ)cos(θ)kj/k2(0≤θ≤π/2) is a useful relation in this regards, while ∂kjφplays no role. As for the |k| integral, we no longer have the derivative of a spectral function, b ut rather a product of two spectral functions; the resulting integrals may be easily evaluated using the method of residu es. The final result reads Eiqj/integraldisplay kRe/bracketleftbig ∝an}b∇acketle{tb,k|S−|a,k∝an}b∇acket∇i}ht∝an}b∇acketle{ta,k|S+∂kj|b,k∝an}b∇acket∇i}ht+(S+↔S−)/bracketrightbig AaAbvi k,aτk,a ≃ −E·q/bracketleftbiggm 32m−λ2k2 F∆2 0 b6/parenleftbigg 1+τ− τ+/parenrightbigg +m 32m+λ2k2 F∆2 0 b6/parenleftbigg 1+τ+ τ−/parenrightbigg/bracketrightbigg +E·q/bracketleftbiggm 4m+λ2k2 F∆2 0 b4τ2 ++m 4m−λ2k2 F∆2 0 b4τ2 −/bracketrightbigg (C4) The first line in Eq. (C4) stems from inter-band transitions, wherea s the second comes from intra-band transitions; bothvanish in absence of SO. Once again we can take τ+≃τ−→τ. Combining Eqs. (C2) and (C4) one can immediately reach Eq. (26). 1L. Berger, J. Appl. Phys. 3, 2156 (1978); ibid.3, 2137 (1979). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 4H. Kubota, A. Fukushima, Y. Ootani, S. Yuasa, K. Ando, H. Maehara, K. Tsunekawa, D.D. Djayaprawira, N. Watanabe and Y. Suzuki, Jap. J. of Appl. 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MacDonald, arXiv:0808.3923. 19Here we assume that the dependence of energy on mag- netization direction which determines Heffis specified as a function of Ω xand Ω yonly with Ω zimplicitly fixed by the constraint Ω z= [1−Ω2 x−Ω2 y]1/2. If the free energy was expressed in a form with explicit Ω zdependence we would find Heff,x=−∂F/∂Ωx−(∂F/∂Ωz)(∂Ωz/∂Ωx) = −∂F/∂Ωx+(∂F/∂Ωz)Ωx, whereFis the free energy of the ferromagnet. Similarly we would find Heff,y=−∂F/∂Ωy+ (∂F/∂Ωz)Ωy. The terms which arise from the Ω zdepen- dence ofthe free energy would more commonly be regarded as contributions to Heff,z. The difference is purely a mat- ter of convention since both results would give the same value for ˆΩ×Heff. 20Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404 (2002). 21O. Gunnarsson, J. Phys. F 6, 587 (1976). 22We assume that magnetic anisotropy and the external magnetic fields are weak compared to the exchange- correlation splitting of the ferromagnet. ¯∆ is the spin- density weighted average of ∆( r) (see Ref. [17]). 23For convenience in Eq. (8) we use /angbracketleftS+S−/angbracketrightresponse func-tions instead of /angbracketleftSxSx/angbracketrightand/angbracketleftSySy/angbracketright. They are related via Sx= (S++S−)/2 andSy= (S+−S−)/2i. 24J. Fernandez-Rossier, M. Braun, A. S. Nunez, A. H. Mac- Donald, Phys. Rev. B 69, 174412 (2004). 25J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag- netic Structures III: Fundamentals of Nanomagnetism (Springer-Verlag, New York, 2005). 26G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008). 27For a theoretical study on how Rashba spin-orbit interac- tion affects domain wall dynamics see K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008). 28T. Jungwirth, J. Sinova, J. Masek, J. Kucera and A.H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). 29For actual ab-initio calculations it may be more con- venient to substitute |/angbracketlefta,k|∆0S+|b,k/angbracketright|2in Eq. (35) by |/angbracketlefta,k|K|b,k/angbracketright|2, where Kis the spin-torque operator dis- cussed in Section VII. In either case we are disregarding impurity vertex corrections, which may become significant in disordered and/or strongly spin-orbit coupled systems. 30K.M.D. Hals, A.K. Nguyen and A. Brataas, arXiv:0811.2235. 31V. Kambersky, Phys. Rev. B 76, 134416 (2007); K. Gilmore, Y.U. Idzerda and M.D. Stiles, Phys. Rev. Lett. 99, 27204 (2007). 32In order to gauge the accuracy of either matrix element, one must obtain an exact evaluation of the non-adiabatic STT, which entails a ladder-sum renormalization18ofS±. This is beyond the scope of the present work. 33S.E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). 34K. Gilmore, Y.U. Idzerda and M.D. Stiles, J. Appl. Phys. 103, 07D303 (2008). 35D. Steiauf and M. Fahnle, Phys. Rev. B 72, 064450 (2005); D. Steiauf, J. Seib and M. Fahnle, Phys. Rev. B 78, 02410(R) (2008). 36This principle states that Wa,b=Wb,aexp((ǫa−ǫb)/T). Since the magnon energy is much smaller than the un- certainty in the quasiparticle energies, we approximate Wa,b≃Wb,a. 37For an analogous observation in the context of electron- phonon interaction see e.g.D. Pines, Elementary Excita- tions in Solids (Benjamin, 1963).

2008-12-13

The motion of simple domain walls and of more complex magnetic textures in the presence of a transport current is described by the Landau-Lifshitz-Slonczewski (LLS) equations. Predictions of the LLS equations depend sensitively on the ratio between the dimensionless material parameter $\beta$ which characterizes non-adiabatic spin-transfer torques and the Gilbert damping parameter $\alpha$. This ratio has been variously estimated to be close to 0, close to 1, and large compared to 1. By identifying $\beta$ as the influence of a transport current on $\alpha$, we derive a concise, explicit and relatively simple expression which relates $\beta$ to the band structure and Bloch state lifetimes of a magnetic metal. Using this expression we demonstrate that intrinsic spin-orbit interactions lead to intra-band contributions to $\beta$ which are often dominant and can be (i) estimated with some confidence and (ii) interpreted using the "breathing Fermi surface" model.

Non-Adiabatic Spin Transfer Torque in Real Materials

0812.2570v1

arXiv:0901.2191v1 [hep-th] 15 Jan 2009The sound damping constant for generalized theories of gravity Ram Brustein Department of Physics, Ben-Gurion University, Beer-Sheva, 84105 Israel, E-mail: ramyb@bgu.ac.il A.J.M. Medved Physics Department, University of Seoul, Seoul 130-743 Korea, E-mail: allan@physics.uos.ac.kr Abstract The near-horizon metric for a black brane in Anti-de Sitter ( AdS) space and the metric near the AdS boundary both exhibit hydro dy- namic behavior. We demonstrate the equivalence of this pair of hy- drodynamic systems for the sound mode of a conformal theory. This is first established for Einstein’s gravity, but we then show how the sound damping constant will be modified, from its Einstein fo rm, for a generalized theory. The modified damping constant is expre ssible as the ratio of a pair of gravitational couplings that are indic ative of the sound-channel class of gravitons. This ratio of couplings d iffers from 1both that of the shear diffusion coefficient and the shear viscos ity to entropy ratio. Our analysis is mostly limited to conformal t heories but suggestions are made as to how this restriction might eve ntually be lifted. 1 Introduction Thenear-horizon geometry of ablack branein an Anti-de Sitt er (AdS) spacetime provides a translationally invariant and therma lly equili- brated background; two of the characteristic features of an y hydrody- namic theory. Indeed, the long-wavelength fluctuations of t he near- horizon metric are known to satisfy equations of motion that are com- pletely analogous to the hydrodynamic equations of a viscou s fluid [1]. The very same statements can be made about the metric nea r the AdS boundary. It is, however, quite remarkable that this pai r of ef- fective theories appears to be described by an equivalently defined set of hydrodynamic parameters [1, 2, 3, 4]; this, in spite of the ir obvious lack of proximity. The relevant thermodynamic and hydrodynamic parameters — such as the entropy density or the various transport coefficie nts — are intrinsic properties of the black brane horizon. Conseq uently, such parameters should be determined by the near-horizon me tric. This makes it all the more phenomenal that the very same param - eters are employed in the AdS boundary theory. In the context of the gauge–gravity duality, this apparent non-locality bec omes much more sensible. The duality relates the AdS boundary hydrody namics to the hydrodynamics of strongly coupled gauge theories [5, 3]. Mean- 2while, thedual gauge theory is supposedto have its thermal p roperties ascribed to it, holographically, by the thermodynamic natu re of the black brane. Most calculations inthis genretake place at theouter bound ary, as this is the most convenient surface for relating the bulk gra vitational theory to its gauge-theory dual. In many ways, however, the m ost natural setting is at the black brane horizon, where the vari ous hydro- dynamicparameters are actually defined. Thegraviton hydro dynamic “fluid” can be interpreted as “living” on the stretched horiz on, and so it would be rather disturbing if the actual calculations c ould only be performed on a surface that is displaced a spacetime away. Our results will make it clear that there is really nothing parti cularly spe- cial about either the stretched horizon or the AdS outer boun dary. Rather, all calculations might as well be done on any radial s hell that is external to the horizon. Strongly coupled gauge theories provide an intriguing theo retical laboratory to investigate the field of relativistic hydrody namics. It is hoped that, by applying the duality to certain calculatio ns on the gravity side, one would be able explain the experimental res ults of — for instance — heavy-ion collisions [6]. However, advance ments in this direction has been somewhat impeded for the following r eason: Studieson thehydrodynamicsof theAdS boundaryhave, for th emost part, been limited to Einstein’s theory of gravity, which — f rom the gauge-theory perspective — corresponds to infinitely stron g ’t Hooft coupling. Insofar as the objective is to apply what can be lea rnt from the duality to physically real systems, one actually requir es knowledge about gauge theories at finitevalues of ’t Hooft coupling. 3As it so happens, a strong-coupling expansion on the gauge-t heory side corresponds to an expansion in the number of derivative s on the gravity side. Since Einstein’s gravity is only a two-deriva tive theory, it should be clear that describing a finitely coupled gauge theo ry neces- sitates some sort of extension of Einstein’s theory. To put i t another way, any discernible progress will depend upon our understa nding of the boundary hydrodynamics for theories of generalized gravity. In a previous paper [7], we were able to establish two relevan t points. First, with the focus on the shear channel of fluctuat ing grav- itational modes, it was shown that the AdS boundary hydrodyn amics can be translated to and localized on any radial shell in the a ccessi- ble spacetime; including at the (stretched) horizon of the b lack brane. Then, by following [8], we explicitly demonstrated how this formalism can be extended to any generalized (or Einstein-corrected) gravita- tional theory. In [8, 7], we used insight from [9] to make a pertinent observa tion: Various hydrodynamic parameters of an AdS brane theory can b e identified with the different components of a (generally) pola rization- dependent gravitational coupling κµν. Meaning that, for a generic theory, differently polarized gravitons will effectively have differing Newton’s constants. As shown in [9], this distinction can be quanti- fied at a remarkably rigorous level. With this prescription, the shear viscosity to entropy density ratio η/sis generalized from its “stan- dard” (Einstein) value of 1 /4πaccording to [8]η s=1 4π(κrt)2 (κxy)2, with the precise meaning of the subscripts to be clarified below. M oreover, the central finding of [7] was that the shear diffusion coefficien tDis modified from its usual expression 1 /4πTinto the form D=κ2 zx κ2 tx1 4πT, 4withTbeing the temperature. Oneshouldtake notethat thecouplingratios for η/sandDinvolve different polarization directions. This is a natural consequ ence of the class of gravitons that is implicated by each of the hydrodyn amic parameters. In the so-called radial gauge, the non-vanishi ng gravitons separate into three decoupled classes or “channels”: scala r, shear and sound [10]. The shear viscosity ηis most directly associated with the first of these classes, whereas the shear diffusion coefficie ntD is a characteristic of the second. As for the third class, one would analogously associate with it the sound damping constant Γ, as well as the sound velocity (squared) c2 s. The purpose of the current paper is to analyze the case of soun d- mode fluctuations. A straightforward extension of previous analyses is inhibited by two technical issues that are intrinsic to al most any rigorous study of the sound channel. First, for a non-confor mal gauge theory, the sound-channel analysis is highly model specific . Second, the same non-conformality induces would-be radial invaria nts to vary with radial position in the bulk. (See [11] for a discussion. ) As a con- sequence, Γ and c2 sare, even for Einstein’s gravity, model-dependent parameters that vary with radial position in a model-specifi c way. We can still be quite definitive by restricting the immediate con- siderations to conformal theories. When conformality is pr otected by effectively “switching off” all massive fields, the above compli cations will no longer be of issue. At the same time, we will still be ab le to make statements about how deviations from conformality s hould influence the ensuing results. In this sense, the current stu dy can be viewed as a significant first step towards a fully generic anal ysis. 5Similarly to [7], we will begin here by establishing a direct con- nection between sound-mode (conformal) hydrodynamics on t he AdS outer boundary and on any other radial shell up to the horizon of the black brane. This will be accomplished by examining the corr elator of an appropriately defined graviton and verifying that its p ole struc- ture, which determines the associated dispersion relation , is a radial invariant. Next, we will determine how this correlator pole is explicit ly modi- fied for a generalized (although still conformal) theory of g ravity. This will enable us to extract the Einstein-corrected form of the damping constant Γ. Additionally, we will confirm that the sound velo cityc2 s remains fixed at its conformal value. As also discussed, the v ery same outcomes can be deduced through an inspection of the conserv ation equation for the dissipative stress tensor. The paper will conclude with a preliminary discussion of pos sible extensions of our analysis to the non-conformal case. Note that, to avoid needless repetition, some salient point s that are already covered thoroughly in [7] (also see [8]) will onl y be glossed over here. 2 Soundmode conformalhydrodynam- ics for Einstein’s gravity Let us first introduce some notation and conventions, as well as es- tablish the basic framework. We will be considering a black p–brane in ad+ 1-dimensional (asymptotically) AdS spacetime. (Note tha t 6d=p+ 1≥4.) Given translational invariance and spatial isotropy on the brane along with a static spacetime, the associated me tric can always be expressed in the generic brane form ds2=−gtt(r)dt2+grr(r)dr2+gxx(r)/parenleftBiggp/summationdisplay i=1dx2 i/parenrightBigg ,(1) wheregtt(r) has a simple zero and grr(r) has a simple pole at the horizonr=rh, whilegxx(rh) is finite and positive. For any r > rh, these metric components are all well-defined and strictly p ositive functions that go asymptotically to their respective AdS va lues (L2/r2 forgrr, otherwise r2/L2)asr→ ∞. (ListheAdSradiusofcurvature.) If the background theory is conformal, then one can be much mo re explicit. Assuming, for the sake of simplicity, that thebra neis electro- magnetically neutral, we obtain the Schwarzschild-like fo rm such that gxx=r2/L2andgtt= 1/grr=gxxf(r), withf(r) = 1−(rh/r)p+1. It is often convenient to re-express this conformal metric b y changing the radial coordinate to u=r2 h/r2; then ds2=−r2 h L2uf(u)dt2+L2 4u2du2 f(u)+r2 h L2u/parenleftBiggp/summationdisplay i=1dx2 i/parenrightBigg ,(2) withf(u) = 1−up+1 2and the horizon (outer boundary) now located atu= 1 (u= 0). When non-conformal theories are discussed, u will refer to a radial coordinate that is appropriately defin ed so as to extend over the same range of values. Brane hydrodynamics entails expanding the metric: gµν→gµν+ hµν, withhµνrepresenting the fluctuations or gravitons. Let us — without loss of generality — specify xpto be the direction of gravi- ton propagation on the brane and re-label it as z. It follows that 7hµν∼exp[−iΩt+iQz] (and, otherwise, depending only on u), where (Ω,0,...,0,Q) is thep+1–momentum of the graviton. The choice of radial gauge, huα= 0 for any α, is known to separate the non-vanishing fluctuations into three decoupled classe s [10]. Our class of current interest — namely, the sound channel — inclu des the non-vanishing diagonal gravitons hαα(α/negationslash=u) along with htz. Let ustake noteof thesound-modedispersionrelation Ω = ±csQ− iΓQ2+O(Q3) or, equivalently (given that the hydrodynamic or long- wavelength limit is in effect), Ω2=c2 sQ2+i2ΓΩQ2+O(Q4). (3) Here,c2 sis the sound velocity (squared) and Γ is the sound damping constant. For a p+ 1-dimensional conformal theory, c2 s= 1/pand Γ is directly proportional to the shear viscosity to entropy density ratio times the inverse temperature: Γ =p−1 p1 Tη s. So that, for a p- brane theory of Einstein’s gravity, one can deduce that Γ =p−1 p1 4πT [12, 13], where Tis the coordinate-invariant Hawking temperature of the brane. For this conformal case, T= (p+1)rh/4πL2. Meanwhile, for a non-conformal theory, both parameters can differ apprec iably from their conformal values. For Γ, this model-specific devi ation is expressible in terms of the bulk viscosity ζ. For a complete derivation of the sound-mode correlator (whi ch is not needed here), one can follow the by-now standard presc rip- tion as documented in, for instance, [14, 15, 16]. The first st ep is to identify a gauge-invariant combination of the sound-mod e fluctu- ationsHtt= (1/gtt)htt,Hzz= (1/gxx)hzz,Htz= (1/gxx)htzand 8HX= (1/gxx)1 p−1/parenleftBig/summationtextp−1 i=1hxixi/parenrightBig : Z=q2gtt gxxHtt+2qωHtz+ω2[Hzz−HX]+q2gtt′ gxx′HX.(4) Here, a prime indicates a differentiation with respect to u; while ω= Ω/2πTandq=Q/2πTrepresent, respectively, a dimension- less frequency and wavenumber. In the hydrodynamic limit, ωandq are both vanishing although not a prioriat the same rate. Theconformalversionofthesolutionfor Zcanreadilybeextracted out of the existent literature — for instance, [15, 16, 17]. F or the appropriately chosen boundary conditions (as discussed be low), one finds that Z=Cf(u)−iω 2/bracketleftbigg Y(u)−ω2 q2p−iω(p−1)f(u)+O(q2,ω2)/bracketrightbigg ,(5) whereCis an integration constant (to be fixed by normalization con- siderations) and we have defined Y(u)≡gtt′/gxx′. For this theory in particular, Y= (f/u)′/(1/u)′=f−uf′, which is everywhere positive, non-vanishing and O(1). Also note that Y= 1 on the outer boundary. A specified pair of boundaryconditions determines the solut ion for Z. At the horizon u= 1, the solution should be that of an incoming plane wave, which determined the form of Eq. (5). In addition , the so- called Dirichlet boundary condition still needs to be impos ed. It has become almost traditional to single out the AdS boundary and choose u∗= 0 as the radius at which this condition is enforced; however , one can freely impose this condition at any fixed radius u∗within 0 ≤ u∗<1. The Dirichlet boundary condition necessitates that Z(u) is, priorto its normalization (see below), vanishing as u→u∗. Applying thiscondition toEq. (5), we promptlyobtain theassociated dispersion 9relation ω2=q21 pY(u∗)+iωq2p−1 pf(u∗)+O(q4), (6) where it is now clear that qandωare of the same order in the hydro- dynamic limit. Let us choose, for instance, the “orthodox” b oundary location of u∗= 0, so that Eq. (6) leads to ω2=q2/p+iωq2(p− 1)/p+O(q4). Comparing this to the standard dispersion relation in Eq. (3), one can readily verify the expected identifications c2 s= 1/p and Γ = ( p−1)/(4πpT) for a conformal theory. One further normalization condition that complements the D irich- let boundary condition is that Z, rather than vanishing at u∗, should ultimately be normalized to unity there. This can be achieve d by the unique choice C−1=/bracketleftbigg Y(u∗)−ω2 q2p−iω(p−1)f(u∗)/bracketrightbigg . (7) Let us take notice that, given the associated dispersion rel ation,C−1 is a vanishing quantity as it must be to obtain a finite value of Z(u∗). The normalized value of the field mode is simply 1, and so one might wonder as to the physical significance of the implied di sper- sion relation. However, we are simply using the standard “tr ick” of field-theoretic calculations to obtain the pole in the corre lator. The properly normalized correlator GZZfor this gauge-invariant variable (up to an inconsequential numerical factor) is given by the b oundary residue of the canonical term in the bulk action or GZZ∼ZZ′|u=u∗. It should not be difficult to convince oneself that, at the lead ing hy- drodynamic order, this quantity goes as GZZ∼C· O(q0), which is notably divergent and of finite hydrodynamic order. A proper ac- counting of the metrical factors in the action — namely, the p roduct 10√−gguu— reveals that there are no other hidden zeros or infinities in this calculation at any permissible value of u∗. What is really significant here is that — from the quasinormal - mode perspective of brane hydrodynamics [15] — the pole in th e cor- relator assigns a clear physical credence to Eq. (6) as the sp ectrum for the dissipative modes of the black brane. However, one co uld (and should!) be rightfully concerned that this dispersion rela tion appears to vary as the Dirichlet-boundary surface is moved radially through the spacetime. This is not only in conflict with intuitive exp ectations but with the analysis of [11], where it is made evident that (i nasmuch as the theory is conformal) both the sound mode and its correl ator should be radial invariants. We can readily account for the undesirable factor of f(u∗) in the second term of Eq. (6): As detailed in [7], ωandqshould natu- rally be sensitive to the the effects of a gravitational redshi ft. It was then argued — in the context of shear modes — that consistency of the hydrodynamic expansion along with protection of the inc oming boundary condition necessitates that ωremains fixed while q2scales asf−1. That is, ω(u) =ωbandq2(u) =q2 b/f(u) (with the subscript bindicating the outer-boundary value of a quantity). Then, s ince the gravitational redshift should not be able to discriminate b etween the different channels being probed, it follows that these same re lations should persist for the current case. This brings us to the first term in Eq. (6), which has the awkwar d appearance of Y(u∗) to be dealt with. Clearly, this will require new inputs. Thekey hereis theassociation of this term with c2 s;cf, Eq. (3). Normally, the sound velocity of a hydrodynamic fluid is prese nted as 11the variation of the pressure with respect to the energy dens ity or c2 s=δP/δǫ. This cannot, however, be a universally accurate account. A closer look at the derivation of the sound dispersion relat ion (see, e.g.,[13])revealsthattheactual variation whichentersunder theguise of the sound velocity comes packaged in the term/parenleftbig δTzz/δTtt/parenrightbig ∂2 zTtt, whereTαβis the stress tensor for the brane theory. For a flat or an effectively flat brane, such as at the AdS outer boundary, thi s distinction is of no consequence, but this is not a general tr uism. On a “warped” brane, rather, Tzz=gzzPandTtt=−gttǫ. Hence, the correct statement about the relevant term is (by way of the ch ain rule) δTzz δTtt∂2 zTtt=gzzδP δǫ∂2 zǫ+gttP ǫ∂ugzz ∂ugtt∂2 zǫ . (8) We will now argue that the first term on the right-hand side of Eq. (8) is parametrically smaller than the second and, thus, the for- mer can be disregarded for current purposes. It follows from the ther- modynamic relation sT=ǫ+Pand the infinite transverse volume of the brane that ∂P/∂ǫ= 0. So, to the leading non-vanishing order, δP=1 2∂2P ∂ǫ2(δǫ)2orδP δǫ∼δǫ ǫ≪1 . On the other hand, P/ǫ=1 pis of the order of unity. Now, comparing gtt∂ugzz ∂ugttwithgzz, one will find that the ratio of these quantities is of O(1). Having deemed the first term in Eq. (8) as inconsequential, we need only to evaluate the second. Since the brane metric is di agonal (so that gαα=g−1 αα), the right-hand side reduces to δTzz δTtt∂2 zTtt=P ǫgtt g2xxY−1∂2 zǫ+···, (9) where we have returned to the brane notation of Eq. (1) (so tha t gtt>0) and recalled the definition of Y(u) beneath Eq. (5). 12Actually, all other terms in thedispersionrelation contai n a spatial component of the brane stress tensor, and so share a common fa ctor ofg−1 xx. Hence, we can strip off one factor of this from the right side o f Eq. (9). Next, let us identify P/ǫas the sound velocity as measured on the outer AdS boundary and everything to the left of ∂2 zǫas the sound velocity as measured on a radial shell of arbitrary rad ius. Then it follows that the sound velocity scales relative to the out er boundary as /bracketleftbig c2 s/bracketrightbig u=Y−1(u)gtt(u) gxx(u)[c2 s]b, (10) where a subscript of udenotes the value of a parameter at that radius. Calling again on our conformal-theory notation, let us take note that, bydefinition, f=gtt/gxx, andsothesoundvelocity equivalently scales asf/Y. Next, let us re-express Eq. (6) in a way that makes the scaling properties of the parameters explicit: ω2 u∗=q2 u∗Y(u∗)[c2 s]u∗+iωu∗q2 u∗p−1 pf(u∗)+O(q4 u∗).(11) Here, we have made the identification1 p→[c2 s]u∗on the basis that the Dirichlet-boundary surface is where the sound velocity should be calibrated to its conformal value — just like it is the Dirich let surface that defines where the field Zis exactly unity. We can now apply the previously discussed scalings ( q2∼1/f,c2 s∼f/Yand an invariant ω) to convert the above expression into one that involves only t he outer- boundary values of the parameters. Also recalling that f=Y= 1 at the AdS boundary u= 0, we then have ω2 b=q2 bY(0)[c2 s]b+iωbq2 bp−1 pf(0)+O(q4 b). (12) 13But this is precisely what would have been obtained had we mad e the choice of u∗= 0 in the first place. Hence, the dispersion relation is indeed a radial invariant and, by direct implication, the co rrelator is as well. 3 Soundmode conformalhydrodynam- ics for generalized theories of gravity Next on the agenda, we will investigate as to how the scenario changes when the theory is extended from Einstein’s gravity. It will be shown that, for a quite general (although still conformal) gravit y theory, the damping constant is modified in a very precise way. Meanwhile , the sound velocity is shown to be unmodified, as must be the case fo r a conformal theory. These tasks will be accomplished by exam ining the (modified) pole of the just-discussed correlator. These general- izations will be further supported by a simple argument that is based upon inspecting the conservation equation that gives rise t o the sound dispersion relation. By ageneralized gravity theory, wehave inmindaLagrangian that can be expressed as Einstein’s form plus higher-derivative s terms. If Einstein’s gravity is “non-trivially” modified by these cor rections — meaning that the general Lagrangian can notbe converted into Ein- stein’s form by a field redefinition — then the gravitational c oupling is no longer as simple as κ2 E= constant. Rather, the coupling (or effective Newton’s constant) can be expected to depend on the p olar- ization of the gravitons being probed. We will denote this de pendence 14by expressing the general couplings as κµν. It is now well understood as to how one should calculate these couplings for a given theory [9, 8, 7]. These formalities nee d not concern the present discussion, although a schematic under standing of how the couplings come about should prove useful. One begi ns by writing the Lagrangian as a perturbative expansion in power s of the metric fluctuations or h’s. Of particular significance are the terms that are quadratic in hand contain exactly two derivatives. For such terms, the gravitational couplings are identified on the pre mise that hµν→κµνhµνleads to a canonical kinetic term for the µν-polarized graviton. As it turns out, the gravitational couplings are expressibl e strictly in terms of the metric at the horizon. Like the metric, they ar e typically radial functions; however, at the level of a two-d erivative expansion of the Lagrangian, the couplings can safely be tre ated as (polarization-dependent) constants. Moreover, sincethe horizon is the true arena for black brane hydrodynamics, this locality is q uite nat- ural and falls in line with other parameters, such as the entr opy and shear viscosity, being intrinsic properties of this specia l surface. Letusre-emphasizethatanygivenhydrodynamicparameters hould be modified according to the class of gravitons that it probes . By working in the radial gauge and then restricting to the decou pled set of modes that defines the sound channel, we are limited to a sel ect class. Namely, the zz,tt, andtz-polarized gravitons, as well as the “trace mode”, which can be identified with HXin Eq. (4). When the theory is conformal, we can anticipate a further lim i- tation. To elaborate, in obtaining the solution for Z(see,e.g., [15, 1516, 17]), one finds that the Httmode makes no direct contribution to Eq. (5). (This is not at all true when conformality is broken. ) Recall- ing that the gauge–gravity duality identifies the tt-polarized gravitons with fluctuations in the energy density, we suspect that this null con- tribution is another manifestation of the suppression of th e variation δP/δǫ(as discussed in the previous section). On this basis, it see ms reasonable to suggest that the ttfluctuations can be excluded from a conformal theory in the hydrodynamic limit. As implied above, the modifications of interest can be extrac ted fromthepolestructureof the(generalized) correlator GZZ. Critical to this procedureis the identification of the gravitational co uplinghµν→ κµνhµν, which persuades us to adapt the gauge-invariant variable Z of Eq. (4) as follows: Z= 2qωκtzHtz+ω2κzzHZ+q2YκzzHX, (13) whereHZ≡Hzz−HX, the non-contributing mode Htthas been droppedand, as before, Y=gtt′/gxx′. Also, the spatial isotropy of the branehasenabledustomaketheconvenientsubstitution1 p−1/summationtextp−1 i=1κxixi→ κzz. The scaling properties of the damping constant can now be de- termined with a methodology akin to dimensional analysis: First, redefine the wavenumber and the frequency (and other paramet ers as necessary) with a scaling operation, second, re-express the solution in terms of these revised parameters and, third, interpret the modified pole structure. With regard to the first step, it is actually n ecessary to fixω, otherwise the incoming boundary condition at the horizon would be jeopardized. We are, however, free at this level of a nalysis 16to change the normalization of Z. On this basis, we arrive at Z= 2qωκtz κzzHtz+ω2HZ+q2YHX (14) or Z= 2/tildewideqωHtz+ω2HZ+/tildewideq2/tildewideYHX, (15) with /tildewideq≡qκtz κzz, /tildewideY≡Yκ2 zz κ2 tz. (16) By invoking Y→/tildewideY, we do not mean to suggest that this function actually gets rescaled. Rather, the presence of Yin Eq. (6) for Z represents a direct contribution from q2HX, which — after rescaling q2— picks up the extra factor κ2 zz/κ2 tz. Since the couplings can be regarded as constants, the soluti on in Eq. (5) is formally unchanged and need only be rewritten in te rms of the rescaled parameter. By this logic, the same can be said ab out the dispersion relation in Eq. (6), which takes on the modified fo rm ω2=/tildewideq21 p/tildewideY+iω/tildewideq2p−1 pfκ2 zz κ2 tz. (17) Takingu∗= 0andthencomparingdirectlytoEq.(3), wecanpromptly extract the damping constant for a generalized (but conform al) theory of gravity: Γ =κ2 zz κ2 tzp−1 p1 4πT(18) and, as advertised, the sound velocity is clearly unmodified . Let us briefly comment upon the significance of this result. It is commonplace, for a conformal theory, to relate the sound dam ping 17constant directly to the shear diffusion coefficient or (equiva lently) the shear viscosity to entropy ratio: Γ =p−1 pDand Γ =p−1 p1 Tη s respectively. This is all indisputably true for an Einstein theory of gravity; however, as we have now shown, these relations can n ot be taken verbatim for a generalized theory. To be clear, let us c ompare Eq. (18) to our prior results from [7] D=κ2 xz κ2 tz1 4πTand from [8] η/s= 1 4πκ2 rt κ2xz(wherexandzcould beany pair of orthogonal directions on the brane and note that, in general, κxz/negationslash=κzz). It should now be evident that both of the above relations for Γ will generally be modifi ed for an Einstein-corrected theory. The very same outcome as in Eq. (18) can be surmised from the z-component of the conservation equation for the dissipativ e stress tensor; with this being the equation that gives rise to the so und-mode dispersion relation (see, e.g., [13]). An inspection of this conservation equation ∂tTtz+∂zTzz= 0 and the steps leading up to the dispersion relation (3) is quite revealing. It is the tzcomponent of the stress ten- sorthataccounts fortheΩ2terminEq.(3), whereasthe zzcomponent gives riseto the Q2Ω term. Now, given a gravitational pedigreefor the hydrodynamic modes, it is natural to associate a coupling of κ2 µνwith theµνcomponent of the stress tensor. Hence, we anticipate that, f or a generalized gravity theory, the conservation equation sh ould really beκ2 tz∂tTtz+κ2 zz∂zTzz= 0. Similarly, we can expect the dispersion relation to take on the modified form Ω2=κ2 zz κ2 tz/bracketleftbig i2ΓΩQ2/bracketrightbig +.... (with the dots referring to the sound-velocity and higher-order t erms). Ab- sorbing this ratio of couplings into the damping constant, w e have precisely the same generalized form as obtained in Eq. (18). Naively, this latter argument would also suggest that c2 sscales in 18the same way as Γ, given that both are associated with the same zzcomponent of the stress tensor. However, this is not really c or- rect: The sound velocity is associated with the variation of the pres- sure; with the pressure having originated from the non-diss ipative background part of the stress tensor. Meanwhile, the other t erms in the dispersion relation are strictly associated with the flu ctuations or leading-order dissipative part. On this basis, we would not anticipate the sound velocity to be scaled for a generalized (conformal ) theory; again in compliance with the previous analysis. 4 Discussion: Some aspectsofthenon- conformal case To summarize, we have demonstrated two important outcomes f or the sound-mode conformal hydrodynamics of an AdS brane theory. First, we have confirmed, for Einstein’s theory, that the hydrodyna mics at the outer boundary is equivalent to that of any other radial s hell up to (and including at) the stretched horizon. Second, we have sh own — quite precisely — how the sound velocity and damping coefficie nt will be modified for a generalized (but conformal) theory of gravi ty. More specifically, c2 sis unaffected, whereas Γ is scaled by a particular ratio of (generalized) gravitational couplings. Further note th at, inasmuch as the couplings can be treated as constants, the former outc ome will carry through unfettered for any Einstein-corrected confo rmal theory. It is also of some interest to reflect upon how a non-conformal theory would impact upon our findings. Let us first consider th e is- 19sue of radial invariance for Einstein’s theory. Clearly, th is invariance for the correlator depended, in large part, on being able to d isregard the first term in Eq. (8). However, the introduction of a massi ve field into the spacetime (a prerequisite for breaking conformali ty) would be tantamount totheinclusionofachemical potential intothe thermody- namics. Such an inclusion would then negate our previous arg ument for the suppression of the scrutinized term; in particular, sT=ǫ+P could no longer be true. Hence, there could no longer be any re ason to expect thatδP δǫis a parametrically small quantity for a non-conformal theory — meaning that the radial scaling of the sound velocit y would certainly be more complicated. However, that this deviatio n from the conformal calculation is seemingly encapsulated in the sin gle varia- tionδP δǫgives one hope of being able to describe even the fully genera l situation by way of a radial “flow” equation. Although, it sho uld be kept in mind that a further breach of radial invariance is pos sible (if not probable) from additional terms that would (almost inev itably) appear in the O(q1) solution for Z. For the case of generalized gravity, the state of affairs can be come significantly more convoluted for a non-conformal theory. H ere, the first order of business is to re-incorporate the previously d isregarded ttmode — but then what? Well, at a first glance, the situation doe s not appear to look too bad. For the reason discussed at the end of the prior section, we would not expect the sound velocity to b e modi- fied irrespective of the generalized gravitational couplin gs. As for the damping coefficient, one can show that Httmakes no contribution to thisparticularterm, soitseemsreasonabletosuggestthat Γmaintains its modified form of Eq. (18). 20It is, however, a nearly certain likelihood that the situati on can not be as simple as so far discussed. For a non-conformal theo ry, there is an inevitable mixing between HXand the massive bulk fields, and it is not yet clear as to how this mixing might effect the scal ing relationsforeitherΓor c2 s(withbothofthesebeingdirectlyimplicated with the “polluted” HXmode). Certainly, a mode formed out of HX and some, for instance, massive scalar field, could no longer have an effective coupling as trivial as κzz. The main issue of non-conformal treatments is that, due to th e high degree of model dependence in the formalism, very littl e can be said in a generic sense. There has, however, been some recent progress in such a direction [18, 19]. These papers indicate that a bet ter start- ing point might be to look at certain classes of non-conforma l theories, as opposed to the “extreme limiting cases” of a specific model or com- pletely generality. Work along this line is only at a prelimi nary stage. Acknowledgments: The research of RB was supported by The Is- rael Science Foundation grant no 470/06. The research of AJM M is supported by the University of Seoul. References [1] P. Kovtun, D. T. Son and A. O. Starinets, “Holography and hydrodynamics: Diffusion on stretched horizons,” JHEP 0310, 064 (2003) [arXiv:hep-th/0309213]. 21[2] A. Buchel and J. T. Liu, “Universality of the shear vis- cosity in supergravity,” Phys. Rev. Lett. 93, 090602 (2004) [arXiv:hep-th/0311175]. [3] See, for a review and more references as well, D. T. Son and A.O.Starinets, “Viscosity, Black Holes, andQuantumField The- ory,” Ann. Rev. Nucl. Part. Sci. 57, 95 (2007) [arXiv:0704.0240 [hep-th]]. [4] A. O. Starinets, “Quasinormal spectrumandthe black hol e mem- brane paradigm,” arXiv:0806.3797 [hep-th]. [5] G. Policastro, D. T.SonandA. O.Starinets, “Theshearvi scosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, ” Phys. Rev. Lett. 87, 081601 (2001) [arXiv:hep-th/0104066]. [6] See, for recent reviews with many references, S. Mrowczynski and M. H. Thoma, “What do electromagnetic plasmas tell us about quark-gluon plasma?,” Ann. Rev. Nucl. Part. Sci. 57, 61 (2007) [arXiv:nucl-th/0701002]; E. Shuryak, “Physics of Strongly coupled Quark-Gluon Plasm a,” arXiv:0807.3033 [hep-ph]. [7] R. Brustein and A. J. M. Medved, “Theshear diffusion coeffici ent for generalized theories of gravity,” to appear in Phys. Let t. B [arXiv:0810.2193 [hep-th]]. [8] R. Brustein and A. J. M. Medved, “The ratio of shear viscos ity to entropy density in generalized theories of gravity,” to a ppear in Phys. Rev. D [arXiv:0808.3498 [hep-th]]. 22[9] R. Brustein, D. Gorbonos and M. Hadad, “Wald’s entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling,” arXiv:0712.3206 [hep-th]. [10] G. Policastro, D. T. Son and A. O. Starinets, “From AdS/C FT correspondence to hydrodynamics,” JHEP 0209, 043 (2002) [arXiv:hep-th/0205052]. [11] N. Iqbal and H. Liu, “Universality of the hydrodynamic l imit in AdS/CFT and the membrane paradigm,” arXiv:0809.3808 [hep- th]. [12] G. Policastro, D. T. Son and A. O. Starinets, “From AdS/C FT correspondence to hydrodynamics. II: Sound waves,” JHEP 0212, 054 (2002) [arXiv:hep-th/0210220]. [13] C. P. Herzog, “The sound of M-theory,” Phys. Rev. D 68, 024013 (2003) [arXiv:hep-th/0302086]. [14] D. T. Son and A. O. Starinets, “Minkowski-space correla tors in AdS/CFT correspondence: Recipe and applications,” JHEP 0209, 042 (2002) [arXiv:hep-th/0205051]. [15] P. K. Kovtun and A. O. Starinets, “Quasinormal modes and holography,” Phys. Rev. D 72, 086009 (2005) [arXiv:hep-th/0506184]. [16] J. Mas and J. Tarrio, “Hydrodynamics from the Dp-brane, ” JHEP0705, 036 (2007) [arXiv:hep-th/0703093]. [17] M. Fujita, “Non-equilibrium thermodynamics near the h orizon and holography,” JHEP 0810, 031 (2008) [arXiv:0712.2289 [hep- th]]. 23[18] S. S. Gubser, S. S. Pufu and F. D. Rocha, “Bulk viscosity o f strongly coupled plasmas with holographic duals,” JHEP 0808, 085 (2008) [arXiv:0806.0407 [hep-th]]. [19] T. Springer, “Sound Mode Hydrodynamics from Bulk Scala r Fields,” arXiv:0810.4354 [hep-th]. 24

2009-01-15

The near-horizon metric for a black brane in Anti-de Sitter (AdS) space and the metric near the AdS boundary both exhibit hydrodynamic behavior. We demonstrate the equivalence of this pair of hydrodynamic systems for the sound mode of a conformal theory. This is first established for Einstein's gravity, but we then show how the sound damping constant will be modified, from its Einstein form, for a generalized theory. The modified damping constant is expressible as the ratio of a pair of gravitational couplings that are indicative of the sound-channel class of gravitons. This ratio of couplings differs from both that of the shear diffusion coefficient and the shear viscosity to entropy ratio. Our analysis is mostly limited to conformal theories but suggestions are made as to how this restriction might eventually be lifted.

The sound damping constant for generalized theories of gravity

0901.2191v1

arXiv:0904.0813v2 [cs.IT] 8 Apr 2009Projective Space Codes for the Injection Metric Azadeh Khaleghi, Frank R. Kschischang Department of Electrical and Computer Engineering, Univer sity of Toronto, Canada Email:{azalea,frank }@comm.utoronto.ca Abstract—In the context of error control in random linear network coding, it is useful to construct codes that compris e well-separated collections of subspaces of a vector space o ver a finite field. In this paper, the metric used is the so-called “injection distance,” introduced by Silva and Kschischang . A Gilbert-Varshamov bound for such codes is derived. Using th e code-construction framework of Etzion and Silberstein, ne w non- constant-dimension codes are constructed; these codes con tain more codewords than comparable codes designed for the sub- space metric. I. INTRODUCTION The problem of error-correction in random network coding has recently become an active area of research [1], [2], [3], [4], [5], [6]. The main motivation for this problem is the phenomenon of error-propagation in the network. Since the received packets are random linear combinations of packets inserted at intermediate nodes, the system is very sensitiv e to transmission errors. Due to the vector-space preserving nature of random linear network coding, it has been shown that codes constructed in the projective space are suitable for error-correction for network coding. Our focus in this paper is on construction of codes in the projective space for adversarial error-correction in r andom networkcoding.Asshownin[7]asuitablemeasureofdistanc e between subspaces for an adversarial error-control model i s given by dI(U,V) = max{dimU,dimV}−dim(U∩V), a measure known as the “ injection metric ”. This choice of dis- tance metric is the main parameter that distinguishes our wo rk from the existing literature on (subspace) codes construct ed for random linear network coding. All existing bounds and constructions are based on a metric known as the subspace distancedSoriginally introduced by K¨ otter and Kschischang in [5]. In the special case where codes are contained in the Grassmannian, codes designed for dIcoincide with those designed for dS. However, as shown in [7], in general non- constant dimension codes designed for dImay have higher rates than those designed for dS. In this paper we present a construction of a class of codes in the projective space for the injection distance. Th is constructionismotivatedbytheworkofEtzionandSilberst ein in [8], with the main difference that the construction in [8] is based on dS. In Section II, we present a brief overview of the projective space and the Grassmannian,as well as rank-metriccodes. We also briefly review Etzion and Silberstein’s “Ferrers diagr am lifted rank-metric codes”, as our work in Section V is relate dto this construction. In Section III, we present a Gilbert- Varshamov-type bound on the size of codes of a certain minimum injection distance in the projective space. As we are precluded by space in this paper, we present this theorem without proof. In Section IV we present a construction for th e Ferrers diagram rank-metric codes as subcodes of linear MRD codes. In Sections V and VI, we provide an algorithm for the construction of a class of non-constant-dimensioncodes in the projective space designed for dI. Finally, in Section VII we present our numerical results. As shown in this section our construction results in codes of slightly higher rates than the codes of [8]. II. PRELIMINARIES A. Notation Letq≥2be a power of a prime. In this paper, all vectorsand matrices are defined over the finite field Fq, unless otherwisementioned.We denoteby Fm×n q, the set of all m×n matrices over Fq. Ifvis a vector then the ithentry ofvis denoted by vi. We denote the logical complement of a binary vectorv= (v1,v2,···,vn)by¯v= (¯v1,¯v2,···,¯vn). The number of non-zero elements of vis denoted by wt(v). We define the support set of a vector v, denoted supp(v)to be the set of indices corresponding to the non-zero entries of v. Let xandybe two binary vectors of the same length. We denote the number of 1→0transitions from xtoybyN(x,y), their Hammingdistanceby dH(x,y)andthelogicalANDoperation betweenxandyby∧. IfXis a matrix then the rank of Xis denoted by rankXand its row space is denoted by /an}bracketle{tX/an}bracketri}ht. Let n >0be an integer. We denote by [n]the set of all positive integers less than or equal to n, i.e.[n] ={0,1,2,···,n}. B. Rank-Metric Codes LetXandYbe two matrices in Fm×n q. Therank dis- tancebetween XandY, denoted dR(X,Y)is defined as dR(X,Y)/definesrank(Y−X). As shown in [9] the rank distance is indeed a metric. Let Fqbe a base field and Fqm withm≥1be an extension of Fq. The rank of a vector v= (v1,v2,···,vn)∈(Fqm)nis the rank of the m×n matrix obtained by expanding each entry of vto anm×1 column vector over Fq. A code CRis a rank-metric code overFqmof minimum distance d, ifCR⊆(Fqm)nand for allX, Y∈CRdR(X,Y)≥d. As shown in [9] CRmust satisfylogq|CR| ≤max{m,n}(min{m,n}−d+1),and rank metric codes achieving this bound with equality are said to be Maximum Rank Distance (MRD) codes. Gabidulin codes, presented by Gabidulin [9] are an extensive class of MRD codes,whichare the analogsof the generalizedReed-Solomo ncodes designed for the rank metric. Efficient polynomial-ti me decoding algorithms exist that correct errors of rank up to/floorleftbiggd−1 2/floorrightbigg . See for example [10], [11], [12]. C. Projective Space LetVbe ann-dimensional vector space over the finite fieldFqof order q. For a non-negative integer k≤n denote by G(n,k)the set of all k-dimensional subspaces ofV. This set is known as a Grassmannian and its cardi- nality is given by the q-aryGaussian coefficient defined as/bracketleftbiggn k/bracketrightbigg q/defines(qn−1)(qn−1−1)···(qn−k+1−1) (qk−1)(qk−1−1)···(q−1). The set of all subspacesof Vformaprojectivespace Pn qofordernoverFq. ThusPn qcan be viewed as a union of the Grassmannians for allk≤n, i.e.Pn q=n/uniondisplay k=0G(n,k). A code Cis an(n,M,d)dS code inPn qif|C|=Mand for all U, V∈ C, dS(U,V)≥d. Similarly, a code C ⊆ Pn qis an(n,M,d)dIcode if|C|=M and for all U, V∈ C, dI(U,V)≥d. A code Cis an (n,M,d,k )constant-dimension code if C ⊆ G(n,k)for some k∈[n]. Since in this case dIanddSare equal up to scale, there is no need to distinguish between (n,M,d,k )dIand (n,M,d,k )dScodes. D. Ferrers Diagram Lifted Rank-Metric Codes In this section we review the code construction of [8] with a slightly different notation. The key idea in this construc tion is the observation that every k-dimensional vector space V inPn qarises uniquely as the row space of a k×nmatrix in Reduced Row Echelon Form (RREF). Let Vbe a vector space inPn qand letE(V)be its corresponding generator matrix in RREF. We define the profile vector ofVdenotedp(V), to be a binary vector of length nwhose non-zero elements appear onlyin positions where E(V)has a leading 1. Consider an equivalence relation ∼onPn qwhere, ∀V1, V2∈ Pn q, V1∼V2↔p(V1) =p(V2).(1) This relation partitions Pn qinto equivalence classes, where V1andV2belong to the same class provided that they are identified by the same profile vector. Let Γdenote the set of all equivalence classes generated in Pn qaccording to (1). Consider an equivalence class γ∈Γwith a profile vector v of length nand weight k. We define the profile matrix PM(v) to be ak×nmatrix in RREF where the leading coefficients of its rows appear in columns indexed by supp(v), and has •’s in all its entries which are not required to be terminal zeros or leading ones. For example if p= (0,1,0,1,1,0,0) thenPM(v) = 0 1•0 0• • 0 0 0 1 0 • • 0 0 0 0 1 • • . Notice that the generator matrices in RREF of the elements of γdiffer only in entries of PM(v)marked as •’s. Letηdenote the number of columns of PM(v)which contain at least a single •. Let S(v)be thek×ηsub-matrix of PM(v)composed of all such columnsof PM(v).Acodeisan [S,κ,δ]Ferrersdiagramrank- metric code if it forms a rank-metric code with dimension κand minimum rank-distance δ, all of whose codewords are m×ηmatrices with zeros in all their entries where S(v)has zeros. In the construction presented in [8], a set Ω⊆Γis constructed in such a way that for all γ1, γ2∈Ωwith γ1/ne}ationslash=γ2and for all V1∈γ1,V2∈γ2, dS(V1,V2)≥d. By Lemma 2 in [8], this is possible by selecting the profile vectorsoftheequivalenceclassesaccordingtoabinarycod eof minimumHammingdistance d. Then within each class γ∈Ω, a Ferrers diagram rank-metric code is used to ensure that for allV1, V2∈γ,dS(V1,V2)≥d. Finally C={V∈γ|γ∈Ω}. III. A G ILBERT-VARSHAMOV -TYPEBOUND ON THE SIZE OFCODES IN THE PROJECTIVE SPACE LetVbe ak-dimensional vector space in Pn q. We define St(V)to be the set of all vector spaces in Pn qat an injection distance at most tfromV. i.e. St(V) ={W∈ Pn|dI(V,W)≤t} We may view St(V)as a hypothetical sphereof radius t centered at V. In Theorem 1 we give the cardinality of St(V)centered at some k-dimensional vector space with k≤n. Since the projective space is non-homogeneous, the size ofSt(V)does not depend merely on its radius, but also on the dimension of its center. In other words for two vector spaces V1andV2withdimV1/ne}ationslash= dimV2, we have |St(V1)| /ne}ationslash=|St(V2)|. Theorem 1. LetVbe ak-dimensional vector space in Pn q, withk≤n, and let N(k,t)denote the cardinality of St(V). Then, N(k,t) =t/summationdisplay r=0qr2/bracketleftbiggk r/bracketrightbigg q/bracketleftbiggn−k r/bracketrightbigg q+ r/summationdisplay j=1qr(r−j)/parenleftBigg/bracketleftbiggk r/bracketrightbigg q/bracketleftbiggn−k r−j/bracketrightbigg q+/bracketleftbiggn−k r/bracketrightbigg q/bracketleftbiggk r−j/bracketrightbigg q/parenrightBigg Using Theorem 1, and following an approach similar to that of Etzion and Vardy in [13], we obtain the following generalized Gilbert-Varshamov-type bound on the size of codes in the projective space. Theorem 2. LetAq(n,d)denote the maximum number of codewords in an (n,M,d)code inPn q. Then, Aq(n,d)≥/vextendsingle/vextendsinglePn q/vextendsingle/vextendsingle2 n/summationdisplay k=0/bracketleftbiggn k/bracketrightbigg N(k,d−1) IV. FERRERSDIAGRAM RANK-METRICCODE CONSTRUCTION Letvbe a binary vector of length nand weight m. LetCF be an[S(v),κ,δ]Ferrers diagram rank-metric code that fits S(v), i.e. every codeword in CFhas zeros in all its entries whereS(v)has zeros. We may view CFas a subcode of a linearrank-metriccode Cofminimumrank-distance dR(C)≥ δ, with a further set of linear constraints ensuring that CFfits S(v).In Theorem 3 we provide a lower bound on the dimension κof the largest [S(v),κ,δ]Ferrers diagram rank-metric code obtained as a subcode of a linear MRD code. Theorem 3. Letvbe a binary vector of weight mand let S(v)be them×ηsub-matrix of PM(v)composed of all the columns of PM(v)that contain at least a single •. Assume thatS(v)contains a total of w•’s. Consider the dimension κ of the largest [S,κ,δ]Ferrer’s diagram rank-metric code CF. We have, κ≥w−max{m,η}(δ−1). Proof:LetV=Fm×η q. Note that Fm×η qis anmη- dimensionalvectorspaceover Fq.LetCbealinearMRDcode withdR(C)≥δ. This code is a k-dimensional subspace of Fm×η qwithk= max{m,η}(min{m,η}−δ+1). There exists a linear transformation Φ :V−→V/CwithkerΦ =C, and by the First Isomorphism Theorem dimV/C=mη−k. Let A={(i,j)|S(v)ij= 0}bethesetof (i,j)indiceswhere S(v) has zeros, and note that |A|=mη−w. Letf:V−→Fmη−w q such that f(x) = (xij),(i,j)∈A. Now any subcode C′ofC satisfying f(c) = 0∀c∈C′is an[S(v),κ,δ]Ferrers diagram rank-metric code. Let CFbethe largest such subcode of C. Define a linear transformation Φ′:V−→V/C×Fmη−w q, by whichx/mapsto→(Φ(x),f(x)). Now by construction kerΦ′=CF. Noting that Φ′(V)⊆V/C×Fmη−w qwe have dimΦ′(V)≤ 2mη−k−w, and by the rank-nullity theorem we obtain dimCF≥w+k−mη=w−max{m,η}(δ−1), and the theorem follows. As an example, given a profile vector vof length n, with wt(v) =mwe may construct an [S(v),κ,d]code by taking a Gabidulin code over Fη qmwithdR≥d, expand the elements of its parity-check matrix Hover the base field Fq, and add appropriate parity-check equations to HinFqto ensure that the resulting code fits S(v). V. FERRERSDIAGRAM LIFTEDRANK-METRICCODES FOR THE INJECTION METRIC Inspiredbytheconstructionof[8],inthissectionweprese nt a scheme for constructing (n,M,d)dIFerrers diagram lifted rank-metric codes in Pn q. The following theorem is key in our construction. Theorem 4. LetUandVbe two vector spaces in Pn q, with profile vectors u, andvrespectively. Then we have, dI(U,V)≥max{N(u,v),N(v,u)}. Proof:First note that the dimension of a vector space is equal to the Hamming weight of its profile vector, i.e. dimU=wt(u)anddimV=wt(v). Now let w=u∧v and observe that dimU∩V≤wt(w). Therefore we have dimU−dim(U∩V)≥wt(u)−wt(w). Similarly, dimV− dim(U∩V)≥wt(v)−wt(w). Taking the maxof both equa- tions we obtain, dI(U,V)≥max{wt(u),wt(v)} −wt(w) = max{N(u,v),N(v,u)}. For two binary vectors xandy, the quantity max{N(x,y),N(y,x)}is a metric, known as the asymmetric distance between xandy. The asymmetric distance was first introduced by Varshamov in [14] for construction of codes for the Z channel. Constructions exist mainly for single-asymmetric error-correcting codes, and some multi -error correcting codes ([15] and references therein). Plea se refer to [16] for a more recent work on general t-asymmetric error-correcting codes. By Theorem 4 two spaces are guaranteed to have an injection distance of at least d, provided that the asymmetric distance between their profile vectors is at least d. Thus to construct a code in Pn qwith minimum injection distance d, we may select a set of subspaces according to an asymmetric code in the Hamming space with minimum asymmetric dis- tancedand follow a procedure similar to that presented in [8]. Construction of our (n,M,d)dIcode can be described algorithmically as follows: 1) Take a binary asymmetric code Aof length nand minimum asymmetric distance d. 2) For each codeword c∈ A, obtainS(c), (composed of the columns of PM(c)with at least one •). 3) Given each k×ηmatrixS(c), use the construction of SectionIVtoobtainan [S(c),κ,d]Ferrersdiagramrank- metric code. 4) Lift each matrix S(c)to its correspondingprofile matrix PM(c), to obtain a generator matrix Gc. 5) Finally C={V∈ Pn q|V=/an}bracketle{tGc/an}bracketri}ht}. Note that a slight modification to Step 1 and Step 3 in the above procedure allows for the construction of an (n,M,d)dS code in the projective space. More specifically, in order to construct an (n,M,2δ)dScode inPn q, we may first take a binarycode Hofminimum Hamming distancedH≥2δ.Then for each codeword c∈ Hwe may construct an [S(c),κ,δ] Ferrers diagram rank-metric code. Following the rest of the steps are described above, we obtain an (n,M,2δ)dScode. VI. PROFILEVECTORSELECTION As suggested by Theorem 3, the dimension of an [S,κ,δ] Ferrers diagram rank-metric code depends not only on the desired minimum distance δ, but also on the number of •’s in S. Since the number of •’s inSis directly related to its corre- sponding profile vector, the choice of the asymmetric code in the first step is crucial to the size of our codes. For instance , the vector v= (1,1,0,0,0)results in a profile matrix with a higher number of •’s than that of v= (1,1,0,1,1). Thus a code of lower rate that contains vectors which potentially result in larger number of •’s in their corresponding profile matrices may be preferable over one with a higher rate, that involves vectors resulting in smaller number of •’s. With this observation, given a minimum asymmetric dis- tancedwe define a scoring function score(v,d)on the set of all binary vectors, which calculates for every v∈ {0,1}n, the lower bound κof the dimension of the largest [S(v),κ,d] Ferrers diagram rank-metric code induced by v. It is easy to observe that score(v,d) =n/summationdisplay i=1i/summationdisplay j=1¯vivj−max{wt(v),η(v)}(d−1) whereη(v) =n−(wt(v)+ min t∈supp(v)t)+1 Now in order to select a set Pof profile vectors at aminimum asymmetric distance d, we use a standard greedy algorithm that maintains a list of available profile vectors A⊆ {0,1}n, (withAinitialized to {0,1}n). At each step an availableprofilevector v∈Awiththelargestscore score(v,d) is added to P, and vectors within asymmetric distance dof vare made unavailable. The algorithm proceeds until A=∅. By a slight modification to this algorithm we may allow for the same greedy selection of a set of profile vectors at a certain minimum Hamming distance as opposedto a minimum asymmetric distance. VII. N UMERICAL RESULTS As constant-dimensioncodes designed for dScoincide with those designedfor dI, we are interested in the analysis of non- constant dimension (n,M,d)dIcodes. A. Our(n,M,d)dIvs.(n,M,d)dSCodes For our(n,M,d)dIcodes we first used the selection algo- rithm presented in Section VI to obtain a set of binary profile vectors at a minimum asymmetric distance da≥d. Using this procedure along with the bound of Theorem 3 we obtained |(n,M,d)dI|. As discussed previously, for every (n,M,d)dI code constructed according to the procedure described in Section V, we may construct an (n,M,2d)dScounterpart through a similar procedure. In order to select a set of profil e vectors for our (n,M,2d)dScodes we used the algorithm of Section VI for dH≥2d. As shown in Table I our (n,M,d)dI codes denoted by C2have a slightly higher rate than their (n,M,2d)dScounterparts, C1. B. Our(n,M,d)dICodes vs. Codes of [8] The best(n,M,d,k )dSconstant-dimensioncodes of [8] are obtainedby using constant-weightlexicodesas profile vect ors. These codes achieve maximum cardinality when k=/floorleftBign 2/floorrightBig . The column corresponding to C3in Table I shows rates of the (n,M,d S,/floorleftbign 2/floorrightbig )dSconstant-dimension codes of [8]. Non-constant-dimension (n,M,d)dScodes of [8] are con- structed by means of a puncturing operation performed on constant-dimension codes. As shown in [8] puncturing an (n,M,d,k )dScode results in an (n−1,M′,d−1)dScode withM′≥M(qn−k+qk−2) qn−1. In Table I, logq|C4|denotes the guaranteed minimum rate of (n,M′,dS)punctured codes obtained from the best (n+ 1,M,dS+ 1,/floorleftbign+1 2/floorrightbig )dScodes of [8]. As shown in the table, our (n,M,d)dIcodes have a slightly higher rate than both constant and non-constant- dimension codes of [8]. VIII. C ONCLUSION We presented a construction for the Ferrers diagram rank- metric codes as subcodes of linear MRD codes, and provided a lower bound on the dimension of the largest such codes. Using a greedy profile vector selection algorithm along with our construction of Ferrers diagram rank-metric codes we presented a class of non-constant dimension lifted Ferrers diagram rank-metric codes for the injection distance. We al so presented a similar construction for non-constant dimensi onTABLE I PARAMETERS OF CODES CONSTRUCTED WITH C1=OUR(n,M,d S)dS CODES,C2=OUR(n,M,d I)dICODES,C3= (n,M,d S,n/2)dSCODES OF[8],C4=PUNCTURED CODES OF [8] q dIdSnlogq|C1|logq|C2|logq|C3|logq|C4| 2 2 4 9 15 .1732 15 .6245 15 .1731 10 .9588 2 2 4 10 20 .1551 20 .3294 20 .1548 13 .5585 2 2 4 12 30 .1561 30 .3346 30 .1559 13 .7676 2 3 6 10 15 .0031 15 .0071 15 .0032 7 .5581 2 3 6 13 28 .0032 28 .0263 28 .0032 21 .9888 3 2 4 7 8 .0177 8 .1331 8 .0170 4 .6210 3 2 4 8 12 .0138 12 .0311 12 .0138 6 .2567 4 2 4 7 8 .0039 8 .0522 8 .0038 4 .4974 4 2 4 8 12 .0031 12 .0068 12 .0031 6 .1599 codes designed for the subspace distance. We observed that our non-constant dimension codes designed for the injectio n distance have a slightly higher rate than their counterpart s designed for the subspace distance. Moreover, comparing our codes designed for the injection distance, with the best subspace codes of [8], we observed a minor improvement in rate. The Ferrers diagram lifted rank-metric codes introdu ced by [8], as well as those presented in our paper achieve higher ratesthantheoriginalliftedrank-metriccodesof[5].How ever, we believe that these rate improvements are minute from a practical perspective. REFERENCES [1] T. Etzion and A. Vardy, “Error-correcting codes in proje ctive space,” IEEE Intern. Symp. on Inform. Theory , pp. 871–875, July 2008. [2] E. Gabidulin and M. Bossert, “Codes for network coding,” IEEE Intern. Symp. on Inform. Theory , pp. 867–870, July 2008. [3] F. Manganiello, E. Gorla, and J. Rosenthal, “Spread code s and spread decoding in network coding,” IEEE Intern. Symp. on Inform. Theory , pp. 881–885, July 2008. [4] A.Kohnertand S.Kurz,“Construction oflargeconstant d imension codes with a prescribed minimum distance,” in MMICS, 2008, pp. 31–42. [5] R. K¨ otter and F. R. Kschischang, “Coding for errors and e rasures in random network coding,” IEEE Trans. on Inform. Theory , vol. 54, no. 8, pp. 3579–3591, Aug. 2008. [6] D. Silva, F. R. Kschischang, and R. K¨ otter, “A rank-metr ic approach to error correction in random network coding,” IEEE Trans. on Inform. Theory, vol. 54, pp. 3951–3967, Sep. 2008. [7] D. Silva and F. R. Kschischang, “On metrics for error corr ection in network coding,” 2008, submitted for publication. [Online ]. Available: http://arxiv.org/abs/0805.3824 [8] Tuvi Etzion and N. Silberstein, “Error-correcting code s in projective spaces via rank-metric codes and Ferrers diagrams,” 2009. [ Online]. Available: http://arxiv.org/abs/0807.4846v4 [9] E.M.Gabidulin, “Theory of codes with maximum rank dista nce,”Probl. Inform. Transm , vol. 21, no. 1, pp. 1–12, 1985. [10] R. Roth, “Maximum-rank array codes and their applicati on to crisscross error correction,” IEEE Trans. on Inform. Theory , vol. 37, no. 2, pp. 328–336, Mar 1991. [11] G. Richter and S. Plass, “Fast decoding of rank-codes wi th rank errors and column erasures,” IEEE Intern. Symp. on Inform. Theory , 2004. [12] P.Loidreau, “A Welch-Berlekamp like algorithm for dec oding Gabidulin codes,” in WCC, 2005, pp. 36–45. [13] L. Tolhuizen, “The generalized Gilbert-Varshamov bou nd is implied by Turan’s theorem [code construction],” IEEE Trans. on Inform. Theory , vol. 43, no. 5, pp. 1605–1606, Sep 1997. [14] R. Varshamov, “A class of codes for asymmetric channels and a problem from the additive theory of numbers,” IEEE Trans. on Inform. Theory , vol. 19, no. 1, pp. 92–95, Jan 1973. [15] T. Kløve, “Error correction codes for the asymmetric ch annel,” Math. Inst. Univ. Bergen, Tech. Rep., 1981. [16] V. P. Shilo, “New lower bounds of the size of error-corre cting codes for the Z-channel,” Cybernetics and Sys. Anal. , vol. 38, pp. 13–16, 2002.

2009-04-05

In the context of error control in random linear network coding, it is useful to construct codes that comprise well-separated collections of subspaces of a vector space over a finite field. In this paper, the metric used is the so-called "injection distance", introduced by Silva and Kschischang. A Gilbert-Varshamov bound for such codes is derived. Using the code-construction framework of Etzion and Silberstein, new non-constant-dimension codes are constructed; these codes contain more codewords than comparable codes designed for the subspace metric.

Projective Space Codes for the Injection Metric

0904.0813v2

arXiv:0905.3242v1 [math.SP] 20 May 2009EIGENVALUE ASYMPTOTICS, INVERSE PROBLEMS AND A TRACE FORMULA FOR THE LINEAR DAMPED WAVE EQUATION DENIS BORISOV AND PEDRO FREITAS Abstract. We determine the general form of the asymptotics for Dirichlet eigenvalues of the one–dimensional linear damped wave operator. Asaconsequence,weobtainthatgivenaspectrumcor re- sponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem. 1.Introduction Consider the one–dimensional linear damped wave equation on the interval (0 ,1), that is, (1.1) wtt+2a(x)wt=wxx+b(x)w, x ∈(0,1), t >0 w(0,t) =w(1,t) = 0, t > 0 w(x,0) =w0(x), wt(x,0) =w1(x), x∈(0,1) The eigenvalue problem associated with (1.1) is given by uxx−(λ2+2λa−b)u= 0, x∈(0,1), (1.2) u(0) =u(1) = 0, (1.3) and has received quite a lot of attention in the literature since the pa - pers of Chen et al. [CFNS] and Cox and Zuazua [CZ]. In the first of these papers the authors derived formally an expression for the a symp- totic behaviour of the eigenvalues of (1.2), (1.3) in the case of a zer o potential b, whichwaslaterprovedrigorouslyinthesecondoftheabove Date: November 17, 2018. 2000Mathematics Subject Classification. Primary 35P15; Secondary 35J05. D.B. was partially supported by RFBR (07-01-00037) and gratefully acknowl- edges the support from Deligne 2004 Balzan prize in mathematics. D.B . is also supported by the grant of the President of Russia for young s cientist and their supervisors (MK-964.2008.1) and by the grant of the Preside nt of Russia for leading scientific schools (NSh-2215.2008.1) P.F. was partially sup ported by FCT/POCTI/FEDER. . 12 DENIS BORISOV AND PEDRO FREITAS papers. Following this, there were several papers on the subject which, among other things, extended the results to non–vanishing b[BR], and showed that it is possible to design damping terms which make the spectral abscissa as large as desired [CC]. In [F2] the second autho r of the present paper addressed the inverse problem in arbitrary dime n- sion giving necessary conditions for a sequence to be the spectrum of an operator of this type in the weakly damped case. As far as we are aware, these are the only results for the inverse problem asso ciated with (1.2), (1.3). Other results for the n−dimensional problem include, for instance, the fact that in that case the decay rate is no longer de- termined solely by the spectrum [L], a study of some particular case s where the role of geometric optics is considered [AL], the asymptotic behaviour of the spectrum [S] and the study of sign–changing damp ing terms [F1]. The purpose of the present paper is twofold. On the one hand, we show that problem (1.2), (1.3) may be addressed in the same way as the classical Sturm–Liouville problem in the sense that, although this is not a self–adjoinf problem, the methods used for the former problemmaybeappliedherewithsimilarresults. Thisideawasalready present in both [CFNS] and [CZ]. Here we take further advantage of this fact to obtain the full asymptotic expansion for the eigenvalue s of (1.2), (1.3) (Theorem 1). Based on these similarities, we were also led to a (regularized) trace formula in the spirit of that for the Stur m– Liouville problem (Theorem 4). On the other hand, the idea behind obtaining further terms in the asyptotics was to use this information to address the associated in verse spectral problem of finding all damping terms that give a certain spe c- trum. Our main result along these lines is to show that in the case of constant damping there is no other smooth damping term yielding the same spectrum (Corollary 2). Namely, we obtain the criterion for th e damping term to be constant. Note that this is in contrast with the inverse (Dirichlet) Sturm–Liouville problem, where for each admissible spectrum there will exist a continuum of potentials giving the same spectrum [PT]. In particular this result shows that we should expect the inverse problem to be much more rigid in the case of the wave equation than it is for the Sturm–Liouville problem. This should be understood in the sense that, at least in the case of constant dam ping, it will not be possible to perturb the damping term without disturbing the spectrum, as is the case for the potential in the Sturm–Liouville problem. The plan of the paper is as follows. In the next section we set the notation and state the main results of the paper. The proof of theEIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 3 asymptotics of the eigenvalues is done in Sections 3 and 4, where in the first of these we derive the form of the fundamental solutions of equation (1.2), while in the second we apply a shooting method to these solutions to obtain the formula for the eigenvalues as zeros o f an entire function – the idea is the same as that used in [CZ]. Finally, in Section 5 we prove the trace formula. 2.Notation and results It is easy to check that if λis an eigenvalue of the problem (1.2), (1.3), then λis also an eigenvalue of the same problem. In view of this property, we denote the eigenvalues of this problem by λn,n/ne}ationslash= 0, and order them as follows .../lessorequalslantImλ−2/lessorequalslantImλ−1/lessorequalslantImλ1/lessorequalslantImλ2/lessorequalslant... while assuming that λ−n=λn. We also suppose that possible zero eigenvalues are λ±1=λ±2=...=λ±p= 0. Ifp= 0, the problem (1.2), (1.3) has no zero eigenvalues. For any function f=f(x) we denote/an}bracketle{tf/an}bracketri}ht:=/integraltext1 0f(x)dx. Theorem 1. Suppose a∈Cm+1[0,1],b∈Cm[0,1],m/greaterorequalslant1. The eigenvalues of (1.2), (1.3) have the following asymptotic b ehaviour as n→ ±∞: λn=πni+m−1/summationdisplay j=0cjn−j+O(n−m), (2.1) were the cj’s are numbers which can be determined explicitly. In par- ticular, c0=−/an}bracketle{ta/an}bracketri}ht, c1=/an}bracketle{ta2+b/an}bracketri}ht 2πi, (2.2) c2=1 2π2/bracketleftbigg /an}bracketle{ta(a2+b)/an}bracketri}ht−/an}bracketle{ta/an}bracketri}ht/an}bracketle{ta2+b/an}bracketri}ht+a′(1)−a′(0) 2/bracketrightbigg . (2.3) A straightforward consequence of the fact that the spectrum d eter- mines the average as well as the L2norm of the damping term (as- sumingbfixed) is that the spectrum corresponding to the constant damping determines this damping uniquely. Corollary 2. Assume that a∈C3[0,1],λnare the eigenvalues of the problem (1.2), (1.3), the function b∈C2[0,1]is fixed, and the formula (2.1) gives the asymptotics for these eigenvalues. Then the function a(x)is constant, if and only if c2 0= 2πic1−/an}bracketle{tb/an}bracketri}ht,4 DENIS BORISOV AND PEDRO FREITAS in which case a(x)≡ −c0. In the same way, the asymptotic expansion allows us to derive other spectral invariants in terms of the damping term a. However, these do not have such a simple interpretation as in the case of the above constant damping result. Corollary 3. Suppose b≡0,ai(x) =a0(x) +/tildewideai(x),i= 1,2, where a0(1−x) =a0(x),/tildewideai(1−x) =−/tildewideai(x),/tildewideai,a0∈C4[0,1], and for a=ai the problems (1.2), (1.3) have the same spectra. Then /an}bracketle{t/tildewidea2 1/an}bracketri}ht=/an}bracketle{t/tildewidea2 2/an}bracketri}ht,/an}bracketle{ta0/tildewidea2 1/an}bracketri}ht=/an}bracketle{ta0/tildewidea2 2/an}bracketri}ht is valid. From Theorem 1 we have that the quantity Re( λn−c0) behaves as O(n−2) asn→ ∞. This means that the series ∞/summationdisplay n=−∞ n/negationslash=0(λn−c0) = 2∞/summationdisplay n=1Re(λn−c0) converges. In the following theorem we express the sum of this ser ies in terms of the function a. This is in fact the formula for the regularized trace. Theorem 4. Leta∈C3[0,1],b∈C2[0,1]. Then the identity ∞/summationdisplay n=−∞ n/negationslash=0(λn−c0) =a(0)+a(1) 2−/an}bracketle{ta/an}bracketri}ht holds. 3.Asymptotics for the fundamental system Inthissectionweobtaintheasymptoticexpansionforthefundame n- tal system of the solutions of the equation (1.2) as λ→ ∞,λ∈C. This is done by means of the standard technique described in, for instan ce, [E, Ch. IV, Sec. 4.2, 4.3], [Fe, Ch. II, Sec. 3]. We begin with the formal construction assuming the asymptotics to be of the form (3.1) u±(x,λ) = e±λx±xR 0φ±(t,λ)dt , where (3.2) φ±(x,λ) =m/summationdisplay i=0φ(±) i(x)λ−i+O(λ−m−1), m/greaterorequalslant1.EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 5 In what follows we assume that a∈Cm+1[0,1],b∈Cm[0,1]. We substitute the series (3.1), (3.2) into (1.2) and equate the coef - ficients of the same powers of λ. It leads us to a recurrent system of equations determining φ(±) iwhich read as follows: φ(±) 0=a, (3.3) φ(±) 1=−1 2(±a′+a2+b), (3.4) φ(±) i=−1 2/parenleftBigg ±φ(±) i−1′+i−1/summationdisplay j=0φ(±) jφ(±) i−j−1/parenrightBigg , i/greaterorequalslant2. (3.5) The main aim of this section is to prove that there exist solutions to (1.2) having the asymptotics (3.1), (3.2). In other words, we are g oing to justify these asymptotics rigorously. We will do this for u+, the case ofu−following along similar lines. Let us write Um(x,λ) = eλx+mP i=0λ−ixR 0φ(+) i(t)dt . In view of the assumed smoothness for aandbwe conclude that Um∈ C2[0,1]. It is also easy to check that (3.6)U′′ m−λ2Um−2λaUm+bUm=λ−meλxfm(x,λ), x∈[0,1], Um(0) = 1, U′ m(0) =λ+m/summationdisplay i=0φ(+) i(0)λ−i, where the function fmsatisfies the estimate |fm(x,λ)|/lessorequalslantCm uniformly for large λandx∈[0,1] We consider first the case Re λ/greaterorequalslant0. Differentiating the function u+ formally we see that u′ +(0,λ) =λ+φ+(0,λ) =λ+m/summationdisplay i=0φ(+) i(0)λ−i+O(λ−m−1). Let A0(λ) =λ+m/summationdisplay i=0φ(+) i(0)λ−i, andu+(x,λ) be the solution to the Cauchy problem for the equation /diamondsolid (1.2) subject to the initial conditions u+(0,λ) = 1, u′ +(0,λ) =A0(λ).6 DENIS BORISOV AND PEDRO FREITAS We introduce one more function wm(x,λ) =u+(x,λ)/Um(x,λ). This function solves the Cauchy problem (U2 mw′ m)′+λ−mUmeλxfmwm= 0, x∈[0,1], wm(0,λ) = 1, w′ m(0,λ) = 0. The last problem is equivalent to the integral equation wm(x,λ)+λ−m(Km(λ)wm)(x,λ) = 1, (Km(λ)wm)(x,λ) :=x/integraldisplay 0U−2 m(t1)t1/integraldisplay 0Um(t2)eλt2fm(t2,λ)wm(t2,λ)dt2dt1. Since Re λ/greaterorequalslant0 for 0/greaterorequalslantt2/greaterorequalslantt1/greaterorequalslant1, the estimate |U−2 m(t1,λ)Um(t2,λ)eλt1|/lessorequalslantCm holds true, where the constant Cmis independent of λ,t1,t2. Hence, the integral operator Km:C[0,1]→C[0,1] is bounded uniformly in λ large enough, Re λ/greaterorequalslant0. Employing this fact, we conclude that wm(x) = 1+O(λ−m), λ→ ∞,Reλ/greaterorequalslant0, in theC2[0,1]-norm. Hence, the formula (3.1), where (3.7) φ+(x,λ) =m−1/summationdisplay i=0φ(+) i(x)λ−i+O(λ−m), gives the asymptotic expansion for the solution of the Cauchy prob lem (1.2), (3.6) as λ→ ∞, Reλ/greaterorequalslant0. Suppose now that Re λ/lessorequalslant0. LetA1(λ),A2(λ) be functions having the asymptotic expansions A1(λ) =λ+m/summationdisplay i=0λ−i1/integraldisplay 0φ(+) i(x)dx, A 2(λ) =λ+m/summationdisplay i=0φ(+) i(1)λ−i. We define the function /tildewideu+(x,λ) as the solution to the Cauchy problem for equation (1.2) subject to the initial conditions /tildewideu+(1,λ) = eA1(λ),/tildewideu′ +(1,λ) =A2(λ)eA1(λ). In a way analogous to the arguments given above, it is possible to check that the function /tildewideu+has the asymptotic expansion (3.1) in the C2[0,1]-norm as λ→+∞, Reλ/lessorequalslant0. Hence,/tildewideu+(0,λ) = 1 +O(λ−m) for eachm/greaterorequalslant1. In view of this identity we conclude that the function u+(x,λ) :=/tildewideu+(x,λ)//tildewideu+(0,λ) is a solution to (1.2), satisfies the condi- tionu+(0,λ) = 1, and has the asymptotic expansion (3.1), where the asymptotics for φ+is given in (3.7).EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 7 For convenience we summarize the obtained results in Lemma 3.1. Leta∈Cm+1[0,1],b∈Cm[0,1]. There exist two linear independent solutions to the equation (1.2) satisfying the initial con- ditionu±(0,λ) = 1and having the asymptotic expansions (3.1) in the C2[0,1]-norm as λ→ ∞,λ∈C, where φ±(x,λ) =m−1/summationdisplay i=0φ(±) i(x)λ−i+O(λ−m). 4.Asymptotics of the eigenvalues This section is devoted to the proof of Theorem 1 and Corollaries 2 and 3. We assume that a∈Cm+1[0,1],b∈Cm[0,1],m/greaterorequalslant1. Letu=u(x,λ) be the solution to (1.2) subject to the initial condi- tionsu(0,λ) = 0,u′(0,λ) = 1. Denote γ0(λ) :=u(1,λ). The function γ0is entire, and its zeros coincide with the eigenvalues of the problem (1.2), (1.3). It follows from Lemma 3.1 that, for λlarge enough the function u(x,λ) can be expressed in terms of u±by u(x,λ) =u+(x,λ)−u−(x,λ) u′ +(0,λ)−u′ −(0,λ). The denominator is non-zero, since due to (3.1) u′ +(0,λ)−u′ −(0,λ) = 2λ+2/an}bracketle{ta/an}bracketri}ht+O(λ−1), λ→ ∞. Thus, for λlarge enough (4.1) γ0(λ) =u+(1,λ)−u−(1,λ) u′ +(0,λ)−u′ −(0,λ). Lemma 4.1. Fornlarge enough, the set Q:={λ:|Reλ|< πn+π/2,|Imλ|< πn+π/2} contains exactly 2neigenvalues of the problem (3.1), (3.3). Proof.Letγ1(λ) :=γ0(λ)eλ+/angbracketlefta/angbracketright. The zeros of γ1are those of γ0(λ). For λlarge enough we represent the function γ1(λ) as /diamondsolid γ1(λ) =γ2(λ)+γ3(λ), γ2:=e2(λ+a(0))−1 2(λ+a(0)), γ3(λ) =−γ2(λ)/tildewideφ+(0,λ)+/tildewideφ−(0,λ)+2(1+ λ−1a(0))(1−eλ−1/angbracketlefteφ+(·,λ)/angbracketright) 2λ(λ+a(0))+/tildewideφ+(0,λ)+/tildewideφ−(0,λ) +eλ−1/angbracketlefteφ+(·,λ)/angbracketright−e−λ−1/angbracketlefteφ−(·,λ)/angbracketright 2(λ+a(0))+λ−1(/tildewideφ+(0,λ)+/tildewideφ−(0,λ)),8 DENIS BORISOV AND PEDRO FREITAS /tildewideφ±(x,λ) :=λ−1(φ±(x,λ)−a(x)). Itisclear thatfor λlargeenoughthefunction γ3(λ) satisfies anuniform inλestimate |γ3(λ)|/lessorequalslantC|λ|−2/parenleftbig |γ2(λ)|+1/parenrightbig . One can also check easily that /diamondsolid |γ2(λ)|/greaterorequalslantC|λ|, λ∈∂K, ifnis large enough. These two last estimates imply that |γ3(λ)|/lessorequalslant |γ2(λ)|asλ∈∂K, ifnislargeenough. ByRouch´ etheoremweconclude that for such nthe function γ1has the same amount of zeros inside Qas the function γ2does. Since the zeros of the latter are given by πni−/an}bracketle{ta/an}bracketri}ht,n/ne}ationslash= 0, this completes the proof. /square Proof of Theorem 1. Assume first that a∈C2[0,1],b∈C1[0,1]. As was mentioned above, the eigenvalues of problem (1.2), (1.3) are th e zeros of the function γ0(λ) = 0. It follows from Lemma 4.1 that these eigenvalues tend to infinity as n→ ∞. By Lemma 3.1, for λlarge enough the equation γ0(λ) = 0 becomes e2λ+/angbracketleftφ+(·,λ)+φ−(·,λ)/angbracketright= 0 which may be rewritten as (4.2) 2 λ+/an}bracketle{tφ+(·,λ)+φ−(·,λ)/an}bracketri}ht= 2πni, n∈Z. If we now replace φ±by the leading terms of their asymptotic expan- sions we obtain 2λ+2/an}bracketle{ta/an}bracketri}ht+O(λ−1) = 2πni, (4.3) λ=πni−/an}bracketle{ta/an}bracketri}ht+o(1), n→ ∞. Hence, the eigenvalues behave as λ∼πni−/an}bracketle{ta/an}bracketri}htfor large n. Moreover, it follows from Lemma 4.1 that it is exactly the eigenvalue λnwhich behaves as λn=πni−/an}bracketle{ta/an}bracketri}ht+o(1), n→ ∞. It follows from this identity and (4.3) that λn=πni−/an}bracketle{ta/an}bracketri}ht+O(n−1), n→ ∞, and we complete the proof in the case m= 1. Ifm= 2, we substitute the above identity and (3.1) into (4.2) and get λn+/an}bracketle{ta/an}bracketri}ht+1 λn/an}bracketle{tφ(+) 1+φ(−) 1/an}bracketri}ht+O(λ−2 n) =πni, λn=πni−/an}bracketle{ta/an}bracketri}ht−/an}bracketle{tφ(+) 1+φ(−) 1/an}bracketri}ht πni+O(n−2).EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 9 The last formula and the identities (3.4) yield formulas (2.2) for c0and c1. Repeating the described procedure one can easily check that the asymptotics (2.1), (2.2) hold true. /square Proof of Corollary 2. The coefficients c0,c1in the asymptotics (2.1) are determined by the formulas (2.2) and, by the Cauchy-Schwarz in- equality, we thus obtain c2 0=/an}bracketle{ta/an}bracketri}ht2/lessorequalslant/an}bracketle{ta2/an}bracketri}ht= 2πic1−/an}bracketle{tb/an}bracketri}ht, with equality if and only if a(x) is a constant function. This fact completes the proof. /square Proof of Corollary 3. Itfollowsfrom(2.2),(2.3)that /an}bracketle{ta2 1/an}bracketri}ht=/an}bracketle{ta2 2/an}bracketri}ht,/an}bracketle{ta3 1/an}bracketri}ht= /an}bracketle{ta3 2/an}bracketri}ht. Now we check that /an}bracketle{ta2 i/an}bracketri}ht=/an}bracketle{ta2 0/an}bracketri}ht+/an}bracketle{t/tildewidea2 i/an}bracketri}ht,/an}bracketle{ta3 i/an}bracketri}ht=/an}bracketle{ta3 0/an}bracketri}ht+3/an}bracketle{ta0/tildewidea2 i/an}bracketri}ht, i= 1,2, and arrive at the statement of the theorem. /square 5.Regularized trace formulas In this section we prove Theorem 4. We follow the idea employed in the proof of the similar trace formula for the Sturm-Liouville operat ors in [LS, Ch. I, Sec. 14]. We begin by defining the function Φ(λ) :=λ2p∞/productdisplay n=p+1/parenleftbigg 1−λ λn/parenrightbigg/parenleftbigg 1−λ λn/parenrightbigg . The above product converges, since /parenleftbigg 1−λ λn/parenrightbigg/parenleftbigg 1−λ λn/parenrightbigg = 1+λ2−2λReλn |λn|2, and by Theorem 1 we have (5.1)|λn|2=π2n2−2πic1+c2 0+O(n−2), Reλn=c0+O(n−2) asn→+∞. Proceeding in the same way as in the formulas (14.8), (14.9) in [LS, Ch. I, Sec. 14], we obtain Φ(λ) =C0Ψ(λ)sinhλ λ, Ψ(λ) :=∞/productdisplay n=1/parenleftbigg 1−π2n2−|λn|2+2λReλn π2n2+λ2/parenrightbigg ,10 DENIS BORISOV AND PEDRO FREITAS C0:= (πn)2p∞/productdisplay n=p+1π2n2 |λn|2. In what follows we assume that λis real, positive and large. In the same way as in [LS, Ch. I, Sec. 14] it is possible to derive the formula (5.2) lnΨ( λ) =−∞/summationdisplay k=11 k∞/summationdisplay n=1/parenleftbiggπ2n2−|λn|2+2λReλn π2n2+λ2/parenrightbiggk . Our aim is to study the asymptotic behaviour of lnΨ( λ) asλ→+∞. Employing the same arguments as in the proof of Lemma 14.1 and in the equation (14.11) in [LS, Ch. I, Sec. 14], we arrive at the estimate ∞/summationdisplay n=1/parenleftbiggπ2n2−|λn|2+2λReλn π2n2+λ2/parenrightbiggk /lessorequalslantckλk∞/summationdisplay n=11 (π2n2+λ2)k /lessorequalslantckλk+∞/integraldisplay 0dt (π2t2+λ2)k=ck λk+∞/integraldisplay 0dz (π2z2+1)k/lessorequalslantck+1 λk, ∞/summationdisplay k=31 k∞/summationdisplay n=1/parenleftbiggπ2n2−|λn|2+2λReλn π2n2+λ2/parenrightbiggk =O(λ−3), λ→+∞, (5.3) wherecis a constant independent of kandn. Let us analyze the asymptotic behaviour of the first two terms in the series (5.2). As k= 1, we have (5.4)∞/summationdisplay n=1π2n2−|λn|2+2λReλn π2n2+λ2=∞/summationdisplay n=1π2n2−|λn|2−2πic1+c2 0 π2n2+λ2 +/parenleftbig 2πic1−c2 0+2λc0/parenrightbig∞/summationdisplay n=11 π2n2+λ2 +2λ−1S−2λ−1∞/summationdisplay n=1π2n2(Reλn−c0) π2n2+λ2, whereS:=∞/summationdisplay n=1(Reλn−c0). Taking into account (5.1), we have /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay n=1π2n2−|λn|2−2πic1+c2 0 π2n2+λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay n=11 n2(π2n2+λ2) =π2 63+λ2−3cothλ λ4/lessorequalslantCλ−2,EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 11 /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay n=1(Reλn−c0)π2n2 π2n2+λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay n=11 π2n2+λ2/lessorequalslantCλ−1, where the constant Cis independent of λ. Here we have also used the formula (5.5)∞/summationdisplay n=11 π2n2+λ2=λcothλ−1 2λ2=λ−1−λ−2 2+O(λ−1e−2λ) asλ→+∞. We employ this formula to calculate the remaining terms in (5.4) and arrive at the identity (5.6) ∞/summationdisplay n=1π2n2−|λn|2+2λReλn π2n2+λ2=c0+/parenleftbigg 2S−c0−c2 0 2+iπc1/parenrightbigg λ−1+O(λ−2), asλ→+∞. Fork= 2 we proceed in the similar way, ∞/summationdisplay n=1/parenleftbiggπ2n2−|λn|2+2λReλn π2n2+λ2/parenrightbigg2 =∞/summationdisplay n=1(π2n2−|λn|2)2 (π2n2+λ2)2 −2λ∞/summationdisplay n=1(π2n2−|λn|2)Reλn (π2n2+λ2)2+4λ2c2 0∞/summationdisplay n=11 (π2n2+λ2)2 +4λ2∞/summationdisplay n=1(Reλn)2−c2 0 (π2n2+λ2)2. By differentiating (5.5) we obtain ∞/summationdisplay n=11 (π2n2+λ2)2=λcothλ−2−λ2(1−coth2λ) 4λ4. This identity and (5.1) yield that as λ→+∞ /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay n=1(π2n2−|λn|2)2 (π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay n=11 (π2n2+λ2)2/lessorequalslantCλ−3, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay n=1(π2n2−|λn|2)Reλn (π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay n=11 (π2n2+λ2)2/lessorequalslantCλ−3, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay n=1(Reλn)2−c2 0 (π2n2+λ2)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslantC∞/summationdisplay n=11 (π2n2+λ2)2/lessorequalslantCλ−3. Hence, (5.7)∞/summationdisplay n=1/parenleftbiggπ2n2−|λn|2+2λReλn π2n2+λ2/parenrightbigg2 =c2 0λ−1+O(λ−2)12 DENIS BORISOV AND PEDRO FREITAS asλ→+∞. It follows from (5.2), (5.3), (5.6), (5.7) that lnΨ(λ) =−c0−(2S−c0+iπc1)λ−1+O(λ−2), Φ(λ) =C0e−c0sinhλ λ/bracketleftbig 1−(2S−c0+iπc1)λ−1+O(λ−2)/bracketrightbig (5.8) asλ→+∞. It follows from (4.1) and Lemma 3.1 that for λlarge enough the estimate /diamondsolid |γ0(λ)|/lessorequalslantC|λ|−1e|λ| holds true. Hence, the order of the entire function γ0(λ) is one. In view of Theorem 1 we also conclude that the series∞/summationtext n=p+1|λn|−2converges and therefore the genus of the canonical product associated wit hγ0is one. We apply Hadamard’s theorem (see, for instance, [Le, Ch. I, Sec. 10, Th. 13]) and obtain that γ0(λ) = eP(λ)Φ(λ), P(λ) =α1λ+α0+2∞/summationdisplay n=p+1|λn|−2Reλn, whereα1,α0are some numbers. Hence, due to (5.8), it follows that γ0 behaves as γ0(λ) =C0eP(λ)sinhλ λ/bracketleftbig 1−(2S−c0+iπc1)λ−1+O(λ−2)/bracketrightbig , asλ→+∞. On the other hand, Lemma 3.1 and (4.1) imply that γ0(λ) =eλ+/angbracketlefta/angbracketright 2λ/bracketleftbig 1+(/an}bracketle{tφ(+) 1/an}bracketri}ht−a(0))λ−1+O(λ−2)/bracketrightbig +O(λ−1e−λ), asλ→+∞. Comparing the last two identities yields α1= 0, C0eα0−c0+2∞P n=1|λn|−2Reλn= e/angbracketlefta/angbracketright and −(2S−c0+iπc1) =/an}bracketle{tφ(+) 1/an}bracketri}ht−a(0). It now follows from (2.2), (3.4) that ∞/summationdisplay n=−∞ n/negationslash=0(λn−c0) = 2S=c0+a(0)−/an}bracketle{tφ(+) 1/an}bracketri}ht−iπc1=a(0)+a(1) 2−/an}bracketle{ta/an}bracketri}ht, completing the proof of Theorem 4. Acknowledgments This work was done during the visit of D.B. to the Universidade de Lisboa; he is grateful for the hospitality extended to him. P.F. would like to thank A. Laptev for several conversations of this topic.EIGENVALUE ASYMPTOTICS FOR THE DAMPED WAVE EQUATION 13 References [AL] M. Asch and G. Lebeau, The spectrum of the damped wave oper ator for a bounded domain in R2,Exp. Math. 12(2003), 227-241. [BR] A. Benaddi and B. Rao, Energy decay rate of damped wave equ ations with indefinite damping, J. Differential Eq. 161(2000), 337–357. [CC] C. Castro and S. Cox, Achieving arbitrarily large decay in the dam ped wave equation, SIAM J. Control Optim. 39(2001), 1748–1755. [CFNS] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun. Exponential d ecay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51(1991), 266–301. [CZ] S. Cox and E. Zuazua, The rate at which energy decays in a damp ed string.Comm. Part. Diff. Eq. ,19(1994), 213–243. [E] A. Erd´ elyi. Asymptotic expansions. Dover Publications Inc., N.Y. 1956. [Fe] M.V. Fedoryuk, Asymptotic analysis: linear ordinary differential equa- tions. Berlin: Springer-Verlag. 1993. [F1] P. Freitas, On some eigenvalueproblems relatedto the waveequ ation with indefinite damping, J. Differential Equations ,127(1996), 320–335. [F2] P.Freitas,Spectralsequencesforquadraticpencilsandthe inversespectral problem for the damped wave equation, J. Math. Pures Appl. 78(1999), 965–980. [L] G. Lebeau, ´Equations des ondes amorties, S´ eminaire sur les ´Equations aux D´ eriv´ ees Partielles, 1993–1994,Exp. No. XV, 16 pp., ´Ecole Polytech., Palaiseau, 1994. [Le] B.Ya. Levin. Distribution of zeros of entire functions. Providen ce, R.I.: American Mathematical Society. 1964. [LS] B.M. Levitan, I.S. Sargsjan. Introduction to spectral theor y: Selfadjoint ordinarydifferentialoperators.TranslationsofMathematicalMo nographs. Vol. 39. Providence, R.I.: American Mathematical Society. 1975. [PT] J. P¨ oschel and E. Trubowitz, Inverse spectran theory, Pu re and Applied Mathematics, Vol. 130, Academic Press, London, 1987. [S] J.Sj¨ ostrand, Asymptoticdistributionofeigenfrequenciesfo rdampedwave equations, Publ. Res. Inst. Math. Sci. 36(2000), 573–611. Department of Physics and Mathematics, Bashkir State Pedag ogi- cal University, October rev. st., 3a, 450000, Ufa, Russia E-mail address :borisovdi@yandex.ru Department ofMathematics, Faculdade de Motricidade Human a (TU Lisbon) andGroup of Mathematical Physics of the University of Lis- bon, Complexo Interdisciplinar, Av. Prof. Gama Pinto 2, P-1 649-003 Lisboa, Portugal E-mail address :freitas@cii.fc.ul.pt

2009-05-20

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.

Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

0905.3242v1

arXiv:0905.4650v2 [math.PR] 8 Nov 2012The Annals of Applied Probability 2011, Vol. 21, No. 3, 1053–1072 DOI:10.1214/10-AAP718 c/circlecopyrtInstitute of Mathematical Statistics , 2011 HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS By J´ozsef Balogh1, B´ela Bollob ´as2, Michael Krivelevich3, Tobias M ¨uller4and Mark Walters University of California and University of Illinois, University of Cambridge and University of Memphis, Tel Aviv U niversity, Centrum voor Wiskunde en Informatica, and Queen Mary, University of London We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbor model, there is a constant κsuch that almost every κ-connected graph has a Hamilton cycle. 1. Introduction. In this paper we mainly consider one of the frequently studied models for random geometric graphs, namely the Gilb ert model. Suppose that Snis a√n×√nbox and that Pis a Poisson process in it with density 1. The points of the process form the vertex set o f our graph. There is a parameter rgoverning the edges: two points are joined if their (Euclidean) distance is at most r. Having formed this graph we can ask whether it has any of the st andard graph properties, such as connectedness. As usual, we shall only consider these for large values of n. More formally, we say that G=Gn,rhas a prop- Received April 2009; revised January 2010. 1This material is based upon work supported by NSF CAREER Gran t DMS-07-45185 and DMS-06-00303, UIUC Campus Research Board Grants 09072 a nd 08086 and OTKA Grant K76099. 2Supported in part by NSF Grants DMS-05-05550, CNS-0721983 a nd CCF-0728928 and ARO Grant W911NF-06-1-0076. 3Supported in part by USA-Israel BSF Grant 2006322, by Grant 1 063/08 from the Israel Science Foundation and by a Pazy memorial award. 4Supported in part by a VENI grant from Netherlands Organisat ion for Research (NWO). The results in this paper are based on work done while a t Tel Aviv University, partially supported through an ERC advanced grant. AMS 2000 subject classifications. 05C80, 60D05, 05C45. Key words and phrases. Hamilton cycles, random geometric graphs. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics inThe Annals of Applied Probability , 2011, Vol. 21, No. 3, 1053–1072 . This reprint differs from the original in pagination and typographic detail. 12 J. BALOGH ET AL. ertywith high probability (abbreviated to whp) if the probability that Ghas this property tends to one as ntends to infinity. Penrose [ 10] proved that the threshold for connectivity is πr2=logn. In fact he proved the following very sharp result: suppose πr2=logn+αfor someconstant α.Thentheprobabilitythat Gn,risconnectedtendsto e−e−α. He also generalized this result to find the threshold for κ-connectivity forκ≥2: namely πr2=logn+(2κ−3)loglog n. [Since the reader may be surprised that this formula does not work for κ=1 we remark that this is due to boundary effects: the threshold for κ-connectivity is the maximum of two quantities: log n+(κ−1)loglog ntoκ-connect the central points and logn+(2κ−3)loglog ntoκ-connect the points near the boundary. If one worked on thetorus instead of thesquare, then these boundar yeffects would disappear.] Moreover, he found the “obstruction” to κ-connectivity. Suppose we fix the vertex set (i.e., the point set in Sn) and “grow” r. This gradually adds edges to the graph. For a monotone graph property PletH(P) denote the smallest rfor which the graph on this point set has the property P. Penrose showed that H(δ(G)≥κ)=H(connectivity( G)≥κ) whp: that is, as soon as the graph has minimum degree κit isκ-connected whp. Healsoconsideredthethresholdfor GtohaveaHamiltoncycle.Obviously anecessary condition isthat thegraphis2-connected. Inth enormal(Erd˝ os– R´ enyi) randomgraphthisisalsoasufficientcondition inthe following strong sense. If we add edges to the graph one at a time, then the graph becomes Hamiltonian exactly when it becomes 2-connected (see [ 5,8,9] and [14]). Penrose asked whether the same is true for a random geometric graph. In this paper we prove the following theorem answering this que stion. Theorem 1. Suppose that G=Gn,ris the two-dimensional Gilbert model. Then H(Gis 2-connected )=H(Ghas a Hamilton cycle ) whp. Combining this with Penrose’s results mentioned above we se e that, if πr2=logn+loglog n+α, then the probability that Ghas a Hamilton cycle tends to e−e−α−√πe−α/2(the second term in the exponent is the contribution from points near the boundary of the square). Somepartialprogresshasbeenmadeonthisquestionpreviou sly.Petit [ 13] showed that if πr2/logntends to infinity, then Gis, whp, Hamiltonian, andHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 3 D´ ıaz, Mitsche and P´ erez [ 7] proved that if πr2>(1+ε)lognfor some ε>0 thenGis Hamiltonian whp. (Obviously, Gis not Hamiltonian if πr2<logn since whp Gis not connected!) Finally usinga similar method to [ 7] together with significant case analysis, Balogh, Kaul and Martin [ 4] proved for the special case of the ℓ∞norm in two dimensions that the graph does become Hamiltonian exactly when it becomes 2-connected. Our proof generalizes to higher dimensions and to other norm s. The Gilbert model makes sense with any norm and in any number of di men- sions: we let Sd nbe thed-dimensional hypercube with volume n. We prove the analog of Theorem 1in this setting. Theorem 2. Suppose that the dimension d≥2and/bar⌈bl·/bar⌈bl, ap-norm for some1≤p≤∞, are fixed. Let G=Gn,rbe the resulting Gilbert model. Then H(Gis 2-connected )=H(Ghas a Hamilton cycle ) whp. The proof is very similar to that of Theorem 1. However, there are some significant extra technicalities. Togiveanideawhytheseoccurconsiderconnectivity intheG ilbertmodel in the cube S3 n(with the Euclidean norm). Let Abe the volume of a sphere of radius r. We count the expected number of isolated points in the proce ss which are away from the boundary of the cube. The probability a point is isolated is e−A, so the expected number of such points is ne−A, so the threshold for the existence of a central isolated point is ab outA=logn. However, consider the probability that a point near a face of the cube is isolated: there are approximately n2/3such points, and the probability that they are isolated is about e−A/2(since about half of the sphere about the point is outside the cube S3 n). Hence, the expected number of such points is n2/3e−A/2, so the threshold for the existence of an isolated point near a face is about A=4 3logn. In other words isolated points are much more likely near the boundary. These boundary effects are the reason for ma ny of the extra technicalities. We remark that Theorem 2is trivially true for d=1: indeed, if Gis 2- connected then there are two vertex disjoint paths from the l eft-most vertex to the right-most vertex. By adding any remaining vertices t o one of these paths these two paths form a Hamilton cycle. Thek-nearest neighbor model. We also consider a second model for ran- dom geometric graphs: namely the k-nearest neighbor graph. In this model the initial setup is the same as in the Gilbert model: the vert ices are given by a Poisson process of density one in the square Sn, but this time each vertex is joined to its knearest neighbors (in the Euclidean metric) in the box. This naturally gives rise to a k-regular directed graph, but we form a4 J. BALOGH ET AL. simple graph G=Gn,kby ignoring the direction of all the edges. It is easily checked that this gives us a graph with degrees between kand 6k. Xue and Kumar [ 15] showed that there are constants c1,c2such that if k < c1logn, then the graph Gn,kis, whp, not connected, and that if k > c2lognthenGn,kis, whp, connected. Balister et al. [ 1] proved reasonably good bounds on the constants: namely c1= 0.3043 and c2= 0.5139, and later [3] proved that there is some critical constant csuch that if k=c′logn forc′<c, then the graph is disconnected whp, and if k=c′lognforc′>c, then it is connected whp. Moreover, in [ 2], they showed that in the latter case the graph is s-connected whp for any fixed s∈N. We would like to prove a sharp result like the above; that is, t hat as soon as the graph is 2-connected it has a Hamilton cycle. However, we prove only the weaker statement that some (finite) amount of connectivi ty is sufficient. Explicitly, we show the following. Theorem 3. Suppose that k=k(n), thatG=Gn,kis the two-dimensional k-nearest neighbor graph (with the Euclidean norm) and that Gisκ-connected forκ=5·107whp. Then Ghas a Hamilton cycle whp. Analogous results could be proved in higher dimensions and f or other norms but we do not do so here. Binomial point process. To conclude this section we briefly mention a closely related model: instead of choosing the points in Snaccording to a Poisson process of density one we choose npoints uniformly at random, and then form the corresponding graph. This new model is very clo sely related to our first model (the Gilbert model). Indeed, Penrose origi nally proved his results for the Binomial Point Process but it is easy to ch eck that this implies them for the Poisson Process. It is very easy to modify our proof to this new model. Indeed, i n very broad terms each of our arguments consists of two steps: first we have an essentially trivial lemma that says the random points are “r easonably” dis- tributed, and then we have an argument saying that if the poin ts are reason- ably distributed and the resulting graph is two-connected t hen the resulting graph necessarily has a Hamilton cycle. The second of these s teps is entirely deterministic, so only the essentially trivial lemma needs modifying. 2. Proof of Theorem 1.We divide the proof into five parts: first we tile thesquare Snwithsmallsquaresinastandardtessellation argument.Sec ond we identify “difficult” subsquares. Roughly, these will be sq uares containing only a few points, or squares surrounded by squares containi ng only a few points. Third we prove some lemmas about the structure of the difficult subsquares. In stage 4 we deal with the difficult subsquares. F inally we use the remaining easy subsquares to join everything together.HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 5 Stage 1: Tessellation. Letr0=/radicalbig (logn)/π(soπr2 0=logn), and let rbe the random variable H(Gis 2-connected). Let s=r0/c=c′√lognwherec is a large constant to be chosen later (1000 will do). We tesse llate the box Snwith small squares of side length s. Whenever we talk about distances between squares we will always be referring to the distance b etween their centers. Moreover, we will divide all distances between squ ares bys, so, for example, a square’s four nearest neighbors all have distanc e one. By Penrose’s result [ 11] mentioned in the Introduction we may assume that (1−1/2c)r0< r <(1+ 1/2c)r0: formally the collection of point sets which do not satisfy this has measure tending to zero as ntends to infinity, and we ignore this set. Hence points in squares at distancer−√ 2s s≥r0−2s s=c−2 are always joined, and points in squares at distancer+√ 2s s≤r0+2s s=c+2 are never joined. Stage 2: The “difficult” subsquares. We call a square fullif it contains at least Mpoints for some Mto be determined later (107will do), and nonfullotherwise. Let N0be the set of nonfull squares. We say two nonfull squares are joined if their ℓ∞distance is at most 4 c−1 and define Nto be the collection of nonfull components. First we bound the size of the largest component of nonfull sq uares (here, and throughout this paper, we use size to refer to the number o f vertices in the component). Lemma 4. For any M, the largest component of nonfull squares in the above tesselation has size at most U=⌈π(c+2)2⌉ whp. Also, the largest component of nonfull squares including a s quare within c of the boundary of Snhas size at most U/2whp. Finally, there is no nonfull square within distance Ucof a corner whp. Proof. We shall make use of the following simple result: suppose tha t Gis any graph with maximal degree ∆, and vis a vertex in G. Then the number of connected subsets of size nofGcontaining vis at most ( e∆)n (see, e.g., Problem 45 of [ 6]). Hence, the number of potential components of size Ucontaining a par- ticular square is at most ( e(8c)2)Uso, since there are less than nsquares, the total number of such potential components is at most n(e(8c)2)U. The probability that a square is nonfull is at most 2 s2Me−s2/M!. Hence, the expected number of components of size at least Uis at most n(2s2Me−s2(e(8c)2)/M!)U≤n/parenleftbigg 2(logn)Me(8c)2 M!/parenrightbiggU exp/parenleftbigg −(c+2)2logn c2/parenrightbigg ,6 J. BALOGH ET AL. which tends to zero as ntends to infinity; that is, whp, no such component exists. For the second part there are at most 4 c√nsquares within distance cof the boundary of Sn, and the result follows as above. Finally, there are only 4 U2c2squares within distance Ucof a corner. Since the probability that a square is nonfull tends to zero we see t hat there is no such square whp. /square Note that this is true independently of Mwhich is important since we will want to choose Mdepending on U. In the rest of the argument we shall assume that there is no non full component of size greater than U, no nonfull component of size U/2 within cof an edge and no nonfull square within Ucof a corner. Between these components of nonfull squares there are numer ous full squares. To define this more precisely let /hatwideGbe the graph with vertex set the small squares, and where each square is joined to all others w ithin (c−2) of this square (in the Euclidean norm). Since the probabilit y a square is in N0(i.e., is nonfull) is 1 −o(1), the graph /hatwideG\N0has one giant component consisting of almost all the squares. We call this component sea. (We give an equivalent formal definition just before Corollary 8.) The idea is that it is trivial to find a cycle visiting every poi nt of the pro- cess in a square in the sea, and that we can extend this cycle to a Hamilton cycle by adding each nonfull component (and any full squares cut off by it) one at a time. However, it is easier to phrase the argument by s tarting with the difficult parts and then using the sea of full squares. Stage 3: The structure of the difficult subsquares. Consider one com- ponentN∈Nof the nonfull squares, and suppose that it has size u. By Lemma4we know u<U. We will also consider N2c: the 2c-blow-up of N: that is the set of all squares with ℓ∞distance at most 2 cfrom a square in N. Now some full squares may be cut off from the rest of the full squ ares by nonfull squares in N. More precisely the graph /hatwideG\Nhas one component A=A(N) consisting of all but at most a bounded number of squares (si nce we have removed at most Usquares from /hatwideG). We call Acthecutoffsquares. We split the cutoff squares into two classes: those with a neig hbor inA (in/hatwideG) which we think of as being “close” to A, and the rest, which we shall call farsquares. All the close squares must be in N(since otherwise they would be part of A). However, we do not know anything about the far squares: they may be full or nonfull. See Figure 1for a picture. Lemma 5. No two far squares are more than ℓ∞distance c/10apart. Remark. This does not say whp since we are assuming this nonfull component has size at most U.HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 7 Fig. 1. A small part of Sncontaining the nonfull component Nand the corresponding setA, far squares and close squares. It also shows the two vertex d isjoint paths from the far squares to Aand the path joining Q2toQ1(see stage 4). Proof. Suppose not. Suppose, first, that no point of Nis within cof the edge of Sn, and that the two far squares are at horizontal distance at least c/10. Then consider the left-most far square. All squares which are to the left of this and with distancetothissquarelessthan( c−2)mustbecloseandthusin N.Similarly with the right-most far square. Also at least ( c−2) squares [in fact nearly 2(c−2)] in each of at least c/10 columnsbetween theoriginal two farsquares must be in N. This is a total of about π(c−2)2+(c−2)c/10>Uwhich is a contradiction (provided we chose creasonably large). If there is a point of Nwithincof the boundary, then the above argument gives more than U/2 nonfull squares. Indeed, either it gives half of each part of the above construction, or it gives all of one end and all th e side parts. This contradicts the second part of our assumption about the size of nonfull components. We do not need to consider a component near two sides: it canno t be large enough to be near two sides. It also cannot go across a co rner, since no square within distance Ucof a corner is nonfull. /square This result can also be deduced from a result of Penrose, as we do in the next section. We have the following instant corollary. Corollary 6. The graph /hatwideGrestricted to the far squares is complete.8 J. BALOGH ET AL. Corollary 7. The set of cutoff squares Acis contained in Nc(the c-blow-up of N). In particular, the set Γ(Ac)of neighbors in /hatwideGofAcis contained in N2c. Proof. Suppose Ac/\⌉}atio\slash⊆Nc. Letxbe a square in Ac\Nc. First,xcannot be a neighbor of any square in Aorxwould also be in A; that is, xis a far square. Now, let ybe any square with ℓ∞distance c/5 fromx. The square y cannot be in Nsince then xwould be in Nc. Therefore, ycannot be a neighbor of any square in Asince then it would be in Aand, since xandy are joined in /hatwideG,xwould be in A; that is, yis also a far square. Hence, xand yare both far squares with ℓ∞distance c/5 which contradicts Lemma 5./square Inparticular,Corollary 7tellsusthatthesetsofsquarescutoffbydifferent nonfull components and all their neighbors are disjoint (ob viously the 2 c- blow-ups are disjoint). We now formally define the sea/tildewideA=/intersectiontext N∈NA(N). We show later (Corol- lary11) that/tildewideAis connected and, thus, that this is the same as our earlier informal definition. The following corollary is immediate f rom Corollary 7. Corollary 8. For any N∈Nwe have /tildewideA∩N2c=A(N)∩N2c. The final preparation we need is the following lemma. Lemma 9. The setN2c∩Ais connected in /hatwideG. Since the proof will be using a standard graph theoretic resu lt, it is con- venient to define one more graph /hatwideG1: again the vertex set is the set of small squares, but this time each square is joined only to its four n earest neigh- bors;that is, /hatwideG1istheordinarysquarelattice. Weneedtwoquickdefinitions . First, for a set E∈/hatwideG1we define the boundary ∂1EofEto be set of vertices inEcthat are neighbors (in /hatwideG1) of a vertex in E. Second, we say a set Ein/hatwideG1isdiagonally connected if it is connected when we add the edges between squares which are diagonally adjacent (i.e., at dis tance√ 2) to/hatwideG. The lemma we need is the following; since its proof is short we include it here for completeness. (It is also an easy consequence of the unicoherence of the square (see, e.g., page 177 of [ 12]).) Lemma 10. Suppose that Eis any subset of /hatwideG1withEandEccon- nected. Then ∂1Eis diagonally connected: in particular, it is connected in /hatwideG. Proof. LetFbe the set of edges of /hatwideG1fromEtoEc, and let F′be the corresponding set of edges in the dual lattice. Consider the setF′as a subgraph of the dual lattice. It is easy to check that every ve rtex has even degree except vertices on the boundary of /hatwideG1. Thus we can decompose F′ into pieces, each of which is either a cycle or a path starting and finishing atHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 9 the edge of /hatwideG1. Any such cycle splits /hatwideG1into two components, and we see that one of these must be exactly Eand the other Ec. ThusF′is a single component in the dual lattice, and it is easy to check that imp lies that ∂1E is diagonally connected. /square Proof of Lemma 9.Consider /hatwideG1\N2c. This splits into components B1,B2,...,B m. By definition each Biis connected. Moreover, each Bc iis also connected. Indeed, suppose x,y∈Bc i. Then there is an xypath in/hatwideG1. If this is contained in Bc iwe are done. If not then it must meet N2c, but N2cis connected. Hence we can take this path until it first meets N2c, go through N2cto the point where the path last leaves N2cand follow the path on toy. This gives a path in Bc i. Hence, by Lemma 10, we see that each ∂1Biis connected in /hatwideGfor each i (where∂1denotes the boundary in /hatwideG1). Obviously ∂1Bi⊂N2c. As usual, for a set of vertices Vlet/hatwideG[V] denote the graph /hatwideGrestricted to the vertices in V. Claim. Any two vertices in/uniontextm i=1∂1Biare connected in /hatwideG[A∩N2c]. Proof. Suppose not. Without loss of generality assume that, for som e k<m,/hatwideG[/uniontextk i=1∂1Bi] is connected and that no other ∂1Biis connected via a path to it. Pick x∈B1andy∈Bm. Bothxandyare inA(since they are not inN2candAc⊂N2cby Corollary 7). Hence there is a path from xtoyinA. Consider the last time it leaves/uniontextk i=1Bi. The path then moves around in N2cbefore entering some Bjwith j >k. This gives rise to a path in A∩N2cfrom a point in/uniontextk i=1∂1Bito a point in ∂1Bj, contradicting the choice of k./square We now complete the proof of Lemma 9. To avoid clutter we shall say that two points are joinedif they are connected by a path. Suppose that x,y∈A∩N2c. SinceAis connected there is a path in Afromxtoy. If the path is contained in N2cwe are done. If not, consider the first time the path leavesN2c. It must enter one of the Bi, crossing the boundary ∂1Bi. Hence xis joined to some w∈∂1BiinA∩N2c. Similarly, by considering the last time the path is not in N2cwe see that yis joined to some z∈∂1Bjfor somej. However, since the claim showed that wandzare joined in A∩N2c, we see that xandyare joined in A∩N2c./square Corollary 11. The set of sea squares /tildewideAis connected in /hatwideG. Proof. Given two squares x,yin/tildewideA, pick a path in /hatwideGfromxtoy. Now for each nonfull component Nin turn do the following. If the path misses N2cdo nothing. Otherwise let wbe the first point on the path in N2candz be the last point in N2c. Replace the xypath by the path xw, any path wz inA(N)∩N2cand then the path zy.10 J. BALOGH ET AL. At each stage the modification ensured that the path now lies i nA(N). Also, the only vertices added to the path are in N2cwhich is disjoint from all the previous N′ 2c, and thus from all previous sets A(N′). Hence, when we have done this for all nonfull components the path lies in eve ryA(N′), that is, in/tildewideA. Hence, /tildewideAis connected. /square Stage 4: Dealing with the difficult subsquares. We deal with each nonfull component N∈Nin turn. Fix one such component N. Let us deal with the far squares first. There are three possibi lities: the far squares contain no points at all, they contain one point i n total or they contain more than one point. In the first case, do nothing and p roceed to the next part of the argument. In the second case, by the 2-connectivity of G, we can find two vertex disjointpathsfromthissinglevertex v1topointsinsquaresin A.Inthethird case pick two points v1andv2in the far squares. Again by 2-connectivity we can find vertex disjoint paths from these two vertices to po ints in squares inA. Suppose that the path from v1meetsAin square Q1at point q1and the other path (either from v2or the other path from v1again) meets A in square Q2at point q2. LetP1,P2be the squares containing the previous points on these paths. Since no two points in squares at (Eucl idean) distance (c+2) are joined we see that P1is within ( c+2) ofQ1. SinceP1/∈Awe have that some square on a shortest P1Q1path in/hatwideG1is inNand thus that Q1∈N2c. Similarly Q2∈N2c. Combining we see that both Q1andQ2are inN2c∩A. By Lemma 9, we know that N2c∩Ais connected in /hatwideGso we can find a path from Q1toQ2inN2c∩Ain/hatwideG. This “lifts” to a path in G going from q2to a point other than q1inQ1using at most one vertex in each subsquare on the way and never leaving N2c. Construct a path starting and finishing in Q1by joining together the following paths: 1. the path from q1tov1; 2. a path starting at v1going round all points in the far region (except any such points on the q1v1orq2v2paths) finishing back at v2. (Corollary 6 guarantees the existence of such a path.) We omit this piece i f there is just one far vertex; 3. the path v2toq2; 4. the path from q2through the sea back to Q1constructed above. SinceQ1∈A∩N2c, by Corollary 8we have that Q1∈/tildewideA. Combining, we have a path starting and finishing in the same subsquare of the sea/tildewideA(i.e., Q1) containing all the vertices in the far region. Next we deal with the close squares: we deal with each close sq uarePin turn. Since Pis a close square we can pick Q∈AwithPQjoined in /hatwideG. InHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 11 the following we ignore all points that we have used in the pat h constructed above and any points already used when dealing with other clo se squares. If the square Phas no point in it we ignore it. If it has one point in it, then join that point to two points in Q. If it has two or more points in it then pick two of them x,y: and pick two pointsuvinQ(we choose Mlarge enough to ensure that we can find these two unused points in Q, see below). Place the path formed by the edge ux round all the remaining unused vertices in the cutoff square fi nishing at y and back to the square Qwith the edge yvin the cycle we are constructing. The square Qis a neighbor of P∈Acso, by Corollary 7is inN2c. Since Qis also in Awe see, by Corollary 8as above, that Q∈/tildewideA. When we have completed this construction we have placed ever y vertex in a cutoff square on one of a collection of paths, each of which starts and finishes at the same square in the sea (although different paths may start and finish in different squares in the sea). We use at most 2 U+2 vertices from any square in A=A(N) when doing this, so, provided that M >2U+2+(2c+1)2, there are at least (2 c+1)2 unused vertices in each square of Awhen we finish this. Moreover, obviously the only squares touched by this construction are in N2c, and for distinct nonfull components these are all disjoint. Hence, when we ha ve done this for every nonfull component N∈Nthere are at least (2 c+1)2unused vertices in each square of the sea /tildewideA. Stage 5: Using the subsquares in the sea to join everything to gether. It just remains to string everything together. This is easy. Si nce, by Corol- lary11, the sea of squares /tildewideAis connected, there is a spanning tree for /tildewideA. By doubling each edge we can think of this as a cycle, as in Figure 2. This cycle visits each square at most (2 c+1)2times. (In fact, by choosing a spanning tree such that the sum of the edge lengths is minimal we could a ssume that Fig. 2. A tree of subsquares and its corresponding tree cycle.12 J. BALOGH ET AL. it visits each vertex at most six times but we do not need this. ) Convert this into a Hamilton cycle as follows. Start at an unused vertex in a square of the sea. Move to any (unused) vertex in the next square in the tree cycle. Then, if this is the last time the tree cycle visits this square, vis it all remaining vertices and join in all the paths constructed in the first par t of the argu- ment, then leave to the next square in the tree cycle. If it is n ot the last time the tree cycle visits this square, then move to any unused ver tex in the next square in the tree cycle. Repeat until we complete the tree cy cle. Then join in any unused vertices and paths to this square constructed e arlier before closing the cycle. 3. Higher dimensions. We generalise the proof in the previous section to higher dimensions and any p-norm. Much of the argument is the same, in particular, essentially all of stages four and five. We inc lude details of all differences but refer the reader to the previous section where the proof is identical. Stage 1: Tessellation. We work in the d-dimensional hypercube Sd nof volumen(for simplicity we will abbreviate hypercube to cube in the f ollow- ing). As mentioned in the Introduction , we no longer have a nice formula for the critical radius: the boundary effects dominate. Instead, we consider the expected number of isolated vertic esE=E(r). We need a little notation: let Ardenote the set {x∈Sd n:d(x,A)≤r}and |·|denote Lebesgue measure. We have E=/integraltext Sdnexp(−|{x}r|)dx. Letr0=r0(n) be such that E(r0)=1. As before fix ca large constant to be determined later, and let s=r0/c. It is easy to see that rd 0=Θ(logn) andsd=Θ(logn). We tile the cube Sd nwith small cubes of side length s. As before, let r=H(Gis 2-connected). By Penrose (Theorems 1.1 and 1.2 of [11] or Theorems 8.4 and 13.17 of [ 12]) the probability that r /∈[r0(1− 1/2c),r0(1+1/2c)] tends to zero and we ignore all these point sets. (Note that these two of Penrose’s results are not claimed for p=1. However, since for anyε >0 we can pick p >1 such that B1(r)⊂Bp(r)⊂B1((1 +ε)r) [whereB1(r) andBpdenote the l1andlpballs of radius r, resp.], the above bound on rforp=1 follows from Penrose’s results for p>1.) This time any two points in cubes at distancer−s√ d s≥r0−ds s=c−dare joined, and no points in cubes at distancer+s√ d s≤r0+ds s=c+dare joined. Stage 2: The “difficult” subcubes. Exactly as before we define nonfull cubes to be those containing at most Mpoints, and we say two are joined if they have ℓ∞distance at most 4 c−1. We wishto prove aversion of Lemma 4. However, we have several possible boundaries: for example, in three dimensions we have the cen ter, the faces, the edges and the corners. We call a nonfull component contai ning a cubeHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 13 Qbadif it consists of at least (1+1 /c)|Qr0|/sdcubes. (Note a component can be bad for some cubes and not others.) Lemma 12. The expected number of bad components tends to zero as n tends to infinity. In particular there are no bad components wh p. Proof. The number of connected sets of size Ucontaining a particular cube is at most ( e(8c)d)U. The probability that a cube is nonfull is at most 2sdMe−sd/M!. Since min {|Qr0|:cubesQ}=Θ(logn) andsd=Θ(logn), the expected number of bad components is at most /summationdisplay cubesQ(2sdMe−sd(e(8c)d)/M!)(1+1/c)|Qr0|/sd =/summationdisplay cubesQ(2sdM(e(8c)d)/M!)(1+1/c)|Qr0|/sdexp(−(1+1/c)|Qr0|) =o(1)/summationdisplay cubesQexp(−|Qr0|) ≤o(1)/integraldisplay Sdnexp(−|{x}r0|)dx =o(1)E(r0) =o(1). /square (Again, note that this is true independently of M.) From now on we assume that there is no bad component. Stage 3: The structure of the difficult subcubes. In this stage we will need one extra geometric result of Penrose, a case of Proposition 5.15 of [12] (see also Proposition 2.1 of [ 11]). Proposition 13. Suppose dis fixed and that /bar⌈bl·/bar⌈blis ap-norm for some 1≤p≤∞. Then there exists η>0such that if F⊂Od(the positive orthant inRd) is compact with ℓ∞diameter at least r/10, andxis a point of Fwith minimal l1norm; then |Fr|≥|F|+|{x}r|+ηrd. We begin this stage by proving Lemma 5for this model. Lemma 14. No two far cubes are more than ℓ∞distance c/10apart. Proof. Supposenot. Then let Fbethe set of far cubes, let xbea point ofFclosest to a corner in the l1norm and let Qbe the cube containing x (or any of the possibilities if it is on the boundary between c ubes). We know that all the cubes within ( c−d) of a far cube are not in A. Hence all such cubes which are not far must be close, and thus nonfull.14 J. BALOGH ET AL. The number of close cubes is at least |F(c−2d)s\F| sd≥|{x}(c−2d)s|+η((c−2d)s)d sdby Proposition 13 ≥|Q(c−3d)s|+ηrd 0/2 sdprovided cis large enough =|Q(1−3d/c)r0|+ηrd 0/2 sd ≥(1−3d/c)d|Qr0|+ηrd 0/2 sd >(1+1/c)|Qr0| sdprovided cis large enough. This shows that the component is bad which is a contradiction ./square Corollaries 6,7and8hold exactly as before. Lemma 9also holds, we just need to replace Lemma 10by the following higher-dimensional analogue. Note that, even in higher dimensions we say two squares are di agonally connected if their centers have distance√ 2. Lemma 15. Suppose that Eis any subset of /hatwideG1withEandEccon- nected. Then ∂1Eis diagonally connected: in particular, it is connected in /hatwideG. Remark. Again the final conclusion of connectivity in /hatwideGis an easy consequence of unicoherence, this time of the hypercube. Proof. LetIbe a (diagonally connected) component of ∂1E. We aim to show the I=∂1Eand, thus, that ∂1Eis diagonally connected. Claim. Suppose that Cis any circuit in /hatwideG1. Then the number of edges ofCwith one end in Eand the other end in Iis even. Proof. We say that a circuit is contractible to a single point using t he following operations. First, we can remove an out and back ed ge. Second, we can do the following two-dimensional move. Suppose that two consecutive edges of the circuit form two sides of a square; then we can rep lace them by the other two sides of the square keeping the rest of the circu it the same. For example, we can replace ( x,y+1,/vector z)→(x+1,y+1,/vector z)→(x+1,y,/vector z) in the circuit by ( x,y+1,/vector z)→(x,y,/vector z)→(x+1,y,/vector z). Next we show that Cis contractible. Let w(C) denote the weight of the circuit: that is, the sum of all the coordinates of all the ver tices inC. We show that, if Cis nontrivial, we can apply one of the above operations and reducew. Indeed, let vbe a vertex on Cwith maximal coordinate sum, andHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 15 suppose that v−andv+are the vertices before and after von the circuit. If v−=v+then we can apply the first operation removing vandv+from the circuit which obviously reduces w. If not, then both v−andv+have strictly smaller coordinate sums than v, and we can apply the second operation reducing wby two. We repeat the above until we reach the trivial circuit . Now, let Jbe the number of edges of Cwith an end in each of EandI. The first operation obviously does not change the parity of J. A simple finite check yields the same for the second operation. Indeed , assume that we are changing the path from ( x,y+1),(x+1,y+1),(x+1,y) to (x,y+ 1),(x,y),(x+1,y). LetFbe the set of these four vertices. If no vertex of Iis inF, then obviously Jdoes not change. If there is a vertex of IinF, then, by the definition of diagonally connected, F∩I=F∩∂1E. Hence the parity of Jdoes not change. [It is even if ( x,y+1) and ( x+1,y) are both inEor both in Ecand odd otherwise.] /square Supposethat thereis some vertex v∈∂1E\Iand that u∈Eis aneighbor ofv. Lety∈Iandx∈Ebeneighbors. Since EandEcare connected we can findpaths PxuandPvyinEandEc, respectively. Thecircuit Pxu,uv,Pvy,yx contains a single edge from EtoIwhich contradicts the claim. /square To complete this stage observe that Corollary 11holds as before. Stage 4: Dealing with the difficult subcubes, and Stage 5: Usin g the sub- cubes in the sea to join everything together. These two stages go through exactly as before [with one trivial change: replace (2 c+1)2by (2c+1)d]. This completes the proof of Theorem 2. 4. Proof of Theorem 3.In this section we prove Theorem 3. Once again, the proof is very similar to that in Section 2. We shall outline the key differences, and emphasise why we are only able to prove the wea ker version of the result. Stage 1: Tessellation. The tessellation is similar to before, but this time some edges may be much longer than some nonedges. Letk=H(Gisκ-connected) be the smallest kthatGn,kisκ-connected. SinceGis connected we may assume that 0 .3logn<k <0.52logn(see [1] and [2]). Letr−be such that any two points at distance r−are joined whp; for example, Lemma 8 of [ 1] implies that this is true provided πr2 −≤ 0.3e−1−1/0.3logn, so we can take r−=0.035√logn. Letr+be such that no edge in the graph has length more than r+. Then, again by Lemma 8 of [ 1], we have πr2 +≤4e(1+0.52)logn whp, so we can take r+=2.3√logn≤66r−. From here on, we ignore all point sets with an edge longer than r+or a nonedge shorter than r−.16 J. BALOGH ET AL. Lets=r−/√ 8. We tessellate the box Snwith small squares of side lengths. (Since we are proving only this weaker result our tesselati on does not need to be very fine.) By the choice of sand the bound on r−any two points in neighboring or diagonally neighbouring squares a re joined in G. Also, by the bound on r+no two points in squares with centers at distance more than (66√ 5+2)s<150sare joined. Let D=104; we have that no two points in squares with centers distance Dsapart are joined. Stage 2: The “difficult” subsquares. We call a square fullif it contains at leastM=109points and nonfullotherwise. We say two nonfull squares are joined if they are at ℓ∞distance at most 2 D−1. First we bound the size of the largest component of nonfull sq uares. Lemma 16. The largest component of nonfull squares has size less than 7000 whp. Proof. The number of connected subgraphs of /hatwideGof size 7000 con- taining a particular square is at most ( e(4D)2)7000, so, since there are less thannsquares, the total number of such connected subgraphs is at m ost n(e(4D)2)7000.Theprobabilitythatasquareisnonfullisatmost2 s2Me−s2/M!. Hence, the expected numberof components of nonfull squares of size at least 7000 is at most n(2s2Me−s2(e(4D)2)/M!)7000 ≤n/parenleftbigg 2/parenleftbigg(0.035)2logn 8/parenrightbiggMe(4D)2 M!/parenrightbigg7000 exp/parenleftbigg−7000(0.035)2logn 8/parenrightbigg , which tends to zero as ntends to infinity [since 7000( −0.035)2/8>1.07>1]; that is, whp, no such component exists. /square In the rest of the argument we shall assume that there is no non full component of size greater than 7000. Stage 3: The structure of the difficult subsquares. As usual we fix one component Nof the nonfull squares, and suppose that it has size u(so we knowu<7000). This time we define /hatwideGto be the graph on the small squares where each square is joined to its eight nearest neighbors (i .e., adjacent and diagonal). Let A=A(N) be the giant component of G\N, and again split the cutoff squares into close and far dependingwhether they h ave a neighbor (in/hatwideG) inA. By the vertex isoperimetric inequality in the square there a re at most u2/2 squares in Ac\Nso|Ac|≤u2/2+u<2.5·107. Next we prove a result similar to Corollary 7. Lemma 17. The set of cutoff squares Acis inND(whereD=104as above).HAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 17 Fig. 3. Two paths from one cutoff square to the sea together with the pa th from the meeting point in Q2to the square Q1. Proof. Suppose not, and that Qis a square in Acnot inND. Then all squares within ℓ∞distance of Qat mostDare not in N. Hence they must be inAc(since otherwise there would be a path from Qto a square in Anot going through any square in N). Hence |Ac|>D2=108which contradicts Lemma16./square Finally, we need the analogue of Lemma 9whose proof is exactly the same. Lemma 18. The setND∩Ais connected in /hatwideG. Stage 4: Dealing with the difficult subsquares. Let us deal with these cut- off squares now. From each cutoff square that contains at least two vertices, pick any 2 vertices, and from each cutoff square that contains a single vertex pick that vertex with multiplicity two. We have picked at mos t 5·107ver- tices, so since Gisκ=5·107connected we can simultaneously find vertex disjoint paths from each of our picked vertices to vertices i n squares in A (two paths from those vertices that are repeated). We remark that these are not just single edges; these paths ma y go through other cutoff squares. Call the first point of such a path which is in Aameeting point , and the square containing this point a meeting square. Fix a cutoff square and let v1,v2be the two vertices picked above from this square (let v1=v2if the square only contains one vertex). This cutoff square has two meeting points, say q1andq2in subsquares Q1andQ2, respectively. Sincethelongest edgeis at most r+, bothQ1andQ2areinND. SinceA∩NDis connected in /hatwideGwe construct a path in the squares in A∩ND from the meeting point in Q2to a vertex in Q1using at most one vertex in each subsquare on the way, and missing all the other meeting p oints. This is possible since each full square contains at least M=109vertices.18 J. BALOGH ET AL. Construct a path starting and finishing in Q1containing all the (unused) vertices in this cutoff square by joining together the follow ing paths: 1. the path from q1tov1; 2. a path starting at v1going round all points in the cutoff square finishing back atv2(omit this piece if there is just one far vertex); 3. the path v2toq2; 4. the path from q2through A∩NDback toQ1constructed above. Do this for every cutoff square. For each cutoff square this con struction uses at most two vertices from any square in A. Moreover, it obviously only touches squares in ND. Since nonfull squares in distinct components are at distance at least 2 Dthe squares touched by different nonfull components are distinct. Thus in total we have used at most 4 ·107vertices in any square in the sea, and since M=109there are many (we shall only need 8) unused vertices left in each full square in the sea. Stage 5: Using the subsquares in the sea to join everything to gether. This is exactly the same as before. 5. Comments on the k-nearest neighbor proof. We start by giving some reasonswhytheproofinthe k-nearestneighbormodelonlyyields theweaker Theorem 3. The first superficial problem is that we use squares in the tes se- lation which are of “large” size rather than relatively smal l as in the proof of Theorem 1, (in other words we did not introduce the constant cwhen settingsdepending on r). Obviously we could have introduced this constant. The difficu lty when trying to mimic the proof of Theorem 1is the large difference between r− andr+, which corresponds to having a very large number of squares ( many timesπc2) in our nonfull component N. This means that we cannot easily prove anything similar to Lemma 5. Indeed, a priori, we could have two far squares with πc2nonfull squares around each of them. A different way of viewing this difficulty is that, in the k-nearest neighbor model, the graph /hatwideGon the small squares does not approximate the real graphGvery well, whereas in the Gilbert model it is a good approxima tion. Thus, it is not surprising that we only prove a weaker result. This is typical of results about the k-nearest neighbor model; the results tend to be weaker than for the Gilbert model. This is primaril y because the obstructions tend to be more complex; for example, the ob struction for connectivity in the Gilbert model is the existence of an i solated vertex. Obviously in the k-nearest neighbor model we never have an isolated vertex; the obstruction must have at least k+1 vertices. Extensions of Theorem 3.When proving Theorem 3we only used two facts about the random geometric graph. First, that any two p oints at dis- tancer−=0.035√lognare joined whp. Secondly, that the ratio of r+(theHAMILTON CYCLES IN RANDOM GEOMETRIC GRAPHS 19 longest edge) to r−(the shortest nonedge) was at most 60 whp. Obviously, we could prove the theorem (with different constants) in any gr aph with r−=Θ(√logn) andr+/r−bounded. This includes higher dimensions and different norms and to different shaped regions instead of Sn(e.g., to disks or toruses). Indeed, the only place we used the norm was in obt aining the bounds on r+andr−in stage 1 of the proof. Indeed, it also generalizes to irregular distributions of v ertices provided that theabove boundson r−andr+hold.For example, it holdsinthesquare Snwhere the density of points in the Poisson Process decrease l inearly from 10 to 1 across the square. 6. Closing remarks and open questions. A related model where the re- sult does not seem to follow easily from our methods is the dir ected version of thek-nearest neighbor graph. As mentioned above, the k-nearest neigh- bor model naturally gives rise to a directed graph, and we can ask whether this has a directed Hamilton cycle. Note that this directed m odel is signif- icantly different from the undirected. For example, it is like ly (see [1]) that the obstruction todirected connectivity (i.e., theexiste nce of adirected path between any two vertices) is a single vertex with in-degree z ero; obviously this cannot occur in the undirected case where every vertex h as degree at leastk. In some other random graph models a sufficient condition for t he existence of a Hamilton cycle (whp) is that there are no verti ces of in-degree or out-degree zero. Of course, in thedirected k-nearest neighbor model every vertex has out-degree kso we ask the following question. Question. Let/vectorG=/vectorGn,kbe the directed k-nearest neighbor model. Is H(/vectorGhas a Hamilton cycle )=H(/vectorGhas no vertex of in-degree zero ) whp? It is obvious that the bound on connectivity in the k-nearest neighbor model can be improved, but the key question is “should it be tw o?” We make the following natural conjecture: Conjecture. Suppose that k=k(n)such that the k-nearest neighbor graphG=G(k,n)is a2-connected whp. Then, whp, Ghas a Hamilton cycle. Acknowledgments. Some of the results published in this paper were ob- tained in June 2006 at the Institute of Mathematics of the Nat ional Univer- sity of Singapore during the program “11 Random Graphs and Re al-world Networks.” J. Balogh, B. Bollob´ as and M. Walters are gratef ul to the Insti- tute for its hospitality. REFERENCES [1]Balister, P. ,Bollob´as, B.,Sarkar, A. andWalters, M. (2005). Connectivity of randomk-nearest-neighbour graphs. Adv. in Appl. Probab. 371–24.MR213515120 J. BALOGH ET AL. [2]Balister, P. ,Bollob´as, B.,Sarkar, A. andWalters, M. (2009). Highly con- nectedrandom geometric graphs. Discrete Appl. Math. 157309–320. MR2479805 [3]Balister, P. ,Bollob´as, B.,Sarkar, A. andWalters, M. (2009). A critical constant for the k-nearest neighbor model. Adv. in Appl. Probab. 411–12. MR2514943 [4]Balogh, J. ,Kaul, H. andMartin, R. A threshold for random geometric graphs with a Hamiltonian cycle. Unpublished Manuscript. [5]Bollob´as, B.(1984). The evolution of sparse graphs. In Graph Theory and Combi- natorics (Cambridge, 1983) 35–57. Academic Press, London. MR0777163 [6]Bollob´as, B.(2006).The Art of Mathematics: Coffee Time in Memphis . Cambridge Univ. Press, New York. MR2285090 [7]D´ıaz, J.,Mitsche, D. andP´erez, X. (2007). Sharp threshold for Hamiltonicity of random geometric graphs. SIAM J. Discrete Math. 2157–65 (electronic). MR2299694 [8]Koml´os, J.andSzemer´edi, E.(1983). Limit distribution for the existence of Hamil- tonian cycles in a random graph. Discrete Math. 4355–63.MR0680304 [9]Korshunov, A. D. (1977). A solution of a problem of P. Erd˝ os and A. R´ enyi abou t Hamilton cycles in non-oriented graphs. Metody Diskr. Anal. Teoriy Upr. Syst., Sb. Trudov Novosibirsk 3117–56 (in Russian). [10]Penrose, M. D. (1997). The longest edge of the random minimal spanning tree . Ann. Appl. Probab. 7340–361. MR1442317 [11]Penrose, M. D. (1999). On k-connectivity for a geometric random graph. Random Structures Algorithms 15145–164. MR1704341 [12]Penrose, M. (2003).Random Geometric Graphs .Oxford Studies in Probability 5. Oxford Univ. Press, Oxford. MR1986198 [13]Petit, J. (2001). Layout Problems. Ph.D. thesis, Univ. Polit` ecnica de Catalunya. [14]P´osa, L.(1976). Hamiltonian circuits in random graphs. Discrete Math. 14359–364. MR0389666 [15]Xue, F. andKumar, P. R. (2004). The number of neighbors needed for connectivity of wireless networks. Wireless Networks 10169–181. J. Balogh Department of Mathematics University of Illinois Urbana, Illinois 61801 USA E-mail: jobal@math.uiuc.eduB. Bollob ´as Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge, CB3 0WB United Kingdom E-mail: b.bollobas@dpmms.cam.ac.uk M. Krivelevich School of Mathematical Sciences Tel Aviv University Ramat Aviv 69978 Israel E-mail: krivelev@post.tau.ac.ilT. M¨uller Centrum voor Wiskunde en Informatica P.O. Box 94079 1090 GB Amsterdam The Netherlands E-mail: tobias@cwi.nl M. Walters School of Mathematical Sciences Queen Mary, University of London London, E1 4NS United Kingdom E-mail: M.Walters@qmul.ac.uk

2009-05-28

We prove that, in the Gilbert model for a random geometric graph, almost every graph becomes Hamiltonian exactly when it first becomes 2-connected. This answers a question of Penrose. We also show that in the k-nearest neighbor model, there is a constant \kappa\ such that almost every \kappa-connected graph has a Hamilton cycle.

Hamilton cycles in random geometric graphs

0905.4650v2

arXiv:0905.4779v2 [cond-mat.mtrl-sci] 15 Jun 2009Beaujour et al. Ferromagnetic resonance linewidth in ultrathin films with p erpendicular magnetic anisotropy J-M. Beaujour,1D. Ravelosona,2I. Tudosa,3E. Fullerton,3and A. D. Kent1 1Department of Physics, New York University, 4 Washington Pl ace, New York, New York 10003, USA 2Institut d’Electronique Fondamentale, UMR CNRS 8622, Universit ´eParis Sud, 91405 Orsay Cedex, France 3Center for Magnetic Recording Research, University of Cali fornia, San Diego, La Jolla, California 92093-0401, USA (Dated: June 15, 2009) Transition metal ferromagnetic films with perpendicular ma gnetic anisotropy (PMA) have ferro- magnetic resonance (FMR) linewidths that are one order of ma gnitude larger than soft magnetic materials, such as pure iron (Fe) and permalloy (NiFe) thin fi lms. A broadband FMR setup has been used to investigate the origin of the enhanced linewidt h in Ni |Co multilayer films with PMA. The FMR linewidth depends linearly on frequency for perpend icular applied fields and increases sig- nificantly when the magnetization is rotated into the film pla ne. Irradiation of the film with Helium ions decreases the PMA and the distribution of PMA parameter s. This leads to a great reduction of the FMR linewidth for in-plane magnetization. These resu lts suggest that fluctuations in PMA lead to a large two magnon scattering contribution to the lin ewidth for in-plane magnetization and establish that the Gilbert damping is enhanced in such mater ials (α≈0.04, compared to α≈0.002 for pure Fe). PACS numbers: 75.47.-m,85.75.-d,75.70.-i,76.50.+g Magnetic materials with perpendicular magnetic anisotropy (PMA) are of great interest in information storage technology, offering the possibility of smaller magnetic bits [1] and more efficient magnetic random ac- cess memories based on the spin-transfer effect [2]. They typically are multilayers of transition metals (e.g., Co |Pt, Co|Pd, Ni |Co) with strong interface contributions to the magnetic anisotropy [3], that render them magnetically hard. In contrast to soft magnetic materials which have been widely studied and modeled [4, 5, 6, 7], such films are poorly understood. Experiments indicate that there are large distributions in their magnetic characteristics , such as their switching fields [1]. An understanding of magnetization relaxation in such materials is of particu- lar importance, since magnetization damping determines the performance of magnetic devices, such as the time- scale for magnetization reversal and the current required for spin-transfer induced switching [2, 8]. Ferromagnetic resonance (FMR) spectroscopy pro- vides information on the magnetic damping through study of the linewidth of the microwave absorption peak, ∆H, when the applied field is swept at a fixed microwave frequency. FMR studies of thin films with PMA show very broad linewidths, several 10’s of mT at low frequen- cies (/lessorsimilar10 GHz) for polycrystalline alloy [9], multilayer [10] and even epitaxial thin films [11]. This is at least one order of magnitude larger than the FMR linewidth found for soft magnetic materials, such as pure iron (Fe) and permalloy (FeNi) thin films [5]. Further, it has recently been suggested that the FMR linewidth of perpendicu- larly magnetized CoCrPt alloys cannot be explained in terms of Landau-Lifshitz equation with Gilbert damping [12], the basis for understanding magnetization dynamicsin ferromagnets: ∂M ∂t=−γµ0M×Heff+α MsM×∂M ∂t. (1) HereMis the magnetization and γ=|gµB//planckover2pi1|is the gy- romagnetic ratio. The second term on the right is the damping term, where αis the Gilbert damping constant. This equation describes precessional motion of the mag- netization about an effective field Heff, that includes the applied and internal (anisotropy) magnetic fields, which is damped out at a rate determined by α. The absorp- tion linewidth (FWHM) in a fixed frequency field-swept FMR experiment is given by µ0∆H= 4παf/γ , i.e., the linewidth is proportional to the frequency with a slope determined by α. This is the homogeneous or intrinsic contribution to the FMR linewidth. However, experi- ments show an additional frequency independent contri- bution to the linewidth: ∆H= ∆H0+4πα µ0γf, (2) where ∆H0is referred as the inhomogeneous contribution to the linewidth. The inhomogeneous contribution is associated with disorder. First, fluctuations in the materials magnetic properties, such as its anisotropy or magnetization, lead to a linewidth that is frequency independent; in a simple picture, independent parts of the sample come into res- onance at different applied magnetic fields. Second, dis- order can couple the uniform precessional mode ( k= 0), excited in an FMR experiment, to degenerate finite- k (k/negationslash= 0) spin-wave modes. This mechanism of relaxation of the uniform mode is known as two magnon scattering2 /s48 /s51/s48 /s54/s48 /s57/s48/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s48 /s49/s40/s97/s41/s40/s98/s41 /s72 /s72/s32/s61/s57/s48/s111/s86/s105/s114/s103/s105/s110 /s73/s114/s114/s97/s100/s105/s97/s116/s101/s100 /s32/s32 /s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101/s32 /s72/s32/s32/s40/s100/s101/s103/s46/s41/s50/s48/s32/s71/s72/s122/s32/s72 /s114/s101/s115/s32/s32/s40/s32/s84/s32/s41/s120/s121/s122 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s48/s52/s48/s54/s48 /s77 /s115 /s72/s32/s32/s102/s32/s32/s40/s71/s72/s122/s41 /s48/s72 /s114/s101/s115/s32/s32/s32/s32/s40/s32/s84/s32/s41/s32/s48/s72 /s32/s32/s40/s84/s41/s32/s70/s77 /s82/s32/s115/s105/s103/s110/s97/s108 FIG. 1: a) The frequency dependence of the resonance field with the applied field perpendicular to the film plane. The solid lines are fits using Eq. 4. Inset: FMR signal of the virgin and irradated films at 21 GHz. b) The resonance field as a function of applied field angle at 20 GHz. The solid lines are fits to the experimental data points. The inset shows the field geometry. (TMS) [13]. TMS requires a spin-wave dispersion with finite-kmodes that are degenerate with the k= 0 mode that only occurs for certain magnetization orientations. In this letter we present FMR results on ultra-thin Ni|Co multilayer films and investigate the origin of the broad FMR lines in films with PMA. Ni |Co mul- tilayers were deposited between Pd |Co|Pd layers that enhance the PMA and enable large variations in the PMA with Helium ion irradiation [14]. The films are |3nm Ta |1nm Pd |0.3nm Co |1nm Pd |[0.8nm Ni |0.14nm Co]×3|1nm Pd |0.3nm Co |1nm Pd |0.2nm Co |3nm Ta |de- posited on a Si-SiN wafers using dc magnetron sputter- ing and were irradiated using 20 keV He+ions at a flu- ence of 1015ions/cm2. The He+ions induce interatomic displacements that intermix the Ni |Co interfaces lead- ing to a reduction of interface anisotropy and strain in the film. The magnetization was measured at room tem- perature with a SQUID magnetometer and found to be Ms≃4.75×105A/m. FMR studies were conducted from 4 to 40 GHz at room temperature with a coplanar waveguide as a function of the field angle to the film plane. The inset of Fig. 1b shows the field geometry. The parameters indexed with ‘⊥’ (perpendicular) and ‘ /bardbl’ (parallel) refer to the applied field direction with respect to the film plane. The absorp- tion signal was recorded by sweeping the magnetic field at constant frequency [15]. FMR measurements were per- formed on a virgin film (not irradiated) and on an irra- diated film. Fig. 1a shows the frequency dependence of the reso- nance field when the applied field is perpendicular to the film plane. The x-intercept enables determination of the PMA and the slope is proportional to the gyromagneticratio. We take a magnetic energy density: E=−µ0M·H+1 2µ0M2 ssin2φ −(K1+ 2K2)sin2φ+K2sin4φ.(3) The first term is the Zeeman energy, the second the mag- netostatic energy and the last two terms include the first and second order uniaxial PMA constants, K1andK2. Takingµ0Heff=−δE/δMin Eq. 1 the resonance con- dition is: f=γ 2π/parenleftbigg µ0H⊥ res−µ0Ms+2K1 Ms/parenrightbigg . (4) From thex-intercepts in Fig. 1a, K1= (1.93±0.07)× 105J/m3for the virgin film and (1 .05±0.02)×105J/m3 for the irradiated film; Helium irradiation reduces the magnetic anisotropy by a factor of two. Note that in the irradiated film the x-intercept is positive ( µ0Ms> 2K1/Ms). This implies that the easy magnetization di- rection is in the film plane. The angular dependence of Hres(Fig. 1b) also illustrates this: the maximum res- onance field shifts from in-plane to out-of-plane on ir- radiation. The gyromagnetic ratio is not significantly changedγ= 1.996±0.009×10111/(Ts) for the vir- gin film and γ= 1.973±0.004×10111/(Ts) for the irradiated film (i.e., g= 2.24±0.01). The second order anisotropy constant K2was obtained from the angular dependence of the resonance field, fitting HresversusφH for magnetization angle φbetween 45oand 90o. For the virgin film, K2= 0.11×105J/m3. Note that when K2 is set to zero, χ2of the fit increases by a factor 30. For the irradiated film, K2= 0.03×105J/m3. HenceK2de- creases upon irradiation and remains much smaller than K1. The solid line in Fig. 1b is the resulting fit. When the field approaches the in-plane direction, the measured resonance field is higher than the fit. The shift is of the order of 0.1 T for the virgin film and 0 .025 T for the ir- radiated film. It is frequency dependent: increasing with frequency. This shift will be discussed further below. Fig. 2a shows the frequency dependence of the linewidth (FWHM) for two directions of the applied field. /s48/s49/s48/s48/s50/s48/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s48/s53/s48/s49/s48/s48/s72/s32 /s124/s124 /s32/s32/s32 /s72/s32 /s32 /s32/s40/s97/s41 /s32/s86/s105/s114/s103/s105/s110/s72/s32 /s32/s40/s109/s84/s41 /s32 /s102/s32/s32/s40/s71/s72/s122/s41/s40/s98/s41 /s32/s73/s114/s114/s97/s100/s105/s97/s116/s101/s100 /s72 /s84/s77/s83 FIG. 2: The frequency dependence of the FMR linewidth with applied field in-plane and perpendicular to the plane. The solid black lines are linear fits that enable determination o fα and ∆ H0from Eq. 2. The dotted lines show the linewidth from TMS and the red lines is the total linewidth.3 /s48/s49/s48/s48 /s48 /s51/s48 /s54/s48 /s57/s48/s48/s49/s48/s48/s72 /s32/s32/s32/s32/s32 /s72 /s105/s110/s104/s32 /s72 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s72 /s84/s77/s83/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s40/s98/s41 /s32/s48/s72 /s32/s40/s32/s109/s84/s32/s41/s40/s97/s41 /s32/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101/s32 /s72/s32/s32/s40/s32/s100/s101/s103/s46/s32/s41 /s32 FIG. 3: Angular dependence of the linewidth at 20 GHz for (a) the virgin and (b) the irradiated film. The solid line (∆H) is a best fit of the data that includes the Gilbert damp- ing (∆ Hα) and the inhomogeneous (∆ Hinh) contributions. Linewidth broadening from TMS (∆ HTMS) is also shown. The total linewidth is represented by the red line. ∆H⊥of the virgin film increases linearly with frequency consistent with Gilbert damping. Fitting to Eq. 2, we findα= 0.044±0.003 andµ0∆H⊥ 0= 15.6±3.6 mT. When the field is applied in the film plane, the linewidth is significantly larger. ∆ H/bardbldecreases with increasing frequency for f≤10 GHz and then is practically inde- pendent of frequency, at ≈140±20 mT. However, for the irradiated film, the linewidth varies linearly with fre- quency both for in-plane and out-of-plane applied fields, with nearly the same slope. The Gilbert damping is α= 0.039±0.004. Note that µ0∆H/bardbl 0is larger than µ0∆H⊥ 0by about 15 mT. The angular dependence of the linewidth at 20 GHz is shown in Fig. 3. The linewidth of the virgin film de- creases significantly with increasing field angle up 30o, and then is nearly constant, independent of field angle. The linewidth of the irradiated film is nearly indepen- dent of the field angle, with a relatively small enhance- ment of ∼15 mT close to the in-plane direction. We fit this data assuming that the inhomogeneous broad- ening of the line is associated mainly with spatial vari- ations of the PMA, specifically local variation in K1, ∆Hinh.(φH) =|∂Hres/∂K1|∆K1. ∆K1= 4×103J/m3 for the virgin film and 3 ×102J/m3for the irradiated film, which corresponds to a variation of K1of 2% and 0.3% respectively. Including variations in K2and anisotropy field direction do not significantly improve the quality of the fit. Such variations in K1produce a zero fre- quency linewidth in the perpendicular field direction, µ0∆H⊥ 0= 16.8 mT, in excellent agreement with linear fits to the data in Fig. 2. However, the combination of inhomogeneous broadening and Gilbert damping can- notexplain the enhanced FMR linewidth observed for in-plane applied fields. The enhanced linewidth observed with in-plane applied fields is consistent with a significant TMS contribution to the relaxation of the uniform mode–the linewidth is en- hanced only when finite- kmodes equi-energy with theuniform mode are present. We derive the spin-wave dis- persion for these films following the approach of [16]: ω2 k=ω2 0−1 2γ2µ0Mskt(Bx0(cos2φ + sin2φsin2ψk)−By0sin2ψk) +γ2Dk2(Bx0+By0), (5) where: Bx0=µ0Hcos(φH−φ)−µ0Meffsin2φ By0=µ0Hcos(φH−φ) +µ0Meffcos2φ +2K2 Mssin22φ.(6) The effective demagnetization field is µ0Meff= (µ0Ms− 2K1 Ms−4K2 Mscos2φ).ω0=γ/radicalbig Bx0By0is the resonance frequency of the uniform mode. Dis the exchange stiff- ness andtis the film thickness. ψkis the direction of propagation of the spin-wave in the film plane relative to the in-plane projection of the magnetization. The in- set of Fig. 4 shows the dispersion relation for the virgin and the irradiated film for an in-plane applied field at 20 GHz. For the virgin film, with the easy axis normal to the film plane ( M/bardbl eff<0) there are degenerate modes available in all directions in k-space. For the irradiated film (M/bardbl eff>0) degenerate modes are only available when ψk/lessorsimilar74o. The spin waves density of states, determined from Eq. 5, is shown as a function of field angle in Fig. 4 at 20 GHz. The DOS of the virgin film is two times larger than that of the irradiated film at φH= 0. Note that for both films, the DOS vanishes at a critical field angle that corresponds to a magnetization angle φ= 45o. For the virgin film, the enhancement of ∆ Hoccurs atφH≃30o (Fig. 3a), at the critical angle seen in Fig. 4. The TMS linewidth depends on the density of states and the disorder, which couples the modes: ∆HTMS=/parenleftbigg∂Hres ∂ω/parenrightbigg|A0|2 2π/integraldisplay Ck(ξ)δ(ωk−ω0)dk,(7) whereA0is a scattering amplitude. Ck(ξ) = 2πξ2/(1 + (kξ)2)3/2is a correlation function, where ξis correlation length, the typical length scale of disorder. Eq. 7 is valid in the limit of weak disorder. We assume that the disorder of our films is associated with spatial variations of the PMA, K1. Then the mag- netic energy density varies as ∆ E(/vector r) =−k1(/vector r)M2 y/M2 s, and the scattering probability is [17]: |A0|2=γ4 4ω2 0(B2 x0sin4φ+B2 y0cos22φ −2(ω0/γ)2sin2φcos2φ)/parenleftbigg2∆k1 Ms/parenrightbigg2 .(8) ∆k1is the rms amplitude of the distribution of PMA, k1(r). Therefore the TMS linewidth broadening scales4 /s48 /s51/s48 /s54/s48 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s50/s48 /s48 /s53/s50/s48 /s32/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115/s32/s32/s40/s97/s46/s117/s41 /s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101 /s72/s32/s32/s40/s100/s101/s103/s46/s41/s107/s61/s57/s48/s111 /s107/s61/s48/s111/s32/s32 /s32/s32/s102/s32/s32 /s40/s71/s72/s122/s41 /s32 /s73/s114/s114/s97/s100/s105/s97/s116/s101/s100 /s32 /s107 /s32/s40/s49/s48/s53 /s32/s114/s97/s100/s47/s99/s109/s41/s86/s105/s114/s103/s105/s110 FIG. 4: The density of spin-waves states degenerate with the uniform mode as a function of field angle at 20 GHz for the virgin film (solid line) and the irradiated film (dashed-dott ed line). Inset: Spin wave dispersion when the dc field is in the film plane. as the square of ∆ k1. Since the variations in PMA of the virgin film are larger than that of the irradiated film the linewidth broadening from the TMS mechanism is ex- pected to be much larger in the virgin film, qualitatively consistent with the data. A best fit of the linewidth data to the TMS model is shown in Fig. 3a. For the virgin film, we find ξ≈44 nm, approximately four times the film grain size, and ∆ k1= 9×103J/m3. The exchange stiffness, D= 2A/µ0Mswith the exchange constant A= 0.83×10−11J/m, is used in the fittings. The cut-off field angle for the enhancement of the field linewidth agrees well with the data (Fig. 3a). For the irradiated film, a similar analysis gives: ξ= 80± 40 nm and ∆ k1= (4±2)×103J/m3. TMS is also expected to shift the resonance position [17]. For applied fields in-plane and f= 20 GHz we estimate the resonance field shift to be ≈33 mT. This is smaller than what is observed experimentally ( ≈93 mT). The deviations of the fits in Fig. 1b may be associated with an anisotropy in the gyromagnetic ratio, i.e. a g that is smaller for Min the film plane. Note that if we assume that the g-factor is slightly anisotropic ( ∼1%), we can fit the full angular dependence of the resonance field of the irradiated film. We note that the TMS model cannot explain the en- hanced linewidth for small in-plane applied fields for the virgin film (Fig. 2a). The FMR linewidth increases dramatically when the frequency and resonance field de- creases. When the applied in-plane field is less than the effective demagnetization field ( −µ0M|| eff= 0.31 T) the magnetization reorients out of the film plane. For fre- quencies less than about 8 GHz this leads to two resonant absorption peaks, one with the magnetization having an out-of-plane component for Hres<−M|| effand one with the magnetization in-plane for Hres>−M|| eff. It may be that these resonances overlap leading to the enhanced FMR linewidth.In sum, these results show that the FMR linewidth in Ni|Co multilayer films is large due to disorder and TMS as well as enhanced Gilbert damping. The latter is an intrinsic relaxation mechanism, associated with magnon- electron scattering and spin-relaxation due to spin-orbit scattering. As these materials contain heavy elements such as Pd and short electron lifetimes at the Fermi level, large intrinsic damping rates are not unexpected. The re- sults indicate that the FMR linewidth of Ni |Co multilay- ers can be reduced through light ion-irradiation and fur- ther demonstrate that the Gilbert damping rate is largely unaffected by irradiation. These results, including the re- duction of the PMA distribution at high irradiation dose, have important implications for the applications of PMA materials in data storage and spin-electronic application s which require tight control of the anisotropy, anisotropy distributions and resonant behavior. ACKNOWLEDGMENTS We thank Gabriel Chaves for help in fitting the data to the TMS model. This work was supported by NSF Grant No. DMR-0706322. [1] T. Thomson, G. Hu, and B. D. Terris, Phys. Rev. Lett. 96, 257204 (2006). [2] S. Mangin et al., Nature Mater. 5, 210 (2006). [3] G. H. O. Daalderop, P. J. Kelly, and F. J. A. den Broeder, Phys. Rev. Lett. 68, 682 (1992). [4] B. Heinrich, Ultrathin Magnetic Structures III (Springer, New York, 2005), p. 143. [5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007). [6] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett.99, 027204 (2007). [7] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 101, 037207 (2008). [8] J. Z. Sun, Phys. Rev. B 62, 570 (2000). [9] T. W. Clinton et al., J. Appl. Phys. 103, 07F546 (2008). [10] S. J. Yuan et al., Phys. Rev. B 68, 134443 (2003). [11] J. BenYoussef et al., J. Magn. Magn. Mater. 202, 277 (2003). [12] N. Mo et al., Appl. Phys. Lett. 92, 022506 (2008). [13] M. Sparks, Ferromagnetic-Relaxation Theory (McGraw- Hill, 1964). [14] D. Stanescu et al., J. Magn. Magn. Mater. 103, 07B529 (2008). [15] J.-M. L. Beaujour et al., Eur. Phys. J. B 59, 475 (2007). [16] P. Landeros, R. E. Arias, and D. L. Mills, Phys. Rev. B 77, 214405 (2008). [17] R. D. McMichael and P. Krivosik, IEEE Trans. Magn. 40, 2 (2004).

2009-05-29

Transition metal ferromagnetic films with perpendicular magnetic anisotropy (PMA) have ferromagnetic resonance (FMR) linewidths that are one order of magnitude larger than soft magnetic materials, such as pure iron (Fe) and permalloy (NiFe) thin films. A broadband FMR setup has been used to investigate the origin of the enhanced linewidth in Ni$|$Co multilayer films with PMA. The FMR linewidth depends linearly on frequency for perpendicular applied fields and increases significantly when the magnetization is rotated into the film plane. Irradiation of the film with Helium ions decreases the PMA and the distribution of PMA parameters. This leads to a great reduction of the FMR linewidth for in-plane magnetization. These results suggest that fluctuations in PMA lead to a large two magnon scattering contribution to the linewidth for in-plane magnetization and establish that the Gilbert damping is enhanced in such materials ($\alpha \approx 0.04$, compared to $\alpha \approx 0.002$ for pure Fe).

Ferromagnetic resonance linewidth in ultrathin films with perpendicular magnetic anisotropy

0905.4779v2

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/B4/BE/BC/BC/BC/B5/BA/CJ/BK℄ /C2/BA /BV/BA /CB/CP/D2/CZ /CT/DD /B8 /C1/BA /C6/BA /C3/D6/CX/DA /D3/D6/D3/D8/D3 /DA/B8 /CB/BA /C1/BA /C3/CX/D7/CT/D0/CT/DA/B8 /C8 /BA /C5/BA /BU/D6/CP/B9/CV/CP/D2 /CP/B8 /C6/BA /BV/BA /BX/D1/D0/CT/DD /B8 /CA/BA /BT/BA /BU/D9/CW/D6/D1/CP/D2/B8 /CP/D2/CS /BW/BA /BV/BA /CA/CP/D0/D4/CW/B8/C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BE /B8 /BE/BE/BG/BG/BE/BJ /B4/BE/BC/BC/BH/B5/BA /CJ/BL℄ /BW/BA /C0/D3/D9/D7/D7/CP/D1/CT/CS/CS/CX/D2/CT/B8 /CD/BA /BX/CQ /CT/D0/D7/B8 /BU/BA /BW/CX/CT/D2 /DD /B8 /C3/BA /BZ/CP/D6/CT/D0/D0/D3/B8 /C2/BA/B9/C8 /BA /C5/CX /CW/CT/D0/B8 /BU/BA /BW/CT/D0/CP/CT/D8/B8 /BU/BA /CE/CX/CP/D0/CP/B8 /C5/BA/B9/BV/BA /BV/DD/D6/CX/D0/D0/CT/B8 /C2/BA /BT/BA /C3/CP/B9/D8/CX/D2/CT/B8 /CP/D2/CS /BW/BA /C5/CP/D9/D6/CX/B8 /C8/CW /DD/D7/CX /CP/D0 /CA/CT/DA/CX/CT/DB /C4/CT/D8/D8/CT/D6/D7 /BD/BC/BE /B8 /BE/BH/BJ/BE/BC/BE/B4/D4/CP/CV/CT/D7 /BG/B5 /B4/BE/BC/BC/BL/B5/BA/CJ/BD/BC℄ /C2/BA /C3/CP/D8/CX/D2/CT /CP/D2/CS /BX/BA /BX/BA /BY /D9/D0/D0/CT/D6/D8/D3/D2/B8 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2009-08-04

Combining multiple ultrafast spin torque impulses with a 5 nanosecond duration pulse for damping reduction, we observe time-domain precession which evolves from an initial 1 ns duration transient with changing precessional amplitude to constant amplitude oscillations persisting for over 2 ns. These results are consistent with relaxation of the transient trajectories to a stable orbit with nearly zero damping. We find that in order to observe complete damping cancellation and the transient behavior in a time domain sampling measurement, a short duration, fast rise-time pulse is required to cancel damping without significant trajectory dephasing.

Time domain detection of pulsed spin torque damping reduction

0908.0481v1

Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 1 Nonlinear viscoelastic wave propagation: an extension of Nearly Constant Attenuation (NCQ) models. Nicolas Delé pine1, Luca Lenti2, Guy Bonnet3, Jean -François Semblat, A.M.ASCE4 Subject headings : Inelasticity; Viscoelasticity; Damping ; Wave propagation ; Earthquake engineering ; Ground motion ; Nonlinear response ; Finite element method . Abstract Hysteretic damping is often model ed by means of linear viscoelastic approaches such as “nearly constant Attenuation (NCQ) ” models. These models do not take into acco unt nonlinear effects either on the stiffness or on the damping, which are well known features of soil dynamic behavior . The aim of this paper is to propose a mechanical model involving nonlinear viscoelastic behavior for isotropic materials. This model simultaneously takes into account nonlinear elasticity and nonlinear damping. On the one hand, t he shear modulus is a f unction of the excitation level ; on the other , the description of viscosity is based on a general ized Maxwell body involving non -linearity . This formulation is implemented in to a 1D finite element approach for a dry soil. The validation of the model shows its ability to retrieve low amplitude ground motion response. For larger excitation levels, the analysis of seismic wave propagation in a n onlinear soil layer over an elastic bedrock leads to results which are physically satisfactory (lower amplitudes, larger time delays, higher frequency content) . 1 Introduction The analysis of seismic wave propagation in alluvial basins is complex since vario us phenomena are involved at different scales (Semblat and Pecker, 2009) : resonance at the scale of the wh ole basin (Bard and Bouchon, 1985; Paoluci, 1999; Semblat et al., 2003), surface waves generation at the basin e dges (Bard and Riepl -Thomas, 2000; Bozzano et al., 2008; Kawase, 2003; Moeen -Vaziri and Trifunac, 1988 ; Semblat et al. , 2000, 2005; Sánchez -Sesma and Luzón, 1995 ), soil nonlinear behavior at the geote chnical scale ( Bonilla et al., 2006 ; Iai et al., 1995; Kramer, 1996 ). Handling these different features of seismic wave propagation at the same time may be important because the interaction between, for instance, surface wave gener ation and shear modulus degradation may be significant . The impact on the amplification process could th us be very larg e and complex . Nonlinear constitutive equations are very important in the case of strong ground moti on since the mechanical behavior of many soils depend s on the excitation level and on the loading history. In this work, the attention is focused on the asp ects of nonlinear behavior of dry isotropic soils submitted to dynamic loadings. Various approaches are available to model the dependence of the mechanical features of soils on the excitation level: equivalent linear model and non linear cyclic constitutive equations (including plasticity). The equivalent linear model approximat es the problem in the linear range using an iterative proce dure (Schnabel et al., 1972) . Since this model leads to over -damped higher frequency components, r ecent researches improved it by introducing both frequency or mean stress dependencies of the soil properties (Sugito, 1995; Kausel and Assimaki, 2002). Several 1 Université Paris -Est, LCPC, presently at: IFP , 1 & 4 av. de Bois -Préau , 92852 Rueil -Malmaison Cedex, France, nicolas.delepine@ifp.fr 2 Université Paris -Est, LCPC, 58 bd Lefebvre, 75732 Paris Cedex 15, France, lenti@lcpc.fr 3 Université Paris -Est, Champs sur Marne, France, bonnet@univ -paris -est.fr 4 Université Paris -Est, LCPC, 58 bd Lefebvre, 75732 Paris Cedex 15, France, semblat@lcpc.fr (correspond . author) Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 2 comparisons involving such models were proposed by Bonilla et al. (2006) and Kwok et al. (2008) . Concerning nonlinear ity, some models are based on both the “hyperbolic law” , for describing shear modulus reduction cu rves, and on the Masing criterion (Masing, 1926) for the description of unloading and reloading phases . Such models have been widely developed (Matasovic, 1993; M atasovic and Vucetic, 1995). However, these models generally need large computational efforts and often lack of a strong mechanical basis , e.g. thermodynamics (Lemaître and Chaboche, 1992 ). Some other models are fully el astoplastic (Aubry et al., 1982 ; Prevost, 1985; Gyebi and Dasgupta , 1992 ) or include dependence on confining pressure (Hashash and Park, 2001; Park and Hashash, 2004) and pore pressure (Bonilla et al., 2005) . However , their use for large scale wave propagation analyses is limited as a conseq uence of the large number of parameters needed and the frequency/wavelength range to investigate . In this paper, a 3D nonlinear viscoelastic model is proposed. This model simultaneously follows a nonlinear elastic law and a nonlinear viscous law to investi gate the ground response to strong seismic excitation . 2 Mechanical formulation of the model 2.1 3D linear viscoelasticity 2.1.1 General formulation The 3D formulation of the viscoelastic model sta rts from the following relation : ij=sij+pij (1) where ij, sij, ij and p are the Cauchy stress tensor, the deviatoric stress tensor, the Kronecker unit tensor and the volumetric tension respectively . For an isotropic material , we can write : p=K ekk (2) where K and ekk are the bulk modulus and the volumetric strain respectively . The relation between the components of the deviatoric stress tensor s and the shear deviatoric strain tensor e in the case of linear viscoelasticity is formulated in the frequency domain as simply as: sij ()=2M()eij() (3) sij(), eij() are the Fourier transforms of the components of the deviatoric stress and strain tensors. M() is the complex -valued , frequency -dependent, viscoelastic modulus from which we can define the specific attenuation Q-1 in the following way (Bourbié et al. , 1987 ; Semblat and Pecker, 2009 ): 2=Q-1() Im(M())/Re(M()) (4) where is the damping ratio and Re and Im are the real and imaginary parts of a complex variable (resp.). 2.1.2 NCQ models This family of models is defined in term of the quality factor Q. A nearly constant Q in a broad frequency range and for a given strain level is introduced . Biot ( 1958) first demonstrated t hat a causal form of hysteretic damping can be simulated by viscoelastic cell s in parallel. Liu et al. (1976) construct ed such model s by direct superposition of Zener cells (standard solid) . Emmerich and Korn (1987 ) improved and extended the Padé approxima tion (Day and Minster, 1984) by considering a generalized n-cells Maxwell body (Fig. 1, left) . Mozco and Kristek (2005) pro ved the equivalence of the models of Liu and Emmerich and Korn. The implementation proposed by Emmerich and Korn is used in the follo wing . Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 3 Rt() MU MRM t=0 timeaMl a M/llM Fig. 1. Generalized Maxwell body with viscosities l lMa/. and elastic moduli Mal. for each rheological cell (left). Typical relaxation function R(t) (right) with MU the unrelaxed modulus and MR=M U-M the relaxed modulus. The generalized Maxwell model leads to the frequency dependent complex modulus (variables with bracket are not tensorial): n lln ll l l U yi y M M 1)0,(1)( )()0,( 1) /( 1 )( (5) MU is the unrelaxed (instantaneous) modulus and MR is the relaxed (long term) modulus (Fig. 1, right ). The y(l,0) variabl es characteri ze the rheological model and are calculated by means of an optimization method in order to obtain a nearly constant attenuation in a given frequency range (see Appendix ). Using Eqs. (4) and (5), the quality factor has the following expression: n l ll ln l ll l yy MMQ 12 )(2 )( )0,(12 )()( )0,( 1 )/(1)/(1)/(1/ )( Re)( Im)( (6) The )(l frequencies characterize each individual rheological cell (see Appendix) . The constitutive equations for the linear viscoelastic model are thus: n ll ij U ij t te M ts 1)()( )( 2)( (7) and )( 1)( )( 1)0,()0,( )( )()( )( te yyt tij n lll l l l l (8) where (l)(t) are rela xation parameters physically related to the anelastic deformation of the lth- cell (Fig. 1, left) . Fig. 2 displays the attenuation curve (a), 2=Q-1, and the phase velocity (b), Vph, as functions of frequency. These graphs are obtained considering 3 Zener’s cells which are equivalent to generalized Maxwell cells (Fig. 1, left). The attenuation is nearly constant, 2=Q-1=0.05 , in the frequency range 0.1 -10Hz . Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 4 00.010.020.030.040.050.06 180190200210220phase velocity (m/s ) atten uation Q-1 f req ue ncy (Hz )10-210-1100101102 Fig. 2. Generalized Maxwell body appro ximating a Nearly Constant Quality factor Q 20 (top), in the frequency range 0.1 -10Hz, and corresponding phase velocity V ph (bottom ). The target phase velocity, |V ph|=200m/s, is chosen at a frequency of 1Hz. 2.2 3D nonlinear viscoelastic model 2.2.1 Principles of th e nonlinear model In order to describe the soil’s shear modulus and damping variations with the excitation level, an elastic potential function and a dissipation function depending on the magnitude of the second invariant of the strain tensor are introduce d. The description of viscosity is based on a Nearly Constant Attenuation model able to fulfil the causality principle for seismic wave propagation (dispersive materials). Owing to the frequent use of this model within the geophysical community, it is usua lly called Nearly Constant Quality Factor (or “NCQ”) model. At the same time, it leads to a constant value of the damping factor at low strains over a broad frequency range of engineering interest (Kjartansson, 1979). The model is well -adapted to time doma in formulations (some alternative numerical strategies are available ( Carcione et al., 2002; Munjiza et al., 1998 ; Semblat, 1997 )). In the NCQ model, we introduce a dependence on the excitation level in order to consider an increasing damping ratio suggest ed from earthquakes records and geotechnical data (Iai et al ., 1995; Vucetic, 1990). This dependence is controlled during the 3D stress -strain path by the variation of the second order invariant of the strain tensor. 2.2.2 Formulation of the e xtended NCQ model (“X-NCQ”) To account for non linear behavior of soils in the case of any 3D stress -strain path, Eq. (7) is extended as fo llows : n ll l ij U ij Jyt teJ M ts 12 )( )( 2 ))(,( )()( 2)( (9) where J2 is the second invariant o f the deviatoric strain tensor, defined from the following relations: 32' 1 ' 2 2II J (10) with the 2 first invariants of the strain tensor: )(' 1 traceI (11) and Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 5 )(212 ' 2 trace I (12) In addition, the shear modulus is assumed to change during the global stress -strain path according to the following relation : )( 1 )(2 0, 2 J M J MU U (13) with 21 221 2 2 1)( JJJ (14) and where MU,0 denotes the unrelaxed modulus characterizing the instantaneous response of the soil at small strains and is a parameter quantifying its nonlinear behavior for larger strains. The octahedral strain oct is now introduced : 21 22Joct (15) It leads to ) ( 1 ) (0, oct U oct U M M (16) where: 2/ 12/) ( octoct oct (17) Such a dependence of the nonlinear elastic modulus on the octahedral strain also implies a strain dependence for the variables y(l) and (l). Determination of damping ratio has been perform ed by Strick (1967) using wave propagation measurements. Formulations for the dependence of the damping ratio on the shear strain modulus have been proposed by Hardin and Drnevich (1972 ). In the case of 3D loadings, di fferent authors (El Hosri et al. , 1984; Heitz, 1992; Bonnet and Heitz , 1994) proposed an extension of , such as: ) () ( ) (oct 0 max 0 oct (18) where 0andmaxcharacterize the dissipated energy in the small and larger strain range s respectively . Typical MU()=G() and ()curves are proposed in Fig. 3. The damping ratio and the attenuation Q-1 are now related by: ) (21 oct Q (19) Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 6 10-510-410-310-210-100.20.40.60.81 shear strain 00 . 0 50 . 10 . 1 50 . 20 . 2 5 G( )/G0 Fig. 3. Typical nonlinear dynamic properties of soils: shear modulus reduction (solid) and damping increase (dashed) with increasing shear strain. 2.2.3 Features of the extended NCQ model In paragraph 2.1.1 , the solution of Eq. (9) in the limit of low excitation levels has been found. For low octahedral strain s, we can consider that: 01 02Q (20) and 0 0, )0 ( G M MU oct U (21) For every other value of the induced strain, the Q-1 facto r increases with strain according to Eq. (19). This change has no influence on the frequenc y range in which Q-1 is constant. In other words, in Eq. (6) only the variables y(l,0) change to account for the variation of the damping with strain. We therefore introduce a strain variation of the variables y(l) with strain in the following form : )0,( )( ) () (l oct oct l y c y (22) Using Eqs. (4), (15) and (17), for every level of induc ed octahedral strain, Eqs. (5) and (34) can be rewritten in the following form , respectively : n ll octn ll l l oct oct U oct y ci y c M M 1)0,(1)( )()0,( ) (1) /( ) ( 1) ( ) ,( n l ll l oct oct y c Q 12 )()( )0,(1 / 1/) () ,( (23) (24) where, using Eq. (18), c(|oct|) is given by: ) ( 1) ( ) ,() ( 00 max 01 01 octoct oct octQQc (25) For every level of induced octahedral strain, Eq. (8) can be written in the more general form : )( ) (1) ()( )( 1)0,()0,( )( )()( )( te y cy ct tij n ll octl oct l l l l (26) The latter expression and Eq. (16) are used to solve Eq. (9) in the time domain. Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 7 2.3 Synthesis : 1D case For a unidirectional propagating shear wave, |oct| is equal to 2 ||, where is the shear strain. Equation (16) can be written in the form: 1)( )(0GG MU (27) In this case, Eq. (27) expresses a hyperbolic law for the reduction of the shear modulus as the one proposed by Hardin and Drnevich (1972). As a cons equence, the following equation for the function c(|oct|) is obtained: 11)( 00 maxc (28) where max and 0are two constant rheological experimental values. At every time, the values associated to the functions (l)(t) are obtained by solving the following equations: )( )(1)()( )( 1)0,()0,( )( )()( )( te y cy ct tn lll l l l l (29) where the variables y(l,0) are known, given by formula (6) for the lower strain Q-1 value. Final ly, the rheological Eq. (9) is used for the considered 1D case : n ll lyt teGts 1)( )(0))(,( )(12)( (30) 3 Validation of the model for cyclic loadings The nonlinear model will be validated for 1D cyclic loadings first (homogeneous stress -strain state) directly solving Eq s (28), (29) and (30) . The analysis of seismic wave propagation will be considered afterwards. The c yclic loading s correspond to sinusoidal excitations at various strain levels. The nonlinear parameter is chosen as =1000 and the elastic shear modulus is G 0=80MPa. The relaxation parameters may then be computed considering Eqs (28) and (29 ) with the following asymptotic damping values : 0=0.025 and max=0.25. In Fig . 4, some of t he results (obtained at 10Hz) are displayed as stress -strain loops for max=10-5, 10-4, 5.10-4 and 10-3. For each case, the secant shear modulus G is calculated and normalized by G0 (the ratio r=G/G0 is given in each curve). The first case (Fig. 4, top left), correspond ing to max=10-5 and r=0.99 , leads to a nearly linear response with an elliptical stress -strain loop. In the 2nd case, max=10-4 and r=0.91 (Fig. 4, top right), the area of the loop is larger and there is a slight decrease of the shear modulus. For the largest excitations (max=5.10-4; r=0.77 ) and (max=10-3; r=0.50 ) (Fig. 4, bottom), the nonlinear effects are obvious since the stress -strain loops are strongly modified (secant modulus , area, etc). Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 8 0 00 0-1000-50005001000 (Pa) -2.10402.104- 10-5 -5.10-4-10-4 -10-310-5 5.10-410-4 10-3 (Pa) - 4.104- 2.10402.1044.104 -500005000 r=0.99 r=0. 91 r=0.77 r=0.5NM 1st loading Fig. 4. Stress -strain curves from cyclic loadings of variable maximum amplitudes (r=G/G 0) at 10Hz: nonlinear extended NCQ model (solid) and 1st loading curve (dashed) . From these loops, it is straightforward to derive the secant shear modulus as a function of maximum shear strain. For each loading level, the dissipation may also be quantified by calculating the ratio between the area of the stress -strain loop and the strain energy estimated from the first loading curve (up to the maximum shear strain max). The damping ratio may be easily derived from this energy ratio as a function of maximum shear strain (Kramer, 1996) . The actual G(max) and (max) curves are then compared to the theoretical curves in Fig. 5. The effective shear modulus (solid) is very close from the theoretical one (dotted). For the damping ratio, the difference is larger for large shear strains, but the effective dissipation increases as expected. 00.10.20.30.4 02468x 107 10-510-410-310-2theoretical G( ) & ( ) NM: actual G( ) & ( ) G( ) (Pa) () Fig. 5. Comparison of the shear modulus and damping values (%) of the extended NCQ model under cyclic loadings (solid) with the theoretical variations predicted by Eq s. (18) and (27) (dashed) . Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 9 Numerical implementation (FEM) The mechanical model described above is introduced into the framework of the Finite el ement method, for the case of a unidirectional s hear loading. Let us consider a homogeneous layer over an elastic bedrock as depicted in Fig. 6. 0 halluvial layer V =200m/s =2000kg/mL L3N N-1 N-2 3 2 1e( N- 1)/ 2 e1=0 V (2v -v )1 =S S input Z(m)bedrock V =400m/s =2000kg/mS S3 Fig. 6. 1D soil layer over an elastic bedrock: finite element discretization an d absorbing boundary condition at the interface. The domain is divided into (N-1)/2 linear quadratic finite elements, each of the N nodes having 1 degree of freedom (horizontal motion) . Using square brackets […] and braces {…} to denote matrices and vecto rs, the discretiz ed equation of motion can be written in the following form at each time step (n+1)t: max 1n )( )()( )(1 1 1 1 1 ,1 ; )u( )( )()]([ ][ ][ l l Ht tF u uK vC aM l l l ln n n n n (31) where [ M], [C] and [K(un+1)] represent the mass, the radiation condition at the bedrock/layer interface (elastic substratum), and the stiffness matrix respectively . {an+1}, {vn+1} and {un+1} are the acceleration, velocity and displacement vector respectively , while { Fn+1} is the vector of external force s at the inter face. (l) and (l) are the relaxatio n parameters and central frequencies of the rheological cells (resp.), H(l)(un+1) corresponds to the right hand -side term in Eq. (29) and lmax is the total number of cells included in the model (lmax=3 herein) . For the time integration, an extension of the Newmark formulation is used, namely an unconditionally stable implicit -HHT scheme (Hughes, 1987) . This scheme allows a control of the highe r frequencies generated during the propagation (Semblat and Pecker, 2009) . At each time step, the Newton -Raphson iterative algorithm is adopted to deal with the nonlinear nature of the f irst equation in system (31). The Crank -Nicolson procedure (Zienkewicz, 2005) is simultaneously used in order to estimate the (l)(t) variables in the first order differential equation s (system (31), bottom ). 4 Modeling wave propagation in the nonlinear range 4.1 Nonlinear layered model We performed two different types of simulations: linear attenuating model ( denoted “ LM”) and nonlinear extended NCQ model ( denoted “ NM”). For the first one (0=max=2.5% and ), the mechanical and dissipative properties of the material do not depend on the excitation level while, in the second case ( 0=2.5%, max=25% and ), both elastic and dissipative properties are function of the induced str ain as shown in Figs 3 and 5. For both models, we performed simulations for a 20 m deep soil layer over a n elastic bedrock, with a velocity contrast of 2 and an absorbing condition at the bottom of the layer (Fig. 6). The Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 10 excitations considered hereafter thus cor respond to the incident wavefield at the top of the bedrock. 4.2 Sinusoidal incident wavefield In this section, the incident wavefield is a double sine-shaped acceleration wavelet similar to that proposed by Mavroeidis and Papageorgiou (Mavroeidis and Papageor giou, 2003; Semblat and Pecker, 2009). It is defined by the following equation: 00sin) sin()(tt ta with 0 02f and Hz f30 (32) The total duration of the resulting signal is about 2 seconds. In Fig. 7, taking into account the velo city contrast, a comparison is shown in terms of acceleration time histories and corresponding Fourier spectra at the top of the soil layer for two excitati on levels (0.5 and 0.75 m/s2). The nonlinear time histories involve propagation time delay s when compared to the linear ones, as it can be easily observed by comparing the peaks arrival ti mes for both models in Fig. 7. In the latter case, the Fourier spectra of t he nonlinear signals indicate: 1) a significant decrease o f the spectral amplitude , with increasing excitation level, for the main frequency components of the input signal; 2) the generatio n of higher frequenc y peaks which are not contained in the input signal (around 3 and 5 times the predominant excitation freque ncy). Such higher frequency components are larger for stronger excitations (bottom) ; 3) a frequency shift of the largest peaks to lower frequencies for increasing excitations . The shear strain at the center of the layer is also plotted in Fig. 8 (left) for b oth excitation levels. Similar time delays are observed in the time -histories. From the stress -strain paths (Fig. 8, right), the reduction of the shear modulus and the energy dissipation are found to be larger for peaks of increasing amplitudes. The larges t effect is obtained for the strongest excitation (Fig. 8, bottom right). -1.5-1-0.500.511.5 0 0.5 1 1.5 2 2.5 3-1.5-1-0.500.511.500.10.20.30.4 0 5 10 1500.10.20.30.4NM: LM: =1000 0=2.5%0=2.5% m a x=25% time (s )accele ration (m/s )2 FFT (m/s) FFT (m/s)acce leration (m/s)2 frequency (Hz) Fig. 7. Accelerations (left) and corresponding Fourier spectra (right) at the top of the soil layer, for 2 values of t he maximum input acceleration on bedrock 0.5 (top) and 0.75 m/s2 (bottom) : linear (LM , dotted ) and nonlinear (NM , solid ) simulations. Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 11 -1-0.500.51x 1 0-3(t) 0 1 2 3-1-0.500.51x 1 0-3 time (s)(t)-505x 104() (Pa) -1 -0.5 0 0.5 1 x 10-3-505x 104 () (Pa)LM NM 1st loading Fig. 8. Shear strains (left) and stress -strain loops (righ t) at the middle of the soil layer, for 2 values of the maximum input acceleration o n bedrock 0.5 (top) and 0.75 m/s2 (bottom) : linear (LM, dotted ) and nonlinear (NM , solid ) simulations. 4.3 Real seismic input 4.3.1 Linear and nonlinear simulations We use the same m odel as in the previous case ( Fig. 6) but the incident wavefield now corresponds to the horizontal acceler ation recorded at Topanga station during the 1994 M6.7 Northridge ea rthquake ( Fig. 9, top). In the linear case , the results are displayed in terms of time history and Fourier spectrum in Fig. 9 (2nd line ). For the nonlinear case, two different values of the nonlinear parameter are chosen: =300 (Fig. 9, 3rd line ) and =600 (Fig. 9, bottom). From the results of the linear case (2nd line), the incident wa vefield is found to be significantly amplified at the free surface in terms of Peak Ground Acceleration (30%) . Comparing the linear and the nonlinear responses, p eak amplitude s in the time histories and the spectra appear to be modified. The results of the nonlinear cases lead to lower amplitudes at intermediate frequencies, whereas nonlinear responses at higher frequencies are generally larger (Fig. 9, right ). It is nevertheless difficult to assess the influence of the nonlinearities for each individual peak. A time -frequency analysis is thus proposed in the next section. In the case of the seismic excitation, the stress -strain loops are plotted in Fig. 10 for the linear and nonlinear models. When compared to the linear case (Fig. 10 left), the nonlinear c ases (Fig. 10 center and right) lead to a strong modulus decrease and a large dissipation increase. The difference between both values is also significant (e.g. larger loops) showing stronger nonlinear effects for the largest value ( r=0.27 for =600 and r=0.47 for =300). Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 12 4 2 0 -2 -4acc. (m /s²) acc. (m/s²) acc. (m/s²) 16 18 20 22 24 26 28 30 time (s)input input linear linear =300 =300 =600 =600 FFT (m/s) FF T (m /s) FFT (m/s) FFT (m/s) 0 2 4 6 8 10 frequency (Hz)4 2 0 -2 -4 4 2 0 -2 -4acc. (m/s²)4 2 0 -2 -43 3 3 32 2 2 21 1 1 10 0 0 0 Fig. 9. Accelerations at the free surface for the M6.7 Northridge earthquake: time -histories (left) and related spectra (right) ; measured signal at Topanga station (top), linear simulation (2nd line ) and nonlinear simulations with =300 ( 3rd line ) and =600 (bottom). -3 -3 -3 -2 -2 -2 -1 -1 -1 0 0 0 1 1 1 2 2 2 3 3 3x 10-3x 10-3x 10-3 -1.5-1-0.500.511.5x 105 (Pa)lin ear =300 =600 Fig. 10. Stres s-strain curves at the middle of the soil layer for the M6.7 Northridge earthquake: linear case (left) and nonlinear cases with =300 (center) and =600 (right). Time -frequency analysis The analysis will now be performed in different f requency bands a s defined in Fig. 11. In this figure, the spectral amplitudes are found to be similar for the linear and nonlinear cases in frequency bands (a) and (c), whereas bands (b) and (d) evidence significant differences. These frequency bands are the following: (a) [0 -2.5Hz], (b) [2.5 -4.3Hz], (c) [4.3 -6.3Hz] and (d) [6.3 - 20Hz]. The time -histories have been (Butterworth -) filtered in each frequency band to make the comparison between the linear and nonlinear cases easier. Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 13 linear nonlinear ( =600)(a) (b) (c) (d) 2.0 1.0 0.00.0 2 . 0 4.0 6.0 8.0 10.0 12.0 1 4.03.0FFT (m/s) frequency (Hz) Fig. 11. Fourier spectra of the accelerations at the top of the soil layer ( Fig. 9) for the M6.7 Northridge earthquake in the case of linear (LM, dotted) and nonlinear simulations ( =600, solid) . The f iltered accelerograms related to each frequency band are displayed in Fig. 12. The filtered linear time -histories are plotted on the left whereas the nonlinear ones ( =600) are located on the right part. The comparison of the filtered accelerograms lead to the following conclusions: 1) Frequency band s (a) and (c) : the peak amplitudes of the filtered time -histories in the linear and nonlinear cases are similar. It may also be noticed in the spectra plotted in Fig. 11. 2) Frequency band (b) : the discrepancy between both time -histories is large since the linear response may be 30% larger than the nonlinear one. Such a difference may be directly seen in the spectra (Fig. 11). 3) Frequency band (d) : the nonlinear response is now larger than the linear one (up to 40%) due to the influence of higher order harmonics generated by nonlinear models (Van Den Abeele , 2000). For strong seismic motion, the nonlinear ground response may then be smaller or larger than the linear one depending on the excitation level as well as the frequency content of the input motion. The nonlinear properties of the soil are also an important governing parameter of its seismic response. 5 Conclusions A 3D nonlinear viscoelastic model (“extended NCQ”) is proposed to approximate the hysteretic behavior of alluvial deposits undergoing seismic excitations. Such nonlinear features as the reduction of shear modulus and the increase of damping are controlled by the variations of the 2nd invariant of the strain tensor during multidimensional loading. In the case of a unidirectional shear loading, nonlinearity is controlled by only one shear strain component: nonlinear elasticity by a hyperbolic law and viscosity by a NCQ model with nonlinear features (nearly frequency constant but strain amplitude depende nt). This model allows to account for the generation of higher order harmonics shown in the nonlinear case for 1D simulations. At the same time, a reduction of the spectral amplitudes and a shift to lower frequencies were found for increasing motion amplit udes. The interest of the simplified nonlinear “X-NCQ” model proposed herein is to reduce the computational cost for the analysis of strong seismic motion in 2D/3D alluvial basins (small number of constitutive parameters) . Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 14 16 162.02.0 1.01.01.0 -0 .5 1.0 0.00.00 .5 -1 .00.0 0.0 -1 .0-1 .0 -2 .0 -1 .0 18 18 20 20 22 22 24 24 26 26 28 28 30 30(a) linear (b) linear (c) linear (d) linear(a) = 600 (b) =600 (c) = 600 (d) =600acceleration (m/s )2acceleration (m/s )2acceleration (m/s )2acceleration (m/s )2 time (s) time (s) Fig. 12. Accelerations at the top of the soil layer for the M6.7 Northridge earthquake in the case of linear (LM, dotted) and nonlinear simulations ( =600, solid) filtered in different frequency bands defined in Fig. 11. For example, in the 1D case, the reduction of she ar modulus is controlled by a hyperbolic law with only one parameter estimated from the experimental knowledge of the G(γ) curve. As a consequence, the dissipation properties are directly derived from the hy perbolic law and from two other characteristic parameters responsible for the min imum and maximum loss of energy at lower and larger strain levels , 0and max. These are sufficient to give an overall description of the unloading and reloading phases durin g the seismic sequence . Combined with the nonlinear properties of the soil in the simplified model , the frequency content of the seismic input has an important influence on strong ground motions . Finally, the proposed model will allow future computations i n the case of 2D or either 3D alluvial basins for which the amplification is generally found to be much larger than predicted through 1D analyses (Chaillat et al., 2009; Chávez -García et al., 1999 ; Fäh et al., 1994; Gélis et al., 2008 ; Lenti et al., 200 9; Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 15 Moeen -Vaziri and Trifunac, 1988 ; Sánchez -Sesma and Luzón, 1995 ; Semblat et al. , 2000 , 2005 ). Several authors proposed some 2D/1D aggravation factors (Makra et al., 2005; Semblat and Pecker, 2009 ), but it is probably not sufficient for strong seismic motion s involving significant nonlinerities in the soil response . APPENDIX : Emmerich and Korn’s method to find the optim al parameters of the linear viscoelastic “NCQ” model is presented in this appendix . We consider the viscoelastic model depic ted in Fig. 1 (left). To estimate the a (l) coefficients, a normal ization condition is introduced : )( )0,( l Rl aMMy (33) The (l)/(l-1) ratio being chosen constant , Eq. (6) is simplified as: n ll ll ly Q2 )()( )0,(1 / 1/)( (34) The y (l,0) quantities are estimated by using Eq. (34): writing it for different and for several fixed values of (l) and taking the first term equal to a given const ant value, the obtained algebraic linear system can be solved by a least -squares algorithm. An example of the result of this pr ocedure is displayed in Fig. 2: in the case of =2.5% (Q=20) and a velocity of 200m/s. A normalization condition allows to choose a target phase velocity (200m/s) at a given reference frequency (1Hz in the example). For more details, the readers may refer to Emmerich and Korn (1987). Acknowledgements : The authors would like to thank Luis F. Bonilla (IRSN) for fruitful discussions. This work was partly funded by the French National Research Agency in the framework of the “ QSHA ” research project (“Quantitative Seismic Hazard Assessment”). Notations: The following symbols are used in this paper : {a} = acceleration vector (FEM) [C] = damping matrix (FEM) c(|oct|) = weighting function for non linear damping e = shear deviatoric strain tensor eij() = Fourier transforms of the components of the deviatoric strain ekk = volumetric strain {F} = external force vector (FEM) f = frequency G0 = (unrelaxed ) shear modulus at low strains I’1 = first invariant of the strain tensor I’2 = second invariant of the strain tensor J2 = second invariant of the deviatoric strain tensor K = bulk modulus [K] = tangent stiffness matrix (FEM) M() = complex vis coelastic modulus [M] = mass matrix (FEM) MR = relaxed modulus MU = unrelaxed modulus Jal of Engineering M ech. (ASCE) , 135(11), pp.1305 -1314, 2009 Delépine, Lenti, Bonnet, Semblat 16 MU,0 = unrelaxed modulus at low strains p = volumetric tension Q = quality factor Q-1 = specific attenuation s = shear deviatoric stress tensor sij = components of the de viatoric stress tensor sij() = Fourier transforms of the components of the deviatoric stress {u} = displacement vector (FEM) {v} = velocity vector (FEM) y(l,0) = relaxation parameters of the viscoelastic cells for low excitation levels = scalar paramete r characterizing the modulus reduction oct = octahedral strain ij = Kronecker unit tensor components M = difference between the relaxed and unrelaxed modul i l(t) = relaxation functions 0 = minimum damping at low strains max = maximum damping at large strains ij = components of the Cauchy stress tensor = function characterizing the modulus reduction = circular frequency REFERENCES Aubry D., Hujeux J.C., Lassoudire F., Meimon Y. 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(1985). “A simple plasticity theory for frictional cohesionless soils ”, Soil Dynamics and Earthquake Eng ., 4(1), 9-17. Sánchez -Sesma F.J. and Luzón F. (1995). “Seismic Response of Three -Dimensional Alluvial Valleys for Incident P, S and Rayleigh Waves”, Bull. of the Seism. Soc. of America , 85, 269 -284. Seed H.B. and Idriss I.M. (1970). Soil moduli and damping factors for dynamic response analysis , Report N°EERC 70 -10. University of California, Berkeley. Semblat J.F. (1997). "Rheological interpretation of Ra yleigh damping", Jal of Sound and Vibration , 206(5), 741 -744. Semblat J.F., Duval A.M., Dangla P. (2000). “Numerical analysis of seismic wave amplification in Nice (France) and comparisons with experiments ”, Soil Dynamics and Earthquake Eng. , 19(5) , 347–362. Semblat J.F., Paolucci R., Duval A.M. (2003). “ Simplified vibratory characterization of alluvial basins”, C.R. Geoscience , 335, 365 -370. 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2009-08-19

Hysteretic damping is often modeled by means of linear viscoelastic approaches such as "nearly constant Attenuation (NCQ)" models. These models do not take into account nonlinear effects either on the stiffness or on the damping, which are well known features of soil dynamic behavior. The aim of this paper is to propose a mechanical model involving nonlinear viscoelastic behavior for isotropic materials. This model simultaneously takes into account nonlinear elasticity and nonlinear damping. On the one hand, the shear modulus is a function of the excitation level; on the other, the description of viscosity is based on a generalized Maxwell body involving non-linearity. This formulation is implemented into a 1D finite element approach for a dry soil. The validation of the model shows its ability to retrieve low amplitude ground motion response. For larger excitation levels, the analysis of seismic wave propagation in a nonlinear soil layer over an elastic bedrock leads to results which are physically satisfactory (lower amplitudes, larger time delays, higher frequency content).

Nonlinear viscoelastic wave propagation: an extension of Nearly Constant Attenuation (NCQ) models

0908.2715v2

Rigorous Theory of Optical Trapping by an Optical Vortex Beam Jack Ng,1 Zhifang Lin,1,2 and C. T. Chan1 1Department of Physics and Willia m Mong Institute of Nano Science & Technology, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China. 2Department of Physics, Fudan University, Shanghai, China. Abstract We propose a rigorous theory for the optical trapping by optical vortices, which is emerging as an important tool to trap me soscopic particles. Th e common perception is that the trapping is solely due to the gradie nt force, and may be characterized by three real force constants. However, we show that the optical vortex trap can exhibit complex force constants, implying that th e trapping must be stabilized by ambient damping. At different damping levels, particle shows remarkably different dynamics, such as stable trapping, periodic and aperiodic orbital motions. Optical tweezers is a powerful tool for trapping mesoscopic objects. Applications range from the trapping and cooling of atoms, to large molecules such as DNA, and to microscopic particles and biol ogical objects. As new methods to create beam profiles are being introduced, more exotic beams are used to trap particles and many of them, known as optical vortex (OV), carry angular momentum (AM). We will show that although the conventional stiffness constant approach works for a Gaussian beam, the theory of trapping by an OV is more co mplex and interesting. In the conventional approach, optical traps are us ually characterized by three stiffness constants along the three principal axes, which are usually taken to be the Cartesian axes (e.g., x-polarized, z-propagating beam ). Although such approach is pre tty accurate in describing the ordinary optical tweezers, a mo re rigorous treatment reveals that the principal axes are not necessarily given by the Cartesian axes and, for an OV , the pr incipal axes are not even “real”. The rigorous theoretical proced ure to obtain the principal axes is to diagonalize the force constant matrix ,/ij light i jKf x=∂∂ at equilibrium, where , light if and xi are, respectively, the i-th Cartesian component of th e optical force and particle displacement away from the equilibrium positio n. It is the eigenvalues of the force constant matrix that give the eigen force constants (EFC, or trap stiffness), while the eigenmodes determine the principal axes. We shall apply the force constant matrix formalism to study OV trapping [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18, 19,20,21,22,23,24], and we see that the difference between the conventional appr oach and the rigorous treatment is a qualitative one, to the extent that the EF Cs can be complex numbers and we have to abandon concepts such as pa rabolic potential for the tr ansverse directions. By analyzing the stability and simulating the dynamics of a particle trapped by OVs, it is found that the trapping stabilit y of OVs generally depends on the ambient damping. In particular, in the presence of AM, the op tical trapping may exhibit a fascinating variety of phenomena ranging from “opt o-hydrodynamic” trapping (where the trapping is stabilized by the ambient damp ing) to supercritical Hopf bifurcation (where a periodic orbit is created as ambient damping decreases). To illustrate the basic idea, let us starts from the linear stability analysis. Consider a trapped particle, near an equilibrium trapping (zero-force) position, the optical and damping forces [25] are 22/ / F m d x dt K x d x dt γ =∆ ≈ ∆ − ∆I K K, where m is the mass of the particle, x∆K is the particle’s displacement from the equilibrium, Kx∆IK is the optical force, and γ is the ambient damping constant. The eigenvaluesiK’s of the force matrix KI are precisely the EFCs and the eigenvectors of KI are the eigenmodes. For a trapping beam propagating along ˆz, the force constant matrix has the general form 0 0ad Kg b efc⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦I , (1) where all elements are real numbers. The elements ,/light xf z∂∂and,/light yf z∂∂ are zeros by symmetry, because th ere is no induced force along the transverse plane as the particle is displaced along ˆz. The diagonal elements a, b, and c characterize three restoring forces, which are usually take n as the three stiffness constant in conventional approach. Off diagonal elements d and g characterize the rotational torques: i.e. as the particle is displaced along x (y), it experiences a torque that manifests as a force along the y (x) direction. As an OV carries orbital AM, the energy of the beam propagates along a helical path. There is a rotating energy flux in the transverse plane that would exert a torque on the particle, implying non-zero d and g. On the other hand, 0 dg== for beams that carry no AM. By diagonalizing Eq. (1), we obtain three EFCs: axialKc= and 2() 4/ 2transKa b a b d g±⎡⎤=+ ± −+⎣⎦. Conservative mechanical systems can be described by a potential energy U, and their force constant ma trix is a real symmetric matrix (i.e.2/ij i j jiKU x x K=−∂ ∂ ∂ = ), which then follows that all eigenvalues are real. However, optical force is non-conservative as the particle can exchange energy with the beam, and its KI is in general a real but non-sy mmetric matrix. Consequently, its eigenvalues can carry conjugate pair of complex numbers. It can be seen that when 24( )dg a b−> − , (2) transK± are a conjugate pair of complex numb ers. Thus, in general, an optically trapped particle may not be characterized by three real force constants if the beam carries AM. For beams that have no AM, 0 dg==, all the EFCs are real numbers. On the other hand, an OV beam, well known for its AM carrying char acteristic, leads to non zero d and g, and thus (2) can be fulfilled under certain conditions. The existence of complex EFCs implies that the common notion of a parabolic potential in an optical trap is no longer meaningful. Wh ether complex EFC occurs depend on the competition between the beam asymmetry and the AM. Equation (2) cannot be fulfilled when | | ab− is large. Since a (b) is the restoring force constant for the x- (y-) direction, a large | | ab− implies a large asymmetry between the two coordinate axes. The underlining physics is that for large asymmetry, the beam can pin a particle to one of its axis, preventing it from “falling” into the OV . On the contrary, if there is weak or no asymmetry (| | 0 ab−), the trapped particle is not tied to the coordinate axis, and will thus be swiped by the OV . As a result, its AM and energy will accumulate. If there is no dissipation ( 0 γ=) the particle will orbit around the beam center with increasing speed and eventually escape from the trap [26]. It is therefore concluded that in general the OV trapping cannot be achieved solely by light. It is the dissipation in the suspending medium that k eeps the particle’s kinetic energy and AM bounded, rendering the particle tr apped [26]. Such a state of OV trapping is believed to be what the experiments observed, instead of the pure gradient force trapping. For the case of circularly polarized beams (L G or Gaussian), cylindrical symmetry mandates that| | 0 ab−= , implying that the transverse EF Cs are always complex. This means that circularly pol arized beam cannot trap without dissipation. More mathematical details can be found in on-line materials [34]. We now proceed to show concrete examples in which the EFCs are indeed complex. We model the incident trappi ng beam by using the highly accurate generalized vector Debye integral [27,28], where the focusing of the incident laser beam by the high numerical aperture (N.A.) object lens is treate d using geometrical optics, and then the focused field near the focal region is obtained using the angular spectrum representations. The use of geom etrical optics in beam focusing is fully justified as the lens is macroscopic in si ze, and all remaining parts of our theory employs classical electromagnetic optics. W ith the strongly focused beam given by the vector Debye integral, the Mie theory is then applied to calculate the scattered field, and then the Maxwell stress tensor formalism is applied to compute the optical force. [29,30] We note that the formalisms we use have been proven to agree well with experiments [28,31,32,33]. Fig. 1 (a) shows transK± for an LG beam focused by a high N.A. water immersion objective. Th e beam is linearly polarized with wavelength 1064 nmλ= , topological charge l=1, N.A.=1.2, and filling factor f=1. The trapping is in water (21.33waterε= ) and the sphere is made of polystyrene (21.57sphereε= and mass density31050 kg mρ−= ). The axial EFC is not plotted, as it is always real and negative, indicating that the particle can be trapped along the axial direction solely by gradient for ces. It can be clearly seen from Fig. 1(a) that at certain ranges of particle sizes, Im{ } 0transK±≠, and this numerical results manifest the assertion that the OV trap cannot be characterized by th ree real force constants in general. At these particle sizes, * trans transKK+ −= , and the two curves corresponding to Re{ }transK± merge together. Note that only the absolute value of Im{ }transK± is plotted in Fig. 1(a). When the EFCs are all real numbers, the behavior of the trapped particle is qualitatively sim ilar to that of the ordinary optical trapping by conventional optical tweezers. This corresponds to the scen ario that either the beam’s AM is weak (small d and g), or the asymmetry of the beam (| | ab−) is large. We note that the existence of region where Im{ } 0transK±≠ implies that in low viscosity media, only particles of certain sizes can be tra pped. For complex EFCs, the eigenmodes corresponding to the co mplex EFCs are [34]: Im( )Re( )sin[Re( ) ]() Im( ) cos[Re( ) ]tVtxt A e Vtφ φ±±± −Ω ±± ±±⎧ ⎫ Ω+ ⎪ ⎪∆= ⎨ ⎬+Ω +⎪ ⎪ ⎩⎭K KK , (3) where VK is the eigenvector co rresponding to eigenvalue Ki of the force constant matrix KI . {},Aφ±± are to be determined from initial conditions, and 22 1 / 4 22 1 / 4Re( ) ( ) sin( / 2) / 2 , Im( ) ( ) cos( / 2) / 2 ,RI RIm mδ γδ± ±Ω=∆+ ∆ Ω=± ∆+ ∆∓ (4) () ()1 1tan / , if 0, tan / , if 0,IR R IR Rδπ− −⎧∆∆∆ > ⎪=⎨−∆ ∆ ∆ <⎪⎩ (5) where 4 Im{ }Ii mK∆= and 24R e {} .Ri mKγ∆= + The modes are stable if and only if both Im( ) 0±Ω> . If{} Re 0iK>, one of the two modes is always unstable such that upon small perturbation, the particle will spiral outward and leave the trap. If{} Re 0iK<, the mode is unstable for {} {} Im / Recritical i i mK K γγ<= . However this equilibrium can be stabilized by increasing γ to beyond criticalγ . We label this kind of mode as quasistable modes, in which the stability of the modes depends on the ambient damping. The complex modes described by Eq. (3) correspond to spiral motions, which means that the particle is absorbing AM from the beam. The converse is also true: these spir al modes can exist only when the particle can absorb AM. We note in Fig. 1 that in our specific example, {} Re 0transK±> for sphere radius 0.36 R mµ< (radius of the intensity ring ~ 0.33 mµ, see Fig. 2(a)), which means that small dielectric particles are uns table, as reported in experiments [14,35]. The small dielectric particles are attrac ted by the high intensity ring and under sufficient damping, these small particles will orbit along the ring [5]. On the other hand, {} Re 0iK< for 0.36R mµ> . The sphere is bigger than the intensity ring, so that the gradient force drives the sphere to the beam center. A phase diagram for the optically trapped particle is given in Fig. 1(b). At 0.36amµ= , criticalγ→∞ as{} Re 0transK±→ . The equilibrium point at (, ) ( 0 , 0 )xy= is unstable for 0.36amµ< at any values of damping. For 0.36µm a> , the white (shaded) region where criticalγγ> (criticalγγ< ) is the regime where the damping is sufficient (insufficient) to stabilize th e particle. We note that when the EFC is real, no damping is required for stability since 0criticalγ=. According to Stoke’s law, the damping constant of water and air are, respectively, 421.9 10 (pN s/ m ) Rµµ × and223.3 10 (pN s/ m ) Rµµ × . The damping of water is much larger than criticalγ plotted in Fig. 1(b), and thus unless one uses high laser power, one shall observe stable trapping in water, in agreement with existing experiments. The damping of air is of the same order of magnitude of criticalγ plotted in Fig. 1(b) , consequently for an experiment conducted in air, one shall be able to see the transition between the stable and unstable state, depending on the laser power employed. It is now clear that a particle trapped by an OV is stable if criticalγγ> and unstable ifcriticalγγ< . Nevertheless, the case of criticalγγ≈ is non-hyperbolic (the linear term vanishes), and thus the higher order terms ar e important. In that case, we numerically integrate th e full equation of motion 22/ /light md x d t f d x d t γ ∆= − ∆KK K, using an adaptive time-step Runge-Kutta-Ver ner algorithm [29]. Fi g. 2(a) shows the field intensity on the focal plane for a right circularly polarized LG beam, with a dark central spot and a high intensity ring of radius ~0.33 µm. Fig. 2(b)-(f) show the trajectories of a 1- µm -diameter particle illuminated by the LG beam (power=550 mW), in the order of decreas ing damping. When there is strong damping, as shown in Fig. 2(b) where 550 pN s/ m γ µµ= , the trapped sphere exhi bits damped oscillation upon small perturbation and settles into a stable equilibrium position. For weaker damping, the sphere initially spirals outward , and then settles into a periodic circular orbit (see Fig. 2(c) where 110 pN s/ m γ µµ= ). Such bifurcation of a stable equilibrium into an unstable equilibrium and a stable periodic orbit is known as a supercritical Hopf bifurcation [29]. If we further reduce the damping, the radius of the circular orbit increases, as show n in Fig. 2(d) where 55 pN s/ m γ µµ= . If the damping decreases further, the particle goes into an exotic orbit around the intensity ring, as shown in Fig. 2(e) where 5.5 pN s/ m γ µµ= . When there is no damping (see Fig. 2(f), 0 pN s/ mγµµ= ), the particle initia lly fluctuates around the equilibrium with increasing amplitude, and eventually escapes from the trap due to the accumulation of AM. If a small imaginary part is introduced into the dielectric constant of the particle, the introduced absorption will compete with the light scattering, reducing the amount of AM that is transferred to the orbital motion of the particle, and the particle will now spin along its own axis. In other words, small absorption may in fact favor the transverse trapping, though it degrades the axial trapping. Our analysis reveals that for an AM carrying beam, its EFCs can be complex numbers. In the case of complex EFCs, when there is sufficient (insufficient) damping a particle can (cannot) be stably trapped. Th ere is an intermediate range of damping in which the particle will be driven into e xotic periodic or aperiodic orbital motions. Finally, we note that as the ambient damping force plays an important role in the OV trapping, it should be more accurately termed “opto-hydrodynamic trapping”. We have also applied the stability analys is to other types of focused beams [34] with different N.A., and we find that we can observe complex EFCs whenever the beam carries AM. This work is supported by Hong Kong RGC grant 600308. ZFL was supported by NSFC (Grant number 10774028), PCSIRT, an d MOE of China (B06011). Jack Ng was partly supported by N SFC (Grant number 10774028). 012-4-20(a) Radius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse+) Re( Ktransverse-) |Im( Ktransverse)| 0120300600 γ ( pN µm-1 µs ) Radius (µm)(b) Fig. 1 The incident beam is a linearly polarized LG beam with 1064 nmλ= , l=1, f=1, and N.A.= 1.2. (a) The transverse EFCs. (b) Phase diagram for a particle trapped at a power of 1W. The white (red) regions are unstable (stable). The black line markscriticalγ . Fig. 2 (a) The focal plane intensity (arbitrary units) of a right polarized LG beam with 1064 nmλ= , l=1, f=1, and N.A.=1.2. (b)-(f) The trajectory (blue) of a 1-micron-diameter particle trapped by a 550 mW beam. Th e red dotted lines are the approximate radius of the intensity ring of the trapping beam. The damping constants γ for each panel, in unit of pN s/ m µµ, are given by (b)550, (c)110, (d)55, (e)5.5, and (f)0. The arrows in (b) and (f) indicate the direction of motion. 1 A. T. O’Neil and M. J. Padgett, Opt. Commun. 185 139 (2000). 2 H. Rubinsztein-Dunlop et al. , Adv. Quantum Chem. 30, 469 (1998). 3 K. T. Gahagan and G. A. Swartzlander, JOSA B 15, 524 (1998) ; ibid, ibid 16, 533 (1999). 4 L. Allen et al. , Phys. Rev. A 45, 8185 (1992). ibid, Optical Angular Momentum (IOP Publishing, London, 2003). 5 V . Garces-Chavez et al. , Phys. Rev. A 66, 063402 (2002). 6 A. T. O’Neil et al. , Phys. Rev. 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Eng. 78-79 , 125-131 (2005). 23 Adrian Alexandrescu, Dan Cojoc, and Enzo Di Fabrizio, Phys. Rev. Lett. 96, 243001 (2006). 24 D. S. Bradshaw and D. L. Andrew, Opt. Lett. 30, 3039 (2005). 25 Here we neglect the thermal fluctuation, as it is small compare to the optical force for an intense laser. 26 N. R. Heckenberg et al. , “Mechanical effects of optical vortices,” M Vasnetsov (ed.) Optical Vortices (Horizons in World Physics) 228 (Nova Science Publishers, 1999) pp 75-105. 27 L. Novotny and B. Hecht, Principles of Nano Optics (Cambridge University Press, New York, 2006). 28 Y. Z h a o et al. , Phys. Rev. Lett. 99, 073901 (2007). 29 J. Ng et al. , Phys. Rev. B 72, 085130 (2005). 30 M. I. Antonoyiannakis and J. B. Pendry, Phys. Rev. B 60, 01631829 (1999). 31 A. Rohrbach, Phys. Rev. Lett. 95, 168102 (2005). 32 N. B. Viana et al. , Appl. Phys. Lett. 88, 131110 (2006); ibid, Phys. Rev. E 75, 021914 (2007). 33 A. A. Neves, et al. , Opt. Exp. 14, 13101 (2006). 34 See EPAPS Document No. [] for a discussion on (I) linear stability analysis, (II) the eigen force constant for various type of trapping beams, and (III) the optical trapping by cylindrically symmetric beams. 35 For the trapping of a strongly absorptive pa rticle, the particle is unstable along the axial direction. Accordingly, other forces, su ch as a repulsive force from a substrate, are needed to stabilize the particle. However, in this letter, we consider the transverse optical trapping, thus our conclusion is va lid irrespective to the nature of the axial trapping. Appendix I: Linear Stability Analysis In this appendix, we give more details a bout the formalism on the linear stability analysis for a particle trapped by an arbitrary incident light beam. A. Linearized equation of motion We denote the displacement of the particle away from the equilibrium position by the position vector (,,)x xyz∆=∆∆∆K. The equation of motion of the particles are given by 2 2()lightdx d xmf xdt dtγ∆ ∆=∆ −K KKK, (I.1) where m is the mass of the particle, ()lightf x∆KK is the optical force, γ is the damping constant for the particle in the suspending medium. The frictional term in (I.1) is added to account for the Stoke’s drag be tween the particle and the suspending medium, and we have deliberately neglected th e Brownian term in (I.1), as it is of significance only at low laser power. If the displacement x∆K is small compare to the wavelength of inci dent light (| | xλ∆< <K), it is possible to simplify (I.1) with a linear approximation with respect to the displacemen t. The linearized equation of motion is 2 2dx d xmK xdt dtγ∆ ∆≈∆ −K KIK, (I.2) where 0()()light j jk kxfKx ∆=∂=∂∆ KKKI (I.3) is the force constant matrix. We note that the zero-th order term in (I.2) vanishes, because we are expanding the force near an equilibrium where (0 ) 0 .lightfx∆= =K KKK Introducing the transformation xVη∆=IKK, (I.4) where 'isη are the normal coordinate s, and the columns of VI are the eigenvectors of KI so that KI is diagonalized with eigenvalues Ki’s: 1ˆˆT ii i iVK V x x K−=∑III . (I.5) Here ˆix is the unit vector along the Cartesian axes. In a conservative mechanical system that can be desc ribed by a potential energy U, the force constant matrix is symmetric (i.e.2/ij i j jiKU x x K=−∂ ∂ ∂ = ) and can only give real negative or real positive eigen force constants. However, optical force is non-conservative, and KI is in general a real valued non-symmetric matrix (i.e.ij jiKK≠ ). As such, its eigenvalues and their corresponding eigenvectors, can be real numbers or a conjugate pair of complex numbers. After substituting (I.4) into (I.2), the equation of motion is now decoupled into three independent equations: 2 2ii iiddmKdt dtη ηηγ=− . (I.6) Equation (I.6) is a second order linear ordi nary differential equation, which may be solved by the standard technique of substituting 0iit ii eηηΩ= . (I.7) where 0iη and iΩ are independent of time. It turns out that the solution to (I.6) can be categorized according to the eigenvalues iK or equivalently the natural frequency of the eigenmode defined as 0 /ii KmΩ=− . (I.8) B. Types of Eigenmodes 1. Unstable mode characterized by an imaginary natural frequency If Ki is real and positive, the corresponding na tural frequency is purely imaginary and the corresponding solution is 22 22(/ 2 ) (/ 2 ) 00/2()mt mt iitm ii i ixt e V A e B eγγγ−+ Ω + Ω−⎡⎤∆= +⎢⎥⎣⎦K L, ( I . 9 ) where Ai and Bi are unknown constants to be determ ined from the initial conditions. The mode is unstable because (I.9) diverges with time. 2. Stable mode characterized by real natural frequency If Ki is real and negative, the natural freque ncy is purely real and the motions of the particles are that of a damp ed harmonic oscillator. For 22 0 (/ 2)i mγ>Ω , 22 22(/ 2 ) (/ 2 ) 00/2()mt mt iitm ii i ixt e V A e B eγγγ−− Ω − Ω−⎡ ⎤∆= +⎢ ⎥⎣ ⎦K K, (I.10) where Ai and Bi are unknown constants to be determined from initial conditions. The oscillation is over damped. For 22 0 (/ 2)i mγ=Ω, []/2()tm ii i ixte V A B tγ−∆= +K K, (I.11) where Ai and Bi are unknown constants to be determined from initial conditions. The oscillation is critically damped. For22 0 (/ 2)i mγ<Ω, /2 2 2 0 () s i n ( / 2 )tm ii i i ixt A e V m tγγ φ− ⎡ ⎤ ∆= Ω − +⎣ ⎦K K, (I.12) where Ai and φi are unknown constants to be determined from initial conditions. The oscillation is under damped. The trajectories of the solutions (I.10), (I.11) and (I.12) ar e all bounded as time increases, accordingly they are all stable. 3. Complex mode characterized by a complex natural frequency As the force constant matrix is non-sy mmetric, a complex conjugate pair of eigenvalues can occur. To obtain the trajector ies associated with the conjugate pair of eigenvalues iK and * iK, it suffices to consider only Ki where Im{ Ki }>0. The solutions associated with * iKare the same as that of Ki. The solutions are [ ] [ ] { }Im( )( ) Re( )sin Re( ) Im( )cos Re( )it ii i i i a i i i axt a e V t V t φ φ+−Ω ++ +∆= Ω + + Ω +KK K(I.13) [ ] [ ] { }Im( )Re( )sin Re( ) Im( )cos Re( )it ii i i i b i i i bxb e V t V t φ φ−−Ω −− −∆= Ω+ + Ω+KK K (I.14) where { } ,, ,i i ia ibabφφ are unknown constants to be determined from initial conditions, {} {} ( ) {} {} ( )1/42 22 2 1/ 42 22 2(4 R e) 1 6 I m s i n / 2 Re( )2 (4 R e) 1 6 I m c o s / 2 Im( )2ii i i ii i imK m K m mK m K mγδ γγ δ± ±⎡⎤++⎣⎦Ω= ⎡⎤±+ +⎣⎦Ω=∓ (I.15) and {} {}{} {} {}{}12 2 12 24I mtan if 4 Re4R e 4I mtan if 4 Re 4R ei i i i i i imKmKmK mKmK mKγγ δ πγ γ− −⎧ > ⎪+⎪=⎨ ⎪−<⎪+⎩ (I.16) a) Complex unstable mode If Re{ Ki}>0, ( )ixt+∆K is spiraling inward to the equilibrium, whereas ( )ixt−∆K is spiraling outward and its displacement diverg es with time. Consequently, an optically trapped particle having a complex Ki with positive real part is unstable and we denote this kind of solution as complex unstable mode. b) Quasi-stable mode If Re{ Ki}<0, ( )ixt+∆K is spiraling inward to the equilibrium. Here ( )ixt−∆K requires some attention. The mode is spiraling outward if Im( ) Re( )i critical imK Kγγ<= , (I.17) but spiraling inward ifcriticalγγ> . We denote this kind of solu tion as quasi-stable, where the stability depends on the damping provided by the environm ent. We note that the point criticalγγ= is non-hyperbolic, which simply means the lin ear term of the equation of motion vanishes, and the higher order terms are need ed. As discussed in the main text, linear stability analysis is not sufficie nt to determine the stability atcriticalγγ≈ . Consequently, real time dynamics simulations are performe d and the results are presented in Fig. 2 of the main text. Appendix II: The eigen force constant for various types of trapping beams A. The general form of force consta nt matrix and eigen force constants For an incident trapping beam, the general form of the force constant matrix for the trapped particle is 0 0ad Kg b efc⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦I , (II.1) where a, b, c, d, e, f, and g are real numbers, and () /ij light i jKf x=∂∂K . Two of the components in Eq. (II.1), Kxz and Kyz, are zero, because there is no induced force along the transverse plane as the particle is displaced along the z axis. Here, we assume that the optical system including th e focusing lens does not change the axial symmetry of the beam. By diagonalizing KI , we obtained the eigen force constants: 2, () 4/ 2 .axial transKc Ka b a b d g±= ⎡ ⎤ =+ ± −−⎣ ⎦ (II.2) When 24( )dg a b−< − , (II.3) the eigen force constants are all real numbers, and thus the nature of the optical trapping by such beam will be qualitatively si milar to that of the conventional optical tweezers, i.e. all the eigen vibrational modes are stable modes (see Appendix I). However, when 24( )dg a b−> − , (II.4) transK± are conjugate pair of complex numbers, and thus they correspond to the complex unstable mode or quasi-stable mode (see Appendix I). B. Numerical computation of eigen force constants for a variety of trapping beams. In this section, we present the numerica lly computed eigen force constants for a particle in water trapped by a variety of different incident trapping beams. The incident trapping beams include (1) a linear polarized Gaussian beam (Fig. II.1), (2) a circularly polarized Gaussian beam (Fig. II.2), (3) a linear polarized Laguerre-Gaussian beam (Fig. II.3), (4) a ri ght circularly polarized Laguerre-Gaussian beam (Fig. II.4), and (5) a left circularly polarized Laguerre-Gaussian beam (Fig. II.5). 012-4-20 Radius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse1) Im(Ktransverse1) Re( Ktransverse2) Im(Ktransverse2) Re( Kaxial) Im(Kaxial) Fig. II.1. The eigen force constants for a particle (21.57sphereε= ) trapped by a linear polarized Gaussian beam with f=1, and N.A.= 1.2 in water (21.33waterε= ). 012-4-20 Radius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse1) Im(Ktransverse1) Re( Ktransverse2) Im(Ktransverse2) Re( Kaxial) Im(Kaxial) Fig. II.2. The eigen force constants for a particle (21.57sphereε= ) trapped by a circularly polarized Gaussian beam with f=1, and N.A.= 1.2 in water (21.33waterε= ). 012-4-20 Radius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse1) Im(Ktransverse1) Re( Ktransverse2) Im(Ktransverse2) Re( Kaxial) Im(Kaxial) Fig. II.3. The eigen force constants for a particle (21.57sphereε= ) trapped by a linear polarized Laguerre-Gaussian beam with l=1, f=1, and N.A.= 1.2 in water (21.33waterε= ). 012-4-202 Radius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse1) Im(Ktransverse1) Re( Ktransverse2) Im(Ktransverse2) Re( Kaxial) Im(Kaxial) Fig. II.4. The eigen force constant for a particle (21.57sphereε= ) trapped by a right circularly polarized Lague rre-Gaussian beam with l=1, f=1, and N.A.= 1.2 in water (21.33waterε= ). 012-4-20 Radius (µm)Ki ( pN µm-1 mW-1) Re( Ktransverse1) Im(Ktransverse1) Re( Ktransverse2) Im(Ktransverse2) Re( Kaxial) Im(Kaxial) Fig. II.5. The eigen force constants for a particle (21.57sphereε= ) trapped by a left circularly polarized Lague rre-Gaussian beam with l=1, f=1, and N.A.= 1.2 in water (21.33waterε= ).C. Trapping beams that carry no angular momentum For an incident trapping beam that carries no angular moment, 0 dg==as there is no rotating energy flux on the transverse plane (see main text). Accordingly, Eq. (II.4) can never be fulfilled, and thus the eigen force constants are always real numbers. From another perspective, the complex eige n force constants can only occur when the particle is allowed to excha nge its angular momentum with the beam (see main text). Since the beam carries no angular moment um, complex eigen force constant should not occur. The force constant matrix reduces to 00 00a Kb efc⎡⎤ ⎢⎥=⎢⎥ ⎢⎥⎣⎦I , (II.5) and the corresponding eige n force constants are 1 2, , ,axial transverse transverseKc KaKb= = = (II.6) which are indeed real. The eigenvalues of a linearly polarized Gaussian beam, which carries no angular momentum, are plotted in Fi g. II.1. Clearly, its eigenvalues are real numbers. The nature of optical trapping by a beam that carries no angular momentum will be qualitatively similar to that of the conventional optical tweezers. D. Cylindrically symmetric trapping beams that carry angular momentum A cylindrically symmetric optical vortex beam propagates along a helical path, which can drive the trapped particle to rotate (so 0 dg=−≠ ). Moreover, owing to the cylindrical symmetry, ab=. As such, the condition Eq. (II.4) is always fulfilled. Consequently, the corresponding transverse eigen force constants are always a conjugate pair of complex numbers. To show this explicitly, consider a cylindr ically symmetric optical vortex, such as a circularly polarized beam (Gaussian or Laguerre-Gaussian ). It can be shown that a = b, as the restoring force acting on the particle when it is displaced along the x axis is equal to that of the y axis. Moreover, dg=− because the induced torque when the particle is displaced along the x axis is equal to that of the y axis. Finally, ef= because the induced force along the z axis when the particle is displaced along the x axis is equal to that of the y axis. Substituting these expressions into (II.1), we obtain 0 0ad Kd a ee c⎡ ⎤ ⎢ ⎥=−⎢ ⎥ ⎢ ⎥⎣ ⎦I , (II.7) and the corresponding eige n force constants are , .axial transKc Ka i d±= =± (II.8) From Eq. (II.8), we see that complex eigen force constants occur whenever 0d≠, as for any angular momentum carrying beam. In fact 0d≠ indicates that there are angular momentum exchange between the beam and the particle, because complex eigenvalues can exist only when the tr apped particle can exchange angular momentum with the beam (see main text). It is clear from (II.8) that the equilibrium cannot be solely characterized by real optic al force constants. Loosely speaking, a particle in a cylindrically symmetric optical vortex can be considered as simultaneously experiencing a radial restoring force characterized by Re( )transKa= and a torque about the beam’s axis characterized by Im( )transKd=. Fig. II.2, Fig. II.4, and Fig. II.5 show Ki versus the radius of the trapped sphere, for a circularly polarized Gaussian beam, a right circularly polarized Laguerre-Gaussian beam, and a left circul arly polarized Laguerre-Gaussian beam, respectively. Before entering the objective lens, the non-fo cused circularly polarized Gaussian beam carries spin angular moment um due to its polarization state, but not orbital angular momentum. After the beam is being strongly focused by the objective lens, part of its spin angular momentum is converted to orbital angular momentum (see main text). The left and right circul arly polarized Laguerre-Gaussian beams have both spin and orbital angular momentum, in the former (later) case, the two forms of angular momentum are in opposite (same) dire ction. After focusing, the spin angular momentum is partially convert ed to orbital angular momentum. In the left (right) circular polarization case, th e resultant angular momentum is small (large), owing to the cancellation (reinforcemen t) between the spin and orbital angular momentum. Consequently, Im{ Ktrans} is the largest (smallest) for the right (left) circularly polarized Laguerre-Gaussian be am in general, because the spin and orbital angular momentum are reinforcing (cancelling) each others. For all three beams, 'axialKs are always real and nega tive, indicating that the particle can be trapped along the axial di rection due to gradient forces. On the contrary, transK± are a conjugate pairs of complex numbers. For the Gaussian beam, {} Re 0transK±<, indicating that the particle can always be stabilized by introducing sufficient damping. On the other hand, for the Laguerre-Gaussian beams, {} Re 0transK±>for particles that are smaller than th e intensity ring of the beam, which means that small dielectric particles are uns table, as reported in experiments. Small dielectric particles are attracted toward in tensity maxima. Under sufficient damping, these small particles will be orbiting along the high intensity ring of the beam. On the other hand, {} Re 0transK±<for large particle. The spheres are bigger than the intensity ring, so that the gradient force drives the spheres to the beam center. A phase diagram is given in Fig. II.6, for a right circularly polarized LG beam with wavelength 1064 nmλ= , topological charge l=1, numerical aperture N.A. =1.2, and filling factor f=1. The trapped sphere is in water (21.33waterε= ) and it has dielectric constant 21.57sphereε= and mass density31050 kg mρ−= . At radius 0.39R mµ= , criticalγ→∞ as {} Re 0iK→. The equilibrium point at (, ) ( 0 , 0 )xy= is unstable for 0.39R mµ< for any values of damping. The particle will be trapped in the ring of the beam instead. For 0.39R mµ> , the white region (criticalγγ> ) is the regime where sufficient damping can stabilize the particle, and the shaded region (criticalγγ< ) is where the damping is insufficient to stabilize the particle. Compare Fig. II.6 with Fig. 2(b) of the main text, there two major differences. Firstly, criticalγ is always greater than zero for the right circular polarization, whereas for the linear polarization, criticalγ can be zero for some particle sizes. This is because the linear polarization does not posse ss cylindrical symmetric, therefore the condition Eq. (II.4) cannot always be fulfilled. Secondly, in general, the magnitude of criticalγ for the right polarization is greater than that of the linear polarization. This is because the angular momentum of the right circularly polarized beam comes from both the spin and orbital angular momentum that are reinforcing each other, whereas that of the linearly polarized beam comes fr om the orbital angular momentum only. In both polarization, the envelope for criticalγ increases linearly for large particle in general (see the blue dotted line in Fig. II.6) . This is because the envelope of the force, and thus that of the eigen for ce constants, are proportional to R2 (i.e. proportional to the geometrical cross section) for large particle. Then, according to (I.17), the envelope of criticalγ increases linearly. 01205001000 γ ( pN µm-1 µs ) Radius (µm)(b) Fig. II.6 Phase diagram for a particle tra pped at a power of 1W. The white (shaded) regions are unstable (stable). The black line iscriticalγ . The incident beam is a right circularly polarized LG beam with 1064 nmλ= , l=1, f=1, and N.A.= 1.2.

2009-08-31

We propose a rigorous theory for the optical trapping by optical vortices, which is emerging as an important tool to trap mesoscopic particles. The common perception is that the trapping is solely due to the gradient force, and may be characterized by three real force constants. However, we show that the optical vortex trap can exhibit complex force constants, implying that the trapping must be stabilized by ambient damping. At different damping levels, particle shows remarkably different dynamics, such as stable trapping, periodic and aperiodic orbital motions.

Rigorous Theory of Optical Trapping by an Optical Vortex Beam

0908.4504v1

arXiv:0910.0163v1 [cond-mat.mes-hall] 1 Oct 2009Spin motive forces and current fluctuations due to Brownian motion of domain walls M.E. Lucassen, R.A. Duine Institute for Theoretical Physics, Utrecht University, Le uvenlaan 4, 3584 CE Utrecht, The Netherlands Abstract We compute the power spectrum of the noise in the current due to s pin mo- tive forces by a fluctuating domain wall. We find that the power spect rum of the noise in the current is colored, and depends on the Gilbert dampin g, the spin transfer torque parameter β, and the domain-wall pinning potential and magnetic anisotropy. We also determine the average current induc ed by the thermally-assisted motion of a domain wall that is driven by an extern al mag- netic field. Our results suggest that measuring the power spectru m of the noise in the current in the presence of a domain wall may provide a new meth od for characterizing the current-to-domain-wall coupling in the system . Keywords: A. Magnetically ordered materials; A. Metals; A. Semicon ductors; D. Noise Pacs numbers: 72.15 Gd, 72.25 Pn, 72.70 +m 1. Introduction Voltage noise has long been considered a problem. Engineers have be en con- cerned with bringing down noise in electric circuits for more than a cen tury. The seminal work by Johnson[1] and Nyquist[2] on noise caused by t hermal ag- itation of electric charge carriers (nowadays called Johnson-Nyqu ist noise) was largely inspired by the problem caused by noise in telephone wires. The experi- mental work by Johnson tested the earlier observations by engine ers that noise increases with increasing resistance in the circuit and increasing tem perature. He was able to show that there would always be a minimal amount of nois e, be- yond which reduction of the noise is not possible, thus providinga ver ypractical tool for people working in the field. At the same time, the theoretica l support for these predictions was given by Nyquist. It is probably not a coinc idence Email address: m.e.lucassen@uu.nl (M.E. Lucassen) Preprint submitted to Elsevier December 5, 2018that, at the time of his research, Nyquist worked for the American Telephone and Telegraph Company. As long as noise is frequency-independent, i.e., white like Johnson-Ny quist noise, it is indeed often little more than a nuisance (a notable exceptio n to this is shot noise[3] at large bias voltage). However, frequency-de pendent, i.e., colored noise can contain interesting information on the system at h and. For example, in a recent paper Xiao et al.[4] show that, via the mechanism of spin pumping[5], a thermally agitated spin valve emits noisy currents with a c olored power spectrum. They show that the peaks in the spectrum coincid e with the precession frequency of the free ferromagnet of the spin valve. This opens up the possibility of an alternative measurement of the ferromagnetic resonance frequencies and damping, where one does not need to excite the sy stem, but only needs to measure the voltage noise power spectrum. Here, we see that properties of the noise contain information on the system. Clearly, this proposal only worksif the Johnson-Nyquistnoise is not too large comparedto the colored noise. Not only precessing magnets in layered structures induce current s: Re- cent theoretical work has increased interest in the inverse effect of current- driven domain-wall motion, whereby a moving domain wall induces an ele ctric current[6, 7, 8, 9]. Experimentally, this effect has been seen recen tly with field- driven domain walls in permalloy wires[10]. These so-called spin motive for ces ultimately arise from the same mechanism as spin pumping induced by th e pre- cessing magnet in a spin valve, i.e., both involve dynamic magnetization t hat induces spin currents that are subsequently converted into a cha rge current. In this paper, we study the currents induced by domain walls at nonz ero temperature. In particular, we determine the (colored) power sp ectrum of the emitted currents due to a fluctuating domain wall, both in the case of an un- pinned domain wall (Sec. 2.2), and in the case of a domain wall that is ex - trinsically pinned (Sec. 2.3). We also compute the average current in duced by a field-driven domain wall at nonzero temperature. We end in Sec. 4 w ith a short discussion and, in particular, compare the magnitude of the c olored noise obtained by us with the magnitude of the Johnson-Nyquist noise. 2. Spin motive forces due to fluctuating domain walls In this section, we compute the power spectrum of current fluctu ations due to spin motive forces that arise when a domain wall is thermally fluctua ting. We consider separately the case of intrinsic and extrinsic pinning. 22.1. Model and approach The equations of motion for the position Xand the chirality φof a rigid domain wall at nonzero temperature are given by[11, 12, 13] ˙X λ=α˙φ+K⊥ /planckover2pi1sin2φ+/radicalbigg D 2η1, (1) ˙φ=−α˙X λ+Fpin+/radicalbigg D 2η2, (2) whereαis Gilbert damping, K⊥is the hard-axis anisotropy, and λ=/radicalbig K/Jis the domain-wall width, with Jthe spin stiffness and Kthe easy-axisanisotropy. We introduce a pinning force, denoted by Fpin, to account for irregularities in the material. We have assumed that the pinning potential only depen ds on the position of the domain wall. Pinning sites turn out to be well-described b y a po- tentialthat isquadraticin X, suchthat wecantake Fpin=−2ωpinX/λ[11]. The Gaussian stochastic forces ηidescribe thermal fluctuations and are determined by /angbracketleftηi(t)/angbracketright= 0 ; /angbracketleftηi(t)ηj(t′)/angbracketright=δijδ(t−t′). (3) They obey the fluctuation-dissipation theorem[12] D=2αkBT /planckover2pi1NDW. (4) Note that in this expression, the temperature Tis effectively reduced by the number of magnetic moments in the domain wall NDW= 2λA/a3, withAthe cross-sectional area of the sample, and athe lattice spacing. Up to linear order inthe coordinate φ, validwhen K⊥> kBT, wecanwritethe equationsofmotion in Eqs. (1) and (2) as ∂t/vector x=M/vector x+N/vector η , (5) where M=2 1+α2 −αωpinK⊥ /planckover2pi1 −ωpin−αK⊥ /planckover2pi1 ;/vector x=/parenleftbiggX λ φ/parenrightbigg , (6) and N=1 1+α2/radicalbigg αkBT NDW/planckover2pi1/parenleftbigg1α −α1/parenrightbigg ;/vector η=/parenleftbiggη1 η2/parenrightbigg . (7) We readily find that the eigenfrequencies of the system, determine d by the eigenvalues Λ ±of the matrix M, are Λ±≡iω±−Γ∓=−α 1+α2/parenleftbigg ωpin+K⊥ /planckover2pi1/parenrightbigg ±α 1+α2/radicalBigg/parenleftbigg ωpin−K⊥ /planckover2pi1/parenrightbigg2 −4 α2ωpinK⊥ /planckover2pi1, (8) 30.0001 0.0002 0.0003 0.0004/HBarΩpin/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee -1.5-1-0.50.511.52xΑ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt1/Plus Α2 /HBarΩ/PlusMinus/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee/HBar/CΑpGΑmmΑ/PlusMinus/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee Figure 1: Values of Γ ±(red curves) and ω±(blue curves) as a function of the pinning for α= 0.02 . with both the eigenfrequencies ω±and their damping rates Γ ±real numbers. Their behavior as a function of /planckover2pi1ωpin/K⊥is shown in Fig. 1. Note that this expression has an imaginary part for pinning potentials that obey /planckover2pi1ωpin/K⊥≥ (α/2)2, and, because for typical materials the damping assumes values α∼ 0.01−0.1, the eigenfrequency assumes nonzero values already for very s mall pinning potentials. Without pinning potential ( ωpin= 0) the eigenvalues are purely real-valued and the motion of the domain wall is overdamped sin ce Γ±≥ 0 andω±= 0. If we include temperature, we find from the solution of Eq. (5) [witho ut loss of generality we choose X(t= 0) =φ(t= 0) = 0] that the time derivatives of the collective coordinates are given by ∂t/vector x(t) =MeMt/integraldisplayt 0dt′e−Mt′N/vector η(t′)+N/vector η(t), (9) for one realization of the noise. By averaging this solution over realiz ations of the noise, we compute the power spectrum of the currentinduced by the domain wall under the influence of thermal fluctuations as follows. It was shown by one of us[8] that up to linear order in time derivatives , the current induced by a moving domain wall is given by I(t) =−A/planckover2pi1 |e|L(σ↑−σ↓)/bracketleftBigg ˙φ(t)−β˙X(t) λ/bracketrightBigg , (10) withLthe length of the sample, and βthe sum of the phenomenological dissi- pative spin transfer torque parameter[14] and non-adiabatic con tributions. The power spectrum is defined as P(ω) = 2/integraldisplay+∞ −∞d(t−t′)e−iω(t−t′)/angbracketleftI(t)I(t′)/angbracketright. (11) 4Note that in this definition the power spectrum has units [ P] = A2/Hz, not to be mistaken with the power spectrum of a voltage-voltage correlat ion, which has units [ P] = V2/Hz. In both cases, however, the power spectrum can be seen as a measure of the energy output per frequency interval. W e introduce now the matrix O=/parenleftbiggA/planckover2pi1 |e|L/parenrightbigg2 (σ↑−σ↓)2/parenleftbiggβ2−β −β1/parenrightbigg , (12) so that we can write the correlations of the current as /angbracketleftI(t)I(t′)/angbracketright=/angbracketleftBig [∂t/vector x(t)]TO∂t′/vector x(t′)/angbracketrightBig = /integraldisplayt 0/integraldisplayt′ 0dt′′dt′′′/angbracketleftBig /vector η(t′′)TNTeMT(t−t′′)MTOMeM(t′−t′′′)N/vector η(t′′′)/angbracketrightBig +/integraldisplayt 0dt′′/angbracketleftBig /vector η(t′′)TNTeMT(t−t′′)MTON/vector η(t′)/angbracketrightBig +/integraldisplayt′ 0dt′′/angbracketleftBig /vector η(t)TNTOMeM(t′−t′′)N/vector η(t′′)/angbracketrightBig +/angbracketleftBig /vector η(t)TNTON/vector η(t′)/angbracketrightBig = θ(t−t′)/braceleftBigg/integraldisplayt′ 0dt′′Tr/bracketleftBig NTeMT(t−t′′)MTOMeM(t′−t′′)N/bracketrightBig +Tr/bracketleftBig NTeMT(t−t′)MTON/bracketrightBig/bracerightBigg +θ(t′−t)/braceleftBigg/integraldisplayt 0dt′′Tr/bracketleftBig NTeMT(t−t′′)MTOMeM(t′−t′′)N/bracketrightBig +Tr/bracketleftBig NTOMeM(t′−t)N/bracketrightBig/bracerightBigg +δ(t−t′)Tr/bracketleftBig NTON/bracketrightBig .(13) We evaluate the traces that appear in this expression to find that t he power spectrum is given by P(ω) = 2/parenleftbiggA/planckover2pi1 |e|L/parenrightbigg2(σ↑−σ↓)2 1+α2αkBT /planckover2pi1NDW×/bracketleftBigg (1+β)2−/braceleftBigg (1+β2)(1+α2)2/parenleftBigg /planckover2pi1ωpin K⊥/parenrightBigg2 −/bracketleftBigg β2−α2+2(1+β2)/planckover2pi1ωpin K⊥+(1−α2β2)/parenleftBigg /planckover2pi1ωpin K⊥/parenrightBigg2/bracketrightBigg/parenleftBigg /planckover2pi1ω K⊥1+α2 2/parenrightBigg2/bracerightBigg/slashBig /braceleftBigg (1+α2)2/parenleftBigg /planckover2pi1ωpin K⊥/parenrightBigg2 +/bracketleftBigg α2−2/planckover2pi1ωpin K⊥+α2/parenleftBig/planckover2pi1ωpin K⊥/parenrightBigg2/bracketrightBigg/parenleftBigg /planckover2pi1ω K⊥1+α2 2/parenrightBigg2 +/parenleftBigg /planckover2pi1ω K⊥1+α2 2/parenrightBigg4/bracerightBigg/bracketrightBigg . (14) 52.2. Domain wall without extrinsic pinning We first consider a domain wall with Fpin= 0. In this case, only the chirality φdetermines the energy, a situation referred to as intrinsic pinning[1 1]. From the result in Eq. (14) we find that the power spectrum is given by P(ω) = 2/parenleftbiggA/planckover2pi1 |e|L/parenrightbigg2(σ↑−σ↓)2 1+α2αkBT /planckover2pi1NDW× /braceleftBigg (1+β)2+β2−α2 α2/bracketleftBig 1+/parenleftBig/planckover2pi1ω K⊥1+α2 2α/parenrightBig2/bracketrightBig−1/bracerightBigg . (15) Indeed, we find that next to a constant contribution there is also a frequency- dependent contribution for β/negationslash=α, i.e., the power spectrum is colored. The fact thatβ=αis a special caseis understood from the fact that in that case we ha ve macroscopic Galilean invariance. This translates to white noise in the c urrent correlations. The power spectrum is a Lorentzian, centered arou ndω= 0 because the domain wall is overdamped in this case, with a width deter mined by the damping rate in Eq. (8) as /planckover2pi1Γ+/K⊥= 2α/(1 +α2). Relative to the white-noise contribution PW= 2(1+β)2/parenleftbiggA/planckover2pi1 |e|L/parenrightbigg2(σ↑−σ↓)2 1+α2αkBT /planckover2pi1NDW, (16) the height of the peak is given by ∆ P=PW(β2−α2)/α2. The behavior of the power spectrum is illustrated in Fig. 2 for several values of β/α. -0.3 -0.2 -0.1 0.1 0.2 0.3/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee1234/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW Β/Slash1Α/EquΑl2Β/Slash1Α/EquΑl1.5Β/Slash1Α/EquΑl1Β/Slash1Α/EquΑl0.6Β/Slash1Α/EquΑl0 Figure 2: The power spectrum for α= 0.02 and several values of β. 2.3. Extrinsically pinned domain wall For extrinsically pinned domain walls the behavior of the power spectr um givenbyEq.(14) isdepicted in Fig.3. We seethat for /planckover2pi1ωpin/K⊥/greaterorsimilarα2the peaks 60.05 0.1 0.15 0.2/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee12345/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW /HBarΩpin/EquΑl0.002 K/UpTee/HBarΩpin/EquΑl0.001 K/UpTee/HBarΩpin/EquΑl0.0005 K/UpTee/HBarΩpin/EquΑl0 (a) 0.05 0.1 0.15 0.2/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee2468/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW /HBarΩpin/EquΑl0.002 K/UpTee/HBarΩpin/EquΑl0.001 K/UpTee/HBarΩpin/EquΑl0.0005 K/UpTee/HBarΩpin/EquΑl0 (b) 0.05 0.1 0.15 0.2/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee5001000150020002500/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW /HBarΩpin/EquΑl0.002 K/UpTee/HBarΩpin/EquΑl0.001 K/UpTee/HBarΩpin/EquΑl0.0005 K/UpTee/HBarΩpin/EquΑl0 (c) Figure 3: The power spectrum as a function of the frequency ωand the pinning potential ωpin forβ=α/2 (a),β= 2α(b) and β= 50α(c), all for α= 0.02. 7in the power spectrum are approximately centered around the eige nfrequencies /planckover2pi1ω/K⊥≃ ±2/radicalbig /planckover2pi1ωpin/K⊥, consistent with Eq. (8). We can discern between two regimes, one where β∼αin figs. 3 (a-d), and one where β≫αin figs. 3 (e-f). In the former regime, the height of the peaks in the power s pectrum depend strongly on the pinning. For small ωpinwe see a clear dependence on the value of β, whereas for large ωpinthis dependence is less significant. In the regime of large β, the height of the peaks hardly depends on the pinning and is approximately given by P≃PWβ2/α2. Note that the width of the peaks is independent of β. For pinning potentials α2</planckover2pi1ωpin/K⊥the width is given by /planckover2pi1Γ/K⊥=α(1+/planckover2pi1ωpin/K⊥)/(1+α2), so for/planckover2pi1ωpin≪K⊥the dependence of the width on the pinning is negligible. 3. Spin motive forces due to thermally-assisted field-drive n domain walls In this section we compute the average current that is induced by a moving domain wall. The domain wall is moved by applying an external magnetic fi eld parallel to the easy axis of the sample. In this section we take into ac count temperature but ignore extrinsic pinning (see Ref. [15] for a calcula tion of spin motive forces in a weakly extrinsically pinned system for β= 0). The equations of motion for a field-driven domain wall are then given by[11, 12, 13] ˙X λ=α˙φ+K⊥ /planckover2pi1sin2φ+/radicalbigg D 2η1, (17) ˙φ=−α˙X λ+gµBBz /planckover2pi1+/radicalbigg D 2η2, (18) wheregBzis the magnetic field in the z direction, and the stochastic forces are determined by Eq.(3) with the strength givenbyEq. (4). In earlier work[13], we computed from these coupled equations the average drift velocity of a domain wall (here, we set the applied spin current zero) α/angbracketleft˙X/angbracketright λ=−/angbracketleft˙φ/angbracketright+gµBBz /planckover2pi1, (19) with (we omit factors 1+ α2≃1) /angbracketleft˙φ/angbracketright=−2παkBT /planckover2pi1NDW/parenleftbig e−2πgµBBzNDW αkBT−1/parenrightbig/slashBig /braceleftBigg/integraldisplay2π 0dφeNDW kBT/parenleftbig gµBBz αφ+K⊥ 2cos2φ/parenrightbig/bracketleftBigg/integraldisplay2π 0dφ′e−NDW kBT/parenleftbig gµBBz αφ′+K⊥ 2cos2φ′/parenrightbig +/parenleftbig e−2πgµBBzNDW αkBT−1/parenrightbig/integraldisplayφ 0dφ′e−NDW kBT/parenleftbig gµBBz αφ′+K⊥ 2cos2φ′/parenrightbig/bracketrightBigg/bracerightBigg .(20) 8Combining Eqs. (10) and (19) gives us the average current straigh tforwardly /angbracketleftI/angbracketright=−A/planckover2pi1 α|e|L(σ↑−σ↓)/bracketleftbigg (α+β)/angbracketleft˙φ/angbracketright−βgµBBz /planckover2pi1/bracketrightbigg . (21) We evaluate this expression for several temperatures in Fig. 4. Th e black curve 0.5 1 1.5 2 2.5 3Bz/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExtBW -3-2-112/LAngleBracket1I/RAngleBracket1/VertBar1e/VertBar1L/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtA/Spaceg/SpaceΜB/SpaceBW/Space/LParen1Σ/UpArrow/MinuΣ Σ /DownArrow/RParen1 kBT/EquΑl10K/UpTeeNkBT/EquΑl0.5K/UpTeeNkBT/EquΑl0.1K/UpTeeNkBT/EquΑl0 Figure 4: The average current induced by a domain wall as a fun ction of the magnetic field applied to this domain wall. We show curves for several tempe ratures, with α= 0.02 and β= 2α. The normalization of the axes BWis the Walker-breakdown field which is given by BW=αK⊥/gµB. in Fig. 4 is computed at zero temperature. It increases linearly with t he field up to the Walker-breakdown field BW=αK⊥/gµB, where it reaches a max- imum current of /angbracketleftI/angbracketright|e|L/Agµ BBW(σ↑−σ↓) =β/α. Then, it drops and even changes sign. This curve is consistent with the curves obtained by o ne of us[8]. There, an open circuit is treated where there is no current but a vo ltage. The voltage is related to the current as ∆ V=/angbracketleftI/angbracketrightL/A(σ↑+σ↓), such that indeed ∆V/V0=/angbracketleftI/angbracketright|e|L/Agµ BBW(σ↑−σ↓), where the normalization is defined as V0=PgµBBW/|e|and the polarization is given by P ≡(σ↑−σ↓)/(σ↑+σ↓). In- creasing temperatures smoothen the peak around the Walker-br eakdown field, and for high temperatures, the peak vanishes altogether. There fore, for fields smaller than the Walker-breakdown field and for small temperature s, the ther- mal fluctuations tend to decrease the average induced current. However, for higher temperatures, the current reverses and increases again . For fields suffi- ciently larger than the Walker-breakdown field, we always find a slight increase of the current as a function of temperature. 4. Discussion and conclusions In Sec. 2 we computed power spectra for domain walls, both with and with- out extrinsic pinning. In ferromagnetic metals, the spin transfer t orque param- eter has values β∼α, for which we see that for large pinning /planckover2pi1ωpin/K⊥/greaterorsimilarα2 9the power spectra only weakly depend on the spin transfer torque parameter β. Therefore, determination of βis only possible for very small pinning potentials. In an ideally clean sample without extrinsic pinning the height and sign of the peak in the power spectrum can give a clear indication whether βis smaller, (approximately) equal or larger than α. In order to perform these experiments, the contributions due to the domain wallshouldnotbecompletelyoverwhelmedbyJohnson-Nyquistnoise . Theratio of the peaks in our power spectrum and the Johnson-Nyquist noise determine the resolution of the experiment, and we want it to be at least of the order of a percent. We use that the resistance of the domain wall is small, a nd the Johnson-Nyquist noise is governed by the resistance of the wir e. For a wire of length Land cross-section A, the power spectrum due to Johnson- Nyquist noise is given by PJN= 4kBT(σ↑+σ↓)A/L. From section (2.2) we can estimate the ratio of the height of the peak and the Johnson-Nyqu ist noise as ∆P/PJN≃[(β/α)2−1]αA/planckover2pi1(σ↑+σ↓)/(4LNDWP2e2). To make a rough estimate of this ratio, we use λ≃20nm such that A/NDW≃2·10−2˚A2fora≃2˚A, a polarization P ≃0.7 and a conductivity σ↑+σ↓≃107A/Vm. We then find ∆P/PJN≃10−2˚A/L, which shows that the wire length must satisfy L <1˚A in order to have a resolution of 1%. We therefore conclude that this effect at zero pinning is impossible to see in experiment, where wires are typically at least of the order of L≃10µm, i.e., five orders of magnitude larger. We can try to increase the signal by turning on a pinning potential. Insertio n of the eigenfrequencies ω±from Eq. (8) in Eq. (14) shows us that we can maximally gain a factor α−2≃104forα= 0.01, as illustrated in Fig. 5. We can also still 20 40 60 80 100/Radical3/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/HBarΩpin/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee0.20.40.60.81/LParen11/PlusΒ/RParen12Α2 P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW Β/EquΑl1Β/EquΑl0.02 Figure 5: The height of the peaks in the power spectrum as a fun ction of the pinning. We usedα= 0.02 and two values for β. We checked that the curves do not differ significantly for different values for αand therefore, the maximal value of the peaks in the power spe ctrum is ∆P(1+β2)/PJN≃α−2. Note that the dependence on βis negligible for large pinning. This curve was obtained by inserting the eigenfrequencies ω±from Eq. (8) in the expression for the power spectrum in Eq. (14). gain some resolution by considering a domain wall in a nanoconstriction , where 10the width can be as small as λ= 1nm[16]. All this adds up to a resolution of ∆P/PJN∼1%. Therefore, with all parameter values set to ideal values it appears to be possible to observe our predictions in ferromagnet ic metals, although the experimental challenges are big. In a magnetic semiconductorlike GaMnAs the number ofmagnetic mom ents in the domain wall is two orders of magnitude smaller than in a ferromag netic conductor. This increases the ratio ∆ P/PJNwith a factor 100. However, the conductivity is also considerably smaller than that in a ferromagnetic metal. The conductivity of GaMnAs depends on many parameters, but a re asonable estimate is that it is at least about three orders of magnitude smaller than that of a metal, although usually even smaller. Therefore, the adva ntage of a small number of magnetic moments is cancelled. Another property, however, of GaMnAs is that the parameter βcan assume considerably higher values[17], up toβ= 1. This does not influence the maximal value of the power spectrum , but it does dramatically increase the value for small pinning. Note that there are contributions to the noise that we did not discu ss in this article. For example, the time-dependent magnetic field caused by a moving domain wall will induce electric currents that contribute to the color ed power spectrum. Distinguishing such contributions from the spin motive fo rces was essential for the experimental results by Yang et al.[10], and would also be important here. In Sec. 3, we calculated the current induced by a field-driven domain wall under the influence of temperature. We estimate that in ferromag netic metals at room temperature that kBT/K⊥N≃10−3, which is indistinguishable from the zero-temperature curve in Fig. 4. However, for magnetic sem iconductors, like GaMnAs, we find kBT/K⊥N≃10−1atT= 100K, which corresponds to the red curve in Fig. 4. We therefore expect finite-temperature e ffects to be important in magnetic semiconductors like GaMnAs. 5. acknowledgement This work was supported by the Netherlands Organization for Scien tific Re- search(NWO) and by the EuropeanResearchCouncil (ERC) under the Seventh Framework Program. We would like to thank Yaroslav Tserkovnyak f or useful discussions. References [1] J.B. Johnson, Nature 119, 50 (1927); Phys. Rev. 29, 367 (1927); Phys. Rev.32, 97 (1928). [2] H. Nyquist, Phys. Rev. 29, 614 (1927); Phys. Rev. 32, 110 (1928). [3] M.J.M. de Jong and C.W.J. Beenakker, in Mesoscopic Electron Transport , edited by L.L. Sohn, L.P. Kouwenhoven and G. Schoen, NATO ASI Ser ies (Kluwer Academic, Dordrecht, 1997), Vol. 345, pp. 225-258. 11[4] J. Xiao, G.E.W. Bauer, S. Maekawa and A. Brataas, Phys. Rev. B 79, 174415 (2009). [5] Y. Tserkovnyak, A. Brataas and G.E.W. Bauer, Phys. Rev. Lett .88, 117601 (2002). [6] S.E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 (2007). [7] W.M. Saslow, Phys. Rev. B 76, 184434 (2007). [8] R.A. Duine, Phys. Rev. B 77, 014409 (2008); R.A. Duine, Phys. Rev. B 79, 014407 (2009). [9] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77, 134407 (2008). [10] S.A. Yang, G.S.D. Beach, C. Knutson, D. Xiao, Q. Niu, M. Tsoi, and J.L. Erskine, Phys. Rev. Lett. 102, 067201 (2009). [11] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004); 96, 189702 (2006). [12] R.A. Duine, A.S. N´ u˜ nez and A.H. MacDonald, Phys. Rev. Lett. 98, 056605 (2007). [13] M.E. Lucassen, H.J. van Driel, C. Morais Smith and R.A. Duine, Phys . Rev. B79, 224411 (2009). [14] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [15] V. Lecomte, S.E. Barnes, J.-P. Eckmann and T. Giamarchi, Phys . Rev. B 80, 054413 (2009). [16] P. Bruno, Phys. Rev. Lett. 83, 2425 (1999). [17] K.M.D. Hals, A.K. Nguyen and A. Brataas, Phys. Rev. Lett. 102, 256601 (2009). 12

2009-10-01

We compute the power spectrum of the noise in the current due to spin motive forces by a fluctuating domain wall. We find that the power spectrum of the noise in the current is colored, and depends on the Gilbert damping, the spin transfer torque parameter $\beta$, and the domain-wall pinning potential and magnetic anisotropy. We also determine the average current induced by the thermally-assisted motion of a domain wall that is driven by an external magnetic field. Our results suggest that measuring the power spectrum of the noise in the current in the presence of a domain wall may provide a new method for characterizing the current-to-domain-wall coupling in the system.

Spin motive forces and current fluctuations due to Brownian motion of domain walls

0910.0163v1

arXiv:0910.1477v2 [cond-mat.mtrl-sci] 2 Feb 2010Fast domain wall propagation under an optimal field pulse in m agnetic nanowires Z. Z. Sun and J. Schliemann Institute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany (Dated: December 2, 2018) Weinvestigatefield-drivendomainwall (DW)propagation in magneticnanowiresintheframework of the Landau-Lifshitz-Gilbert equation. We propose a new s trategy to speed up the DW motion in a uniaxial magnetic nanowire by using an optimal space-de pendent field pulse synchronized with the DW propagation. Depending on the damping parameter, the DW velocity can be increased by about two orders of magnitude compared to the standard case o f a static uniform field. Moreover, under the optimal field pulse, the change in total magnetic en ergy in the nanowire is proportional to the DW velocity, implying that rapid energy release is ess ential for fast DW propagation. PACS numbers: 75.60.Jk, 75.75.-c, 85.70.Ay Recently the study of domain wall (DW) motion in magnetic nanowires has attracted a great deal of atten- tion, inspired both by fundamental interest in nanomag- netism as well as potential industrial applications. Many interesting applications like memory bits[1, 2] or mag- netic logic devices[3] involve fast manipulation of DW structures, i.e. a large magnetization reversal speed. Ingeneral,themotionofaDWcanbedrivenbyamag- netic field [4–6] and/or a spin-polarized current[7–11]. Although the DW dynamics in systems of higher spatial dimension can be very complicated, some simple but im- portant results were obtained by Schryer and Walker for effectively one-dimensional (1D) situations[12]: At low field (or current density), the DW velocity vis linear in the field strength HuntilHreaches a so-called Walker breakdown field Hw[12]. Within this linear regime, DW propagates as a rigid object. For H > H w, the DW loses its rigidity and develops a complex time-dependent internal structure. The velocity can even oscillate with time due to the “breathing” of the DW width. The time- averaged velocity ¯ vdecreases with the increase of H, re- sulting in a negative differential mobility. ¯ vcan be again linear with Happroximately when H≫Hw. The pre- dictedv-Hcharacteristic is in a good agreement with experimental results on permalloy nanowires[4–6]. Re- cently a general definition of the DW velocity proper for any types of DW dynamics has been also introduced[13]. For a single-domain magnetic nanoparticle (called Stoner particle), an appropriate time-dependent but spa- tially homogeneous field pulse can substantially lower the switching field and increase the reversal speed since it acts as an energy source enabling to overcome the energy barrier for switching the spatially constant magnetization[14, 15]. In the present letter, we inves- tigate the dynamics of a DW in a magnetic nanowire under a field pulse depending both on time and space. As a result, such a pulse, synchronized with the DW propagation, can dramatically increase the DW velocity by typically two orders compared with the situation of a constant field. Moreover, the total magnetic energy typi- cally decreases with a rate being proportional to the DWvelocity, i.e. the external field source can even absorb energy from the nanowire. A Bzx y z x DWy FIG. 1: A schematic diagram of two dynamically equivalent 1D magnetic nanowire structures. (A) Easy axis is along the wire axis (z-axis); (B) Easy axis (z-axis) ⊥the wire axis (x- axis). The region between two dashed lines denotes the DW region. A magnetic nanowirecan be described as an effectively 1D continuum of magnetic moments along the wire axis direction. Magnetic domains are formed due to the com- petitionbetweentheanisotropicmagneticenergyandthe exchangeinteractionamongadjacentmagneticmoments. Let us first concentrate on the case of a uniaxial mag- netic anisotropy: Two dynamically equivalent configura- tions of 1D uniaxial magnetic nanowires are schemati- cally shown in Fig. 1. Type A shows the wire axis to be also the easy-axis (z-axis). Type B shows the easy axis (z-axis) is perpendicular to the wire axis (x-axis). Although our results described below apply to both con- figurations, we will focus in the following on type B. The spatio-temporal dynamics of the magnetization den- sity/vectorM(x,t) is governed by the Landau-Lifshitz-Gilbert (LLG) equation[16] ∂/vectorM ∂t=−|γ|/vectorM×/vectorHt+α Ms/parenleftBigg /vectorM×∂/vectorM ∂t/parenrightBigg ,(1) where|γ|= 2.21×105(rad/s)/(A/m) the gyromag- netic ratio, αthe Gilbert damping coefficient, and Ms is the saturation magnetization density. The total ef- fective field /vectorHtis given by the variational derivative of the total energy with respect to magnetization, /vectorHt=2 −(δE/δ/vectorM)/µ0, whereµ0the vacuum permeability. The total energy E=/integraltext∞ −∞dxε(x) can be written as an inte- gral over an energy density (per unit section-area), ε(x) =−KM2 z+J/bracketleftBigg/parenleftbigg∂θ ∂x/parenrightbigg2 +sin2θ/parenleftbigg∂φ ∂x/parenrightbigg2/bracketrightBigg −µ0/vectorM·/vectorH, (2) wherexis the spatial variable in the wire direc- tion. Here K,Jare the coefficients of energetic anisotropy and exchange interaction, respectively, and /vectorHis the external magnetic field. Moreover, we have adopted the usual spherical coordinates, /vectorM(x,t) = Ms(sinθcosφ,sinθsinφ,cosθ) where the polar angle θ(x,t) andthe azimuthalangle φ(x,t) depend onposition and time. Hence, the total field /vectorHtconsists of three parts: the external field /vectorH, the intrinsic uniaxial field along the easy axis /vectorHK= (2KMz/µ0)ˆz, and the exchange field /vectorHJwhich reads in spherical coordinates as[12, 17], HJ θ=2J µ0Ms∂2θ ∂x2−Jsin2θ µ0Ms/parenleftbigg∂φ ∂x/parenrightbigg2 , HJ φ=2J µ0Mssinθ∂ ∂x/parenleftbigg sin2θ∂φ ∂x/parenrightbigg . (3) Following Ref. [12], let us focus on DW structures ful- filling∂φ/∂x= 0, i.e. all the magnetic moments rotate around the easy axis synchronously. Then the dynamical equations take the form Γ˙θ=α/parenleftbigg Hθ−KMs µ0sin2θ+2J µ0Ms∂2θ ∂x2/parenrightbigg +Hφ, Γsinθ˙φ=αHφ−Hθ+KMs µ0sin2θ−2J µ0Ms∂2θ ∂x2,(4) where we have defined Γ ≡(1 +α2)|γ|−1, andHi(i= r,θ,φ) are the three components of the external field in spherical coordinates. In the absence of an exter- nal field, an exact solution for a static DW is given by tanθ(x) 2= exp(x/∆)where∆ =/radicalbig J/(KM2s)isthewidth of the DW. We note that a static DW can exist in a con- stant field only if the field component along the easy axis is zero,Hz= 0. In fact, according to Eqs. (4) static solu- tions need to fulfill Hφ= 0 [implying φ= tan−1(Hy/Hx) is spatially constant] and 2J µ0Ms∂2θ ∂x2−KMs µ0sin2θ+Hθ= 0 (5) or, upon integration, J µ0Ms/parenleftbigg∂θ ∂x/parenrightbigg2 +KMs 2µ0cos2θ+Hr(θ) = constant .(6) Considering the two boundaries at θ= 0(x→ −∞) and θ=π(x→+∞) for the DW, we conclude Hr(0) =Hr(π), which requires Hz= 0. In this case, the station- ary DW solutions under a transverse field are described asx=/integraltext [/radicalBig (KM2ssin2θ−µ0MsHsinθ)/J]−1dθ. Thus, when an external field with a component along the easy axis is applied to the nanowire, the DW is ex- pected to move. We use a travelling-wave ansatzto de- scribe rigid DW motion[12], tanθ(x,t) 2= exp/parenleftbiggx−vt ∆/parenrightbigg , (7) where the DW velocity vis assumed to be constant. Sub- stituting this trial function into Eq. (4), the dynamic equations become Γsinθv=−∆(αHθ+Hφ),Γsinθ˙φ=αHφ−Hθ.(8) Eq. (8) describes the dependence of the linear velocity v and the angular velocity ˙φon the external field /vectorH. Our following results discussion will be based on Eqs. (8). Let us firstturn to the caseofastatic field caseapplied along the easy axis (z-axis in type B of Fig. 1), Hθ= −Hsinθ,Hφ= 0. Here we recover the well-known static solution for a uniaxial anisotropy[18], v=|γ|∆H α+α−1, (9) where the azimuthal angle φ(t) =φ(0)+|γ|Ht/(1+α2) is spatially constant (i.e. ∂φ/∂x= 0) and increases linearly with time. Let us now allow the applied external field to depend both on space and time. Our task is to design, under a fixed field magnitude H, an optimal field configuration /vectorH(x,t) to increase the DW velocity as much as possible. From Eqs. (8), we find a manifold of solutions of specific space-time field configurations described by a parameter u, Hr(x,t) =Hcosθ, H θ(x,t) =−Hsinθ//radicalbig 1+u2, Hφ(x,t) =−uHsinθ//radicalbig 1+u2.(10) The velocities vand˙φreads v=|γ|∆H 1+α2α+u√ 1+u2,˙φ=|γ|H 1+α21−αu√ 1+u2.(11) The previous static field case is recovered for u= 0. The maximum of the velocity vmwith regard to uus reached foru= 1/α, vm=|γ|∆H√ 1+α2, (12) where the angular velocity is zero, ˙φ= 0. On the other hand,˙φattains a maximum for u=−α, where, in turn, the linear velocity vanishes. In Fig. 2 we have plotted3 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48 /s61/s50/s48/s110/s109 /s72/s61/s49/s48/s48/s79/s101 /s32/s32/s118/s32/s40/s109/s47/s115/s41 /s117/s32 /s61/s48/s46/s49 /s32 /s61/s48/s46/s50 /s32 /s61/s48/s46/s53 /s32 /s61/s48/s46/s56 FIG. 2: (Color online) The DW propagation velocity vver- sus the parameter uat the different damping values α= 0.1,0.2,0.5,0.8. The other parameters are chosen as ∆ = 20nmandH= 100Oe. the dependence of the velocity on the parameter ufor different damping strengths and typical values for the DW width ∆ and the magnitude Hof the external field. To understand the physical meaning of the maximum velocityvm, we note that, according to Eqs. (8), the field components HθandHφare required to be proportional to sinθto ensure the constant velocity under the rigid DW approximation. Moreover,at u= 1/αwe haveHθ= αHφ, and from the identity (αHθ+Hφ)2+(αHφ−Hθ)2= (1+α2)(H2−H2 r),(13) we conclude that the term ( αHθ+Hφ) is maximal re- sulting in a maximal velocity according to Eqs. (8). As a result, the new velocity under the optimal field pulse is largerby a factor of vm/v=√ 1+α2/α≈1/αcompared to a constant field with the same field magnitude. To give a practical example, the typical value for the damp- ing parameter in permalloy is α= 0.01 which results in an increase of the DW velocity by a factor of 100. It is instructive to also analyze the optimal field pulse according to Eq. (10) with u= 1/αin its cartesian com- ponents, Hx(x,t) =Hsin2θ(1−α//radicalbig 1+α2)/2, Hy(x,t) =−Hsinθ//radicalbig 1+α2, (14) Hz(x,t) =H(cos2θ+αsin2θ//radicalbig 1+α2), whereθfollowsthewave-likemotiontanθ(x,t) 2= exp(x ∆− |γ|H√ 1+α2t). In Fig. 3 we plotted these quantities at t= 0 around the DW center where the main spatial variation of the pulse occurs. Note that the space-dependent field distribution should move with the same speed vmsyn- chronizedwiththeDWpropagation. NeartheDWcenter the components HxandHzare (almost) zero whereas a largetransversecomponent Hyis required to achievefast DW propagation. Qualitatively speaking, the transverse field causes a precessionof the magnetization resulting inits reversal. This finding is consistent with recent micro- magnetic simulations showing that the DW velocity can be largely increased by applying an additional transverse field[19]. /s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48 /s61/s48/s46/s48/s49 /s61/s50/s48/s110/s109 /s72/s61/s49/s48/s48/s79/s101 /s32/s32/s72 /s120/s44/s121/s44/s122/s32/s40/s79/s101/s41 /s120/s32/s40/s110/s109/s41/s32/s72 /s120 /s32/s72 /s121 /s32/s72 /s122 FIG. 3: (Color online) The x,y,zcomponents of the optimal field pulse. The parameters are chosen as α= 0.01, ∆ = 20nmfor permalloy[6]. The field magnitude is H= 100Oe. It is also interesting to study the energy variation un- der the optimal field pulse, dE dt=−µ0/integraldisplay+∞ −∞dx/parenleftBigg ∂/vectorM ∂t·/vectorHt+/vectorM·∂/vectorH ∂t/parenrightBigg ≡Pα+Ph. (15) The first term Pαis the intrinsic damping power due to all kinds of damping mechanisms described by the phe- nomenologicalparameter α. According to the LLG equa- tionPα=−µ0α |γ|Ms/integraltext+∞ −∞dx(∂M ∂t)2is always negative[14], implying an energy loss. Phis the external power due to the time-dependent external field. From Eq. (11), both powers are obtained as Pα=−2α 1+α2µ0|γ|Ms∆H2, (16) Ph=2α(√ 1+u2−1)−2u (1+α2)√ 1+u2µ0|γ|Ms∆H2,(17) such that the total energy change rate is dE dt=−2µ0MsHv=−2(α+u)µ0|γ|Ms∆H2 (1+α2)√ 1+u2.(18) Note that the intrinsic damping power is independent of the parameter uand always negative, whereas the total energy change rate is proportional to the negative DW velocity. Thus, for positive velocities ( u >−α) the total magnetic energy decreases while it grows for negative ve- locities (u <−α). In the former case energy is absorbed by the external field source while in the latter case the field source provides energy to the system. The optimal field source helps to rapidly release or gain magnetic en- ergywhichisessentialforfastDWmotion. Thisaspectis verydifferent from the reversalof a Stoner particle where the time-dependent field is always needed to provide en- ergy to the system to overcome the energy barrier[14].4 Moreover, our new strategy of employing space- dependent field pulses can also be applied to uniaxial anisotropies of arbitrary type: Let w(θ) be the uniaxial magnetic energy density. The static DW solution in the absence of an external field reads x=/integraltext χ−1(θ)dθ, where χ(θ) =/radicalbig [w(θ)−w0]/J. (19) Herew0is the minimum energy density for magneti- zation along the easy axis. By performing analogous steps as before, we obtain the the optimal velocity as vm=|γ|H√ 1+α2χmax, whereχmaxdenotes the maximum of χ(θ) throughout all θ. On the other hand, our approach is not straightfor- wardlyextended tothecaseofamagneticwirewith biax- ial anisotropy. To see this, consider, a biaxial anisotropy εi=−KM2 z+K′M2 xwhere the coefficients K,K′cor- respond to the easy and hard axis, respectively[12]. The LLG equations read Γ˙θ=α/parenleftbigg Hθ−KMs µ0sin2θ−K′Ms µ0sin2θcos2φ +2J µ0Ms∂2θ ∂x2/parenrightbigg +Hφ+K′Ms µ0sinθsin2φ, Γsinθ˙φ=αHφ−Hθ+KMs µ0sin2θ−2J µ0Ms∂2θ ∂x2 +K′Ms µ0sin2θcos2φ+αK′Ms µ0sinθsin2φ.(20) Let us assume φ(x,t) =φ0is a constant deter- mined by the applied field. Substituting the travelling- waveansatztanθ(x,t) 2= exp/parenleftbigx−vt ∆/parenrightbig , where now ∆ =/radicalbig J/(K+K′cos2φ0)/Ms, into Eqs. (20) we obtain Γsinθv=−∆(αHθ+Hφ+K′Mssinθsin2φ0/µ0), (21) αK′Mssinθsin2φ0/µ0+(αHφ−Hθ) = 0.(22) For a static field along z-axis Hθ=−Hsinθ,Hφ= 0, the solution is just the Walker’s result v=|γ|∆H/α(Note here ∆ also depends on H)[12]. To implement our new strategy, we need to find the maximum of the right-hand side of Eq. (21) under two constraints of Eq. (22) and Eq. (13) with HθandHφbeing proportional to sin θ. The unique solution to this problem is indeed a constant field along the z-axis which is thus the optimal field con- figuration. In summary, our theory is general and can be applied to a magnetic nanowire with a uniaxial anisotropy which can be from shape, magneto-crystalline or the dipolar interaction. The experimental challenge of our proposal is obviously the generation of a field pulse focused on the DW region and synchronized with its motion. How- ever, the field sourcesynchronizationvelocity can be pre- calculated from the material parameters. As for the re- quired localized field (See Fig. 3), we propose to employa ferromagnetic scanning tunneling microscope (STM) tip to produce a localized field perpendicular to the wire axis[20] and use a localized current to produce an Oer- sted field along the wire axis[21]. Moreover, such re- quired localized fields may also be produced by nano- ferromagnetswithstrongferromagnetic(orantiferromag- netic) coupling to the nanowire. We also point out that, although the field source typically does not consume en- ergy but gain energy from the magnetic nanowire, the pulse source may still require excess energy to overcome effects such as defects pinning, which is not included in our model. At last, the generalization of the strategy be- yond the rigid DW approximation, and to DW motion induced by spin-polarized current will also be attractive direction of future research. Z.Z.S. thanks the Alexander von Humboldt Founda- tion (Germany) for a grant. This work has been sup- ported by Deutsche Forschugsgemeinschaft via SFB 689. [1] R. P. Cowburn, Nature (London) 448, 544 (2007). [2] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). [3] D. A. Allwood, et al., Science 309, 1688 (2005). [4] T. Ono, et al., Science, 284, 468 (1999). [5] D. Atkinson, et al., Nature Mater. 2, 85 (2003). [6] G. S. D. Beach, et al., Nature Mater. 4, 741 (2005); G. S. D. Beach, et al., Phys. Rev. Lett. 97, 057203 (2006); J. Yang, et al., Phys. Rev.B 77014413 (2008). [7] M. Klaui, et al., Phys. Rev. Lett. 94, 106601 (2005). [8] M. Hayashi, et al., Phys. Rev. Lett. 96, 197207 (2006); L. Thomas, et al., Nature (London) 443, 197 (2006); M. Hayashi, et al., Phys. Rev. Lett. 98, 037204 (2007). [9] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [10] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). [11] A. Thiaville, et al., Europhys. Lett. 69, 990 (2005). [12] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). [13] X.R.Wang, P.Yan, andJ.Lu, Europhys.Lett. 86, 67001 (2009); X. R. Wang, et al., Ann. Phys. (N.Y.) 324, 1815 (2009). [14] Z. Z. Sun and X.R. Wang, Phys. Rev. Lett. 97, 077205 (2006); Phys. Rev. B 73, 092416 (2006); ibid.74, 132401 (2006).; X. R. Wang and Z. Z. Sun, Phys. Rev. Lett. 98, 077201 (2007). [15] X. R. Wang, et al., Europhys. Lett. 84, 27008 (2008). [16] Z. Z. Sun and X. R. Wang, Phys. Rev. B 71, 174430 (2005), and references therein. [17] M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008). [18] A. P. Malozemoff and J. C. Slonczewski, Domain Walls in Bubble Materials , (Academic, New York, 1979). [19] M. T. Bryan, et al., J. Appl. Phys. 103, 073906 (2008). [20] T. Michlmayr, et al., J. Appl. Phys. 99, 08N502 (2006). [21] T. Michlmayr, et al., J. Phys. D: Appl. Phys. 41, 055005 (2008).

2009-10-08

We investigate field-driven domain wall (DW) propagation in magnetic nanowires in the framework of the Landau-Lifshitz-Gilbert equation. We propose a new strategy to speed up the DW motion in a uniaxial magnetic nanowire by using an optimal space-dependent field pulse synchronized with the DW propagation. Depending on the damping parameter, the DW velocity can be increased by about two orders of magnitude compared the standard case of a static uniform field. Moreover, under the optimal field pulse, the change in total magnetic energy in the nanowire is proportional to the DW velocity, implying that rapid energy release is essential for fast DW propagation.

Fast domain wall propagation under an optimal field pulse in magnetic nanowires

0910.1477v2

arXiv:0910.3776v1 [cond-mat.other] 20 Oct 2009APS/123-QED Bifurcation and chaos in spin-valve pillars in a periodic ap plied magnetic field S. Murugesh1∗and M. Lakshmanan2† 1Department of Physics & Meteorology, IIT-Kharagpur, Khara gpur 721 302, India and 2Centre for Nonlinear Dynamics, School of Physics, Bharathi dasan University, Tiruchirapalli 620 024, India (Dated: November 10, 2018) We study the bifurcation and chaos scenario of the macro-mag netization vector in a homogeneous nanoscale-ferromagnetic thin film of the type used in spin-v alve pillars. The underlying dynamics is described by a generalized Landau-Lifshitz-Gilbert (LL G) equation. The LLG equation has an especially appealing form under a complex stereographic projection, wherein the qualitative equivalence of an applied field and a spin-current induced to rque is transparent. Recently chaotic behavior of such a spin vector has been identified by Zhang and Li using a spin polarized current passing through the pillar of constant polarization direct ion and periodically varying magnitude, owing to the spin-transfer torque effect. In this paper we sho w that the same dynamical behavior can be achieved using a periodically varying applied magnet ic field, in the presence of a constant DC magnetic field and constant spin current, which is technic ally much more feasible, and demon- strate numerically the chaotic dynamics in the system for an infinitely thin film. Further, it is noted that in the presence of a nonzero crystal anisotropy fie ld chaotic dynamics occurs at much lower magnitudes of the spin-current and DC applied field. Bolstered by the importance of Giant Magneto Resistance (GMR), a sequence of experimental and theoretical developments in the last few years on current induced switching of magnetization in nanoscale ferromagnets has thrown open several prospects in next generation magnetic memory devices. The direct role of spin polarized current, as against the traditional applied field, in control- ling spin dynamics has brought in the possibility of new types of current-controlled memory de- vices and microwave resonators. The system un- der consideration is primarily a nanoscale spin- valve pillar structure, with one freeferromagnetic layer and another pinnedlayer separated by a non- ferromagnetic conducting layer. The behavior of the dynamical quantity of interest, the magneti- zation field in the free layer, is well modeled by an extended Landau-Lifshitz equation with Gilbert damping, which is a fascinating nonlinear dynam- ical system. The free layer is usually assumed to be of single magnetic domain. Owing to the highly nonlinear nature of the LLG equation it is imperative to study the chaotic dynamical regime of the magnetization field. Indeed, several recent experiments have exclusively focused on chaos as- pect of the system. In this article we have shown that a small applied periodically varying (AC) magnetic field, in the presence of a constant spin- current and a steady applied magnetic field, can induce parametric regimes displaying a broad va- riety of dynamics and period doubling route to chaos. A numerical study of the effects of a non- ∗Electronic address: murugesh@phy.iitkgp.ernet.in †Electronic address: lakshman@cnld.bdu.ac.inzero anisotropy field reveals chaotic dynamics at much lower magnitudes of the spin-current and applied DC field. This could be an important factor to consider in microwave resonator appli- cations of spin-valve pillars. I. INTRODUCTION Following the success of GMR, the next major devel- opment in classical computer technology is expected to be through MRAMs (1; 2; 3). Apart from a manifold reduction in power consumption, being inherently non- volatile in nature, they also bring in the prospect of computers that need not be rebooted. Understandably, fabrication of MRAMs has been a major thrust area of research in the last two decades. Typically, the mem- ory unit consists of two nanoscale magnetic films sepa- rated by a spacer conductor/semiconductor medium and works on the principle of GMR. The imminent prospects in the recording media industry has prompted a breadth of development in the field. Besides the eminent role as a memory unit, the possibility of a single MRAM as a logical unit has also been proposed (4; 5). One no- table technological hiccup in fabricating large MRAM grids is the extent to which the applied magnetic field can be localized. This imposes limitations on the ef- ficiency with which an individual unit can be manipu- lated. The applied field required for the purpose is in most cases the Oersted field generated through electrical currents. Asignificantstepforward,inbypassingthelim- itation on field localization, occurred when Slonczewski andBergerindependentlyenvisagedamoredirectrolefor (spin polarized) current on magnetization(6; 7). They predicted that the angular momentum acquired by the spin polarized current can interact with the magnetiza- tion vector, and thus a suitable spin polarized current2 can possibly flip its direction. This phenomenon has been well established in a series of experiments in the last decade and referred to as the spin-torque effect in the literature(8; 9; 10; 11; 12; 13; 14; 15). Interest- ingly, the effect has a simple semiclassical description in the form of an extended Landau-Lifshitz equation with Gilbert damping (6; 7; 16). Several proposals have ap- peared in the last few years suggesting an increased role of the spin current and the associated torque in the even- tually expected version of the MRAM. Oneimportantassumptionoftenmadein mostofthese studies is that the magnetization in the film is homoge- neous, or at least enough well defined that one can con- sider the film to be a mono-domain layer. This homo- geneity assumption effectively nulls the Heisenberg in- teraction between neighboring spins, and allows one to treat the system as a single spin unit. As the size of the magnetic film is increased, this approximation is ex- pected to fail. Indeed, chaotic behavior has been ob- served due to spin field inhomogeneity at lateral sizes of order 60 −130nm(17; 18). Besides, it is well realized experimentally that spin-transfer can induce microwave oscillations(17; 19; 20; 21). The possible role of such current controlled microwave oscillators in the nanoscale has been well realized (17). For higher power levels that are practically desired it is natural to look at a series of such coupled spin-valve oscillators. Studies on synchro- nization of networks of such coupled nano-spin trans- fer oscillators, each one modeled on the extended LLG equation have been carried out both experimentally and numerically in an effort to enhance the emitted power (22; 23; 24; 25). Since theextended LLGequationis highlynonlinearin nature, a detailed study of the underlying nonlinear dy- namics in spin-valve structures becomes inevitable. The stability diagram based on Melnikov theory in the space of the external field along the direction of anisotropy and the strength of spin torque due to a DC current was ob- tained in detail by Bertotti et. al.,in (26). Following this, it was shown by Z. Li, Y. C. Li and S. Zhang that an applied alternating current can induce three broad types of dynamics, vis−a−visChaos, Multiply periodic and Periodic (27). Qualitatively similar features are also noted when the effects of nearest neighbor interactions are included in a long one dimensional ferromagnet with a spin torque term. Indeed the ferromagnetic chain ex- hibits both periodic and chaotic behavior in the presence of an applied AC spin current. (28). The multiply pe- riodic behavior refers to the case where, with change in parameter, the system moves from periodic to multiply periodic behavior but not leading to chaos. It must be mentioned at this point that the last two types ‘multiply periodic’ and ‘periodic’ are also referred to in the litera- ture as ‘modification’ and ‘synchronization’(27; 29). The usage here is deliberate as the periods do not have a di- rect relation to the period of the applied magnetic field. It was also shown that in the space of the applied exter- nal field and the strength of the spin current the systemexhibits chaotic behavior for a range of values within the boundary predicted by Melnikov theory (29). The possibility of chaotic oscillations in monodomain single nanoscale spin-valve requires further in depth study of underlying bifurcation and chaos scenarios, in- cluding the detailed phase diagrams, at least for two rea- sons: (i) Chaotic oscillations in spin transfer oscillators may be unfavorable from a practical/technological point of view and such oscillations need to be suppressed by minimal intervention using one of the several control- ling techniques developed in the field of chaotic dynam- ics (30; 31), where already such a study has been made (32). (ii) The previously mentioned possibility of syn- chronization of coupled spin-valve oscillators raises com- plex problems which are new in spintronics and is related to the topic of chaos synchronization in dynamical sys- tems (30; 31; 33), similar to synchronization of chaoti- cally evolving networks of Josephson junctions, laser sys- tems and nonlinear oscillators. As synchronization of chaotic oscillators is considered as a potential technique for secure communication including cryptography, there existspossibleapplicationsofnetworksofnanoscalespin- valve structures along these lines, although this may be complicated due to the presence of several parameters in the system. Consequently the study of the full nonlinear dynamics of spin transfer oscillators will be of consider- able significance. It may be noted that chaosin magnetic systems, such as yttrium garnet, driven by external fields have been extensively studied in the past (34; 35; 36). Numerical experiments on a model of thin magnetic film wherein spin wave excitations induced by spin-current lead to chaos have also been studied in (37). However, chaosinnano-spinvalvegeometryisreasonablynew with potential new applications as discussed above. In this paper, we study the chaotic dynamics of the magnetization vector in a single domain current driven spin-valve pillar, induced by a periodically varying (AC) applied magnetic field in the presence of a constant spin current and steady (DC) magnetic field, using the ex- tended LLG equation as the model for the system. Mak- ing use of a complex stereographic variable, we observe that the spin current induced torque is effectively equiv- alent to an applied magnetic field. Following this ob- servation, we show numerically that a periodically alter- nating field can lead to chaotic behavior of the magne- tization vector, which is similar to that of an alternat- ing spin-current induced torque, studied in Li, Li, and Zhang (27). It will be shown that the order of magni- tude of the applied alternating field required for chaotic motion is substantially lower, within practically achiev- able limits, compared to the alternating current magni- tudes shown in (27; 29). This is expected to be helpful in applications such as resonators, as an AC magnetic field is much more feasible practically than a AC spin- polarized current (although, when it comes to DC fields, the current induced DC Oersted fields are more cumber- some to produce in spin valve geometries compared to DC spin currents). We study the dynamics in an infi-3 nite thin film, both with a vanishing and non-vanishing anisotropy field. In the setup we shall consider the mon- odomain spin layer influenced by a DC spin-current, and both a DC and AC applied magnetic field. Although the applied magnetic field is qualitatively equivalent to a suitable spin-current(38), we employ both a DC spin- current and applied magnetic field as the magnitudes at which chaos is observed is quite high, and hence difficult to achieve exclusively using either of the two. It may be noted that the chaotic dynamics studied here is induced by an AC applied magnetic field, and is phenomenologi- cally different from the spin field inhomogeneity induced chaotic behavior that is observed in (17). Periodic dy- namics is possible even in the absence of an alternating current, or field, as has been noticed in ferromagnetic films induced by a spin-current (19). Futher it has been reported therein that the current magnitudes at which periodic behavior is seen share a linear relation of nega- tive slope with the oscillationfrequencies. Our numerical results based on the extended LLG model are shown to be in agrement with these observations. Interestingly, in the presence of a non-zero in plane easy-axis crystal anisotropy field (taken along the zdirection), the chaotic dynamical regime is observed at much lower magnitudes of the DC applied field and spin-current. This could be an aspect to factor in while designing spin-valve based microwave resonators. The paper is organized as follows. In Section 2, we detail the various interactions, including spin-transfer torque, that make up the extended LLG equation. Further, we introduce a complex stereographic vari- able equivalent to the macro-magnetization vector, and rewrite the LLG equation in this variable. As will be shown, this elucidates the role of the spin transfer torque as equivalent to an applied magnetic field. In Section 3, we present our numerical results which demonstrate chaoticdynamics ofthe magnetizationvectorin the pres- ence of a periodic applied field. Here we shall consider twocases,namelyresponseinthepresenceandabsenceof a crystal anisotropy field. As will be noticed, the chaotic regimes in the two cases are significantly different. In Section 4 we conclude with a discussion on our results. II. THE EXTENDED LANDAU-LIFSHITZ EQUATION AND COMPLEX REPRESENTATION A typical spin-valve pillar used in most experiments is schematicallyshowninFIG 1. Acurrentpassingthrough this ferromagnet acquires a spin polarization in the ˆ zdi- rection. The thickness of the spacer conductor medium should be less than the spin diffusion length of the polar- ized current. The polarized current then passing through the free layer causes a change in the magnetization vec- torˆS, an effective torque referred to as the spin transfer torque. Interestingly, it has been realized that semiclas- sically the phenomenon can be described by an extensionj Pinned layer Conductor Thin film ConductorS S p xz y~100 nm 2−10 nm FIG. 1 A schematic figure of a spin-valve pillar. The cross section of the free layer is roughly 5000 nm2.ˆSis the magne- tization vector in the free layer, and is the dynamical quant ity of interest. ˆSPis the direction of polarization of the spin cur- rent. to the LLG equation, (6; 7) dˆS dt=−γˆS×/vectorHeff+λˆS×dˆS dt−γ aˆS×(ˆS׈Sp),(1) Here,ˆS={S1,S2,S3}is the unit vector along the di- rection of the magnetization vector in the ferromagnetic film, which isthe dynamicalvariableofinterest, γthe gy- romagnetic ratio, and λthe dissipation coefficient. /vectorHeff in Eq. (1) is the effective field due to exchange inter- action, anisotropy, demagnetization and an applied field (see (38) for details): /vectorHeff=DS0∇2ˆS+κ(ˆS·ˆ e/bardbl)ˆ e/bardbl+/vectorHdemag+/vectorBapplied,(2) ˆ e/bardblbeing the unit vector along the direction of the anisotropy axis. The demagnetization field is obtained as a solution of the differential equation ∇·/vectorHdemag=−4πS0∇·ˆS, (3) and/vectorBappliedis the applied magnetic field on the sample. The last term in Eq. (1) is the additional term describing the spin-transfer torque, and the parameter ‘ a’ depends linearly on the current density j.ˆSpis the direction of magnetization of the polarizer, i.e., the polarization of the spin current. In this study we shall assume the magnetization to be homogeneous. This effectively nulls the exchange inter- action, while Eq. (3) can be immediately solved to give the demagnetization field as /vectorHdemagnetization =−4πS0(N1S1ˆ x+N2S2ˆ y+N3S3ˆ z), (4) whereNi,i= 1,2,3 are conveniently chosen such that N1+N2+N3= 1 (after suitable rescaling of the magni- tude of the spin). For a spherical sample N1=N2=N3, and the demagnetization term is effectively null in Eq. (1). In the next sections we study chaotic dynamics shown by the system when an alternating magnetic field is applied. Rewriting Eq. (1) using the complex stereographic variable (39; 40) Ω≡S1+iS2 1+S3, (5)4 providesamorecomprehensiblepictureoftheroleofspin transfer torque. For the spin valve system, the direction of polarization of the spin-polarized current ˆSpremains a constant, and lies in the plane of the film. Without loss of generality, we call this the direction ˆ zin the internal spin space, i.e., ˆSp=ˆ z. Asmentioned in Sec. 2, wedisre- gard the exchange term. We choose the applied externalfield also in the ˆ zdirection, i.e., /vectorBapplied={0,0,ha3}. Defining ˆ e/bardbl={sinθ/bardblcosφ/bardbl,sinθ/bardblsinφ/bardbl,cosθ/bardbl}(6) and upon using Eq. (5) in Eq. (1), we get (1−iλ)˙Ω =−γ(a−iha3)Ω+iS/bardblκγ/bracketleftBig cosθ/bardblΩ−1 2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig −iγ4π S0 (1+|Ω|2)/bracketleftBig N3(1−|Ω|2)Ω −N1 2(1−Ω2−|Ω|2)Ω−N2 2(1+Ω2−|Ω|2)Ω−(N1−N2) 2¯Ω/bracketrightBig ,(7) whereS/bardbl=ˆS·ˆ e/bardbl. Using Eq. (5) and Eq. (6), S/bardbl, and thus Eq. (7), can be written entirely in terms of Ω. For further details of derivation of Eq. (7) see Ref. (38). It is interesting to note that in this representation the spin-transfertorque,proportionaltotheparameter a, ap- pears only in the first term in the right hand side of Eq. (7) as an addition to the applied magnetic field ha3but with a prefactor i. Thus the spin torque term can be considered as an effective applied magnetic field (38). It was further explicitly shown in (38), in the absence of the crystal anisotropy field, that the switching time due to the spin-torque will be shorter by a factor λ, compared to that of magnetic field induced switching. Further, the spin-torque produced the dual effect of precession and dissipation even in the presence of the external applied field. The nature of switching of magnetization for other types of materials can be investigated by analyzing Eq. (7), and for typical materials this has been carried out in (38). III. CHAOTIC DYNAMICS Magnetization reversal in a spin mono-domain layer in thepresenceofbothasteadyappliedmagneticfieldanda steady polarized current corresponds to a rather compli- cated phasediagram,as revealedin (26). Using Melnikov theory, it was also shown therein that the magnetization vector also has limit cycles for a range of values of the parameters, with frequency in the microwave range. The dynamical quantity in question, the magnetization vec- torˆS, is two dimensional, owing to its constant (unit) magnitude. Hence, chaotic behavior is ruled out. How- ever, making the applied field, or current, time depen- dent is one way of increasing the effective dimensionality of the system to three and hence introduce a possibility of chaotic dynamics. Following (26) it has been shown in (27) that a small alternating current can produce a vari- ety of dynamics, namely, Multiply periodic, Periodic and Chaos. It is also noticed that, along with a steady spincurrent of order 250 Oeand a steady applied magnetic field of the same order, inclusion of a small alternating spin polarized current leads to chaotic dynamics (29). The dynamical similarity of the applied field and the spin-torque was noted in Section II. In this section we show numerically that an applied AC field can also pro- duce diverse dynamical scenarios and point out the ad- vantagesin using a periodically varyingapplied magnetic field instead of an alternating current. i.e., in Eq. (1) (or equivalently Eq. (7)) we take /vectorBapplied={0,0,ha3+haccosωt}. (8) It will be notedthat chaoticdynamicsis possibleatmuch lower DC applied field strengths and spin current in the presence of an anisotropy field. For the film geometry, the film is taken to be in the y−zplane, and the demagnetization field perpendicular to the film, the ˆ xdirection, i.e., Hdemagnetization =−4πS0S1ˆ x. (9) The saturation magnetization is taken to be that of permalloy, so that 4 πS0= 8400Oe. Two different scenar- ios, one without anisotropy, and anotherwith an in-plane easy axis anisotropy of strength κ= 45Oealong the z direction, are investigated. All the numerical results that follow have been obtained by directly simulating Eq. (1) for the vector ˆS. The same results were confirmed using Eq. (7) as well. A. Regions of Chaos in the presence of an applied alternating field As a first step we show below (FIG 2) the regions of chaos, or regions of positive Lyapunov exponent, in the space of the DC magnetic field and the DC current for an alternating applied magnetic field of magnitude 10Oeand frequency 15 ns−1, first without an anisotropy field,κ= 0, (FIG 2a), and then with anisotropy field,5 (a) ha3(Oe)a (Oe) 800700600500400300200100280 270 260 250 240 230 220 210 (b) ha3(Oe)a (Oe) 1800 1400 1000 600 200400 350 300 250 200 150 100 50 0 FIG. 2 Regions of chaos in the a−ha3space, for an applied alternating magnetic field of amplitude hac= 10Oeand fre- quencyω= 15ns−1, a) without anisotropy field, κ= 0, and b) with anisotropy field of strength κ= 45Oealong the zdirection. The dark regions indicate values for which the dynamics is chaotic, i.e, regions where the largest Lyapuno v exponentispositive. Ina)chaosis rarelynoticedforlower val- ues ofha3. The other parameters are N1= 1,N2=N3= 0, 4πS0= 8400Oe. The points are plotted at intervals of 5 Oe along both axes, and hence the figure offers only limited res- olution in the dark(chaotic) regions. κ= 45Oe, (FIG 2b). The regions are obtained by di- rect numerical simulation using the model in Eq. (1), or equivalentlyEq. (7). ThedarkregionsinFIG2represent parameter values when the largest Lyapunov exponent is positive. 1. Case a: No anisotropy ( κ= 0) The similarity of FIG 2a with that of regions of chaos for an alternating spin current must be noted (see FIG 1 in (29)). The figure is a demonstration of the qualitative equivalence of the applied field and current induced spin- torque in describing the gross dynamical scenario. From FIG 2a, chaotic behavior of spins is observed for spin- current magnitudes in the range of a= 200−300Oe, and DC magneticfieldsabove100 Oe. These valuesof‘ a’ cor- respond to spin-current magnitudes of over 1012A/cm2, which is at the higher end of the presently achievable levels. In FIG 3 we present the bifurcation diagram showingthe period doubling route to chaos as the DC current is varied. These diagrams are obtained by plotting the minimum values for one of the components of the spin, in this case S1, over severalperiods of time for each value ofain the given range. From the data in FIG 3a, the first five period doubling bifurcations are seen to occur atan= 268.4845,267.7723,267.6055,267.5685,267.5605. Consequently, the ratio δn= (an−an−1)/(an−1−an+1) takesvalues4 .2698,4.5081,4.625,clearlyapproachingthe Feigenbaum constant. The Lyapunov spectrum for this range of ‘ a′is shown as inset. On comparison, we notice the largest Lyapunov exponent is positive for values of awhere the dynamics is chaotic in FIG 3a. A similar check has been made for FIG 3b, which again shows the period doubling route in the presence of an anisotropy field. Period doubling route to chaos is also noticed as the strength of the steady applied magnetic field is var- ied(keeping the current and frequency fixed), and when a (Oe)λ 268.6268267.40 -4 -8 -12(a) a (Oe)S1minimum 268.6268.4268.2268267.8267.6267.4-0.994 -0.995 -0.996 -0.997 -0.998 (b) a (Oe)S1minimum 250.5250 249.5249 248.5-0.984 -0.986 -0.988 -0.99 -0.992 -0.994 -0.996 FIG. 3 Period doubling route to chaos as ais varied. The figure is a plot of the minimum values of S1over several pe- riods for the given parameter values a) without anisotropy, and b) with anisotropy of κ= 45Oealong the zdirection. The applied DC field ha3= 200Oe. All the other parameters remain the same as in FIG 2. The corresponding Lyapunov spectrum is shown as an inset.6 the frequency of the applied field is varied (keeping cur- rent and magnetic field strength steady), respectively. (a) a (Oe)ω(ns−1) 60055050045040035030025045 40 35 30 25 20 15 10 5 (b) a (Oe)ω(ns−1) 60055050045040035030025045 40 35 30 25 20 15 10 5 FIG. 4 Regions of chaos(red dots) and periodicity (blue wings) in theparameter space of DCcurrent‘ a′andfrequency ‘ω′, a) without anisotropy and b)with anisotropy ( κ= 45Oe) along the zdirection. The left over regions show multiply pe- riodicbehavior. All other parameters remain the same as in FIG 3. The power spectrum at the two dark points in (a) (255,25) and (280 ,25) are shown in FIG 5 (a) and (b), re- spectively. 2. Case b: Non-zero anisotropy( κ= 45Oe) Contrary to the observation in FIG 2a, in the pres- ence of a non-zero easy axis anisotropy, chaos is noted at much lower values of the spin current and DC applied field (FIG 2b). FIGs 3b shows period doubling bifur- cation scenario in the presence of anisotropy. In FIG 3b the current magnitude is varied, keeping the mag- netic field strength and frequency of the AC component fixed. Similar period doubling route to chaos is also no- ticed as a) the magnitude of the steady magnetic field is varied (keeping current and frequency fixed), and when the frequency of the AC component of the applied mag-netic field is changed (keeping the current and magnetic field strengths constant). In either case, an easy axis anisotropy of magnitude κ= 45Oeis chosen along the z direction, which is also the polarization direction of the spin current. The figures corresponding to these results are, however, not presented in here. B. Periodic, Multiply periodic and Chaotic dynamics In the presence of an AC spin-current induced torque, it is known that as the frequency of the spin-current is varied the system exhibits three distinct phases wherein the dynamicsispredominantlyeither- Periodic, Multiply periodicorChaotic(27). Herein we show that an applied AC magnetic field of magnitude 10 Oe, instead of a AC current, results in the three dynamical phases as the fre- quency of the applied field is varied for a constant value of the applied DC field. FIG 4a shows regions of periodic (blue wings), and chaotic (red stem) dynamics in the parameter space of the spin current magnitude and the frequency of the ap- pliedmagneticfield. Multiplyperiodicbehaviorisseenin theunshadedregions. Hereagainthesimilaritywith that ofFIG1in(27)maybenoted. Thisfurtherillustratesthe qualitative similarity of the spin current induced torque with that of the applied field. Our numerical results fur- ther show a number of wing like bands in the ω−aspace where periodic behavior is noticed. This is clearly ab- sent with a periodic spin-current as seen in (27). The powerspectrum correspondingto two specific points, one chaotic and the other periodic (indicated with dark dots in FIG 4a), are shown in FIG 5. Periodic behavior is no- ticed even in the limit ω→0 forcertain values of a, while multiply periodic behavior is noticed for the other inter- mediate values. The power spectrum corresponding to a = 280Oe 20010008 4 0 -4 -8 a = 255Oe ω(ns−1)log(Power) 2502001501005008 6 4 2 0 -2 -4 -6 -8 FIG. 5 The power spectrum distribution corresponding tope- riodic,a= 280Oe(inset), and chaotic, a= 255Oe, scenarios in FIG 4a. The first peak in the inset is seen at ω= 25ns−1. The anisotropy is taken zero, and all other parameters are th e same as in FIG 4a.7 these current values are shown in FIG. 6. Such a behav- ior, induced by a spin-currentof constantmagnitude, has been noticed in (19). Indeed, the current magnitudes, a, where periodic behavior is seen to vary linearly (with a negative slope) with the corresponding periods (see FIG. 6 inset), in further agreement with (19). However,in the presenceof the anisotropyfield chaotic regime is much more pronounced and wider, as seen in FIG 4b. For a much lower value of the DC applied field (ha3= 20Oe) there appear wide bands in the ω−aspace where periodic behavior is exclusively noted at low fre- quencies (FIG 7). As the frequency is increased multiple periodicity islands appear in between periodicbands, and for even higher frequencies the dynamics is largely one of multiply periodic type. No chaotic dynamics is noted in the parameter range chosen. These thick periodicity bands present better regions to choose in applications such as the microwave resonator discussed earlier, while chaos synchronization studies can be carried out in the chaos regimes. IV. DISCUSSION AND CONCLUSION Using the complex stereographic variable to represent the spin vector, and rewriting the modified Landau- Lifshitz-Gilbert equation, we have shown that the spin- current induced torque is qualitatively equivalent to an applied field. Using this equivalence we have shown that an applied AC magnetic field in the presence of con- stant spin current and DC applied magnetic field can lead to varied dynamical scenarios including chaos. We have explicitly demonstrated numerically the chaotic be- havior for a range of values of the parameters. The sys- 182Oe190Oe198Oe199Oe 207Oe 215Oe 241OeAng.freq. (ns−1)a(Oe) 2.31.81.3240 220 200 180 Ang.freq. (ns−1)Power 2.42.221.81.61.41.210.25 0.2 0.15 0.1 0.05 0 FIG. 6 The power spectrum distribution in the limit ω= 0, at certain values of a(indicated on each spectrum) where periodic behavior is noted. Multiply periodic behavior is n o- ticed for other values of ain the range shown. The current magnitudes vary linearly and decrease with the frequency of oscillation ( inset).hac= 0, while all other parameters are the same as in FIG. 3.a (Oe)ω(GHz) 4003503002502001501005045 40 35 30 25 20 15 10 5 FIG. 7 Regions of multiply periodic dynamics for the system with the DC applied field fixed at ha3= 20Oe, and non-zero anisotropy. All the other parameters remain the same as in FIG 4b. Synchronization is noted in the unshaded regions, while chaotic dynamics is not noticed in the parameter range shown in the figure. Islands of multiply periodic behavior ap - pear between regions of periodic behavior for low frequenci es. For higher frequencies, the dynamics is exclusively multip ly periodic. tem also exhibits regular periodic behavior for a different range of values. It is now realized that such nanoscale monodomain layers can find application as resonators, through periodic oscillations induced by an alternating spin-current. The resultspresentedhereprovideanalter- nativemethod throughoscillationsinduced byanapplied magnetic field. It is further noticed that the range of the chaotic regime strongly depends on the presence of a crystal anisotropy field. In the presence of an anisotropy field chaotic behavior is noticed for much lower values of the DC field and spin-current, which are more suited for chaos synchronization studies. However, there are re- gions in the ω−aspace where regular periodic motion is more robust and presents a better alternative in applica- tions. In a future study we will present the possibility of chaos synchronization in spin-valve structures. ACKNOWLEDGMENTS S.M. wishes to thank DST, India, for funding through the FASTTRACK scheme. The work of M.L. forms part of a DST-IRHPA research project and is supported by a DST-Ramanna fellowship. References [1] S. A. Wolf, A. Y. Chtchelkanova and D. M. Treger, IBM J. Res. Dev. 50, 101 (2006). [2] R. K. Nesbet, IBM J. Res. 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2009-10-20

We study the bifurcation and chaos scenario of the macro-magnetization vector in a homogeneous nanoscale-ferromagnetic thin film of the type used in spin-valve pillars. The underlying dynamics is described by a generalized Landau-Lifshitz-Gilbert (LLG) equation. The LLG equation has an especially appealing form under a complex stereographic projection, wherein the qualitative equivalence of an applied field and a spin-current induced torque is transparent. Recently chaotic behavior of such a spin vector has been identified by Zhang and Li using a spin polarized current passing through the pillar of constant polarization direction and periodically varying magnitude, owing to the spin-transfer torque effect. In this paper we show that the same dynamical behavior can be achieved using a periodically varying applied magnetic field, in the presence of a constant DC magnetic field and constant spin current, which is technically much more feasible, and demonstrate numerically the chaotic dynamics in the system for an infinitely thin film. Further, it is noted that in the presence of a nonzero crystal anisotropy field chaotic dynamics occurs at much lower magnitudes of the spin-current and DC applied field.

Bifurcation and chaos in spin-valve pillars in a periodic applied magnetic field

0910.3776v1

arXiv:0910.4684v2 [physics.class-ph] 20 May 2011Two bodies gravitational system with variable mass and damping-anti damping effect due to star wind G.V. L´ opez∗and E. M. Ju´ arez Departamento de F´ ısica de la Universidad de Guadalajara, Blvd. Marcelino Garc´ ıa Barrag´ an 1421, esq. Calzada Ol´ ım pica, 44430 Guadalajara, Jalisco, M´ exico PACS: 45.20.D−,45.20.3j, 45.50.Pk, 95.10.Ce, 95.10.Eg, 96.30.Cw, 03.67.Lx, 03.67.Hr, 03.67.-a, 03.65w July, 2010 Abstract We study two-bodies gravitational problem where the mass of one of the bodies varies and suffers a damping-anti damping effect due to st ar wind during its motion. A constant of motion, a Lagrangian and a Ha miltonian are given for the radial motion of the system, and the period o f the body is studied using the constant of motion of the system. An applic ation to the comet motion is given, using the comet Halley as an example. ∗gulopez@udgserv.cencar.udg.mx 11 Introduction There is not doubt that mass variable systems have been relevant s ince the founda- tion of the classical mechanics and modern physics too [0] which have been known as Gylden-Meshcherskii problems [1]. Among these type of systems on e could mention: the motion of rockets [2], the kinetic theory of dusty plasma [3], prop agation of elec- tromagnetic waves in a dispersive-nonlinear media [4], neutrinos mass oscillations [5], black holes formation [6], and comets interacting with solar wind [7]. T his last system belong to the so called ”gravitational two-bodies problem” w hich is one of the most studied and well known system in classical mechanics [8]. In t his type of system, oneassumes normally that themasses of the bodiesarefix ed andunchanged during the dynamical motion. However,when one is dealing with comets , beside to consider its mass variation due to the interaction with the solar wind, one would like to have an estimation of the the effect of the solar wind pressure on the comet motion. This pressure may produces a dissipative-antidissipative eff ect on its mo- tion. The dissipation effect must be felt by the comet when this one is a pproaching to the sun (or star), and the antidissipation effect must be felt by t he comet when this one is moving away from the sun. In previous paper [14]. a study was made of the two-bodies gravitat ional prob- lem with mass variation in one of them, where we were interested in the difference of the trajectories in the spaces ( x,v) and (x,p). In this paper, we study the two- bodies gravitational problem taking into consideration the mass var iation of one of them and its damping-anti damping effect due to the solar wind. Th e mass of the other body is assumed big and fixed , and the reference system of motion is chosen just in this body. In addition, we will assume that the mass los t is expelled from the body radially to its motion. Doing this, the three-dimensiona l two-bodies problem is reduced to a one-dimensional problem. Then, a constant of motion, the Lagrangian, and the Hamiltonian are deduced for this one-dimension al problem, where a radial dissipative-antidissipative force proportional to th e velocity square is chosen. A model for the mass variation is given, and the damping-a nti damping effect is studied on the period of the trajectories, the trajector ies themselves, and the aphelion distance of a comet. We use the parameters associate d to comet Halley to illustrate the application of our results. 22 Equations of Motion. Newton’s equations of motion for two bodies interacting gravitation ally, seen from arbitrary inertial reference system, and with radial dissipative-a ntidissipative force acting in one of them are given by d dt/parenleftbigg m1dr1 dt/parenrightbigg =−Gm1m2 |r1−r2|3(r1−r2) (1a) and d dt/parenleftbigg m2dr2 dt/parenrightbigg =−Gm1m2 |r2−r1|3(r2−r1)−γ |r1−r2|/bracketleftbiggd|r1−r2| dt/bracketrightbigg2 (r2−r1),(1b) wherem1andm2are the masses of the two bodies, r1= (x1,y1,z1) and r2= (x2,y2,z2) are their vectors positions from the reference system, Gis the gravitational constant ( G= 6.67×10−11m3/Kg s2),γis the nonnegative constant parameter of the dissipative-antidissipative force, and |r1−r2|=|r2−r1|=/radicalbig (x2−x1)2+(y2−y1)2+(z2−z1)2 is the Euclidean distance between the two bodies. Note that if γ >0 and d|r1−r2|/dt>0one has dissipation since the force acts against the motion of the body, and for d|r1−r2|/dt<0one has anti-dissipation since the force pushes the body. Ifγ <0 this scheme is reversed and corresponds to our actual situation with the comet mass lost. It will be assumed the mass m1of the first body is constant and that the mass m2of the second body varies. Now, It is clear that the usual relative, r, and center of mass,R, coordinates defined as r=r2−r1andR= (m1r1+m2r2)/(m1+m2) are not so good to describe the dynamics of this system. However, one can consider the case form1≫m2(which is the case star-comet), and consider to put our referenc e system just on the first body ( r1=˜0). In this case, Eq. (1a) and Eq. (1b) are reduced to the equation m2d2r dt2=−Gm1m2 r3r−˙m2˙r−γ/bracketleftbiggdr dt/bracketrightbigg2 ˆr, (2) where one has made the definition r=r2= (x,y,z),ris its magnitude, r=/radicalbig x2+y2+z2, andˆr=r/ris the unitary radial vector. Using spherical coordinates (r,θ,ϕ), x=rsinθcosϕ , y=rsinθsinϕ , z=rcosθ , (3) 3one obtains the following coupled equations m2(¨r−r˙θ2−r˙ϕ2sin2θ) =−Gm1m2 r2−˙m2˙r−γ˙r2, (4) m2(2˙r˙θ+r¨θ−r˙ϕ2sinθcosθ) =−˙m2r˙θ , (5) and m2(2˙r˙ϕsinθ+r¨ϕsinθ+2r˙ϕ˙θcosθ) =−˙m2r˙ϕsinθ . (6) Taking ˙ϕ= 0 as solution of this last equation, the resulting equations are m2(¨r−r˙θ2) =−Gm1m2 r2−˙m2˙r−γ˙r2, (7) and m2(2˙r˙θ+r¨θ)+ ˙m2r˙θ= 0. (8) From this last expression, one gets the following constant of motion (usual angular momentum of the system) lθ=m2r2˙θ , (9) and with this constant of motion substituted in Eq. 7, one obtains th e following one-dimensional equation of motion for the radial part d2r dt2=−Gm1 r2−˙m2 m2/parenleftbiggdr dt/parenrightbigg −γ m2˙r2+l2 θ m2 2r3. (10) Now, let us assume that m2is a function of the distance between the first and the second body, m2=m2(r). Therefore, it follows that ˙m2=m′ 2˙r , (11) wherem′ 2is defined as m′ 2=dm2/dr. Thus, Eq. (10) is written as d2r dt2=−Gm1 r2+l2 θ m2 2r3−m′ 2+γ m2/parenleftbiggdr dt/parenrightbigg2 , (12) which, in turns, can be written as the following autonomous dynamica l system dr dt=v;dv dt=−Gm1 r2+l2 θ m2 2r3−m′ 2+γ m2v2. (13) Note from this equation that m′ 2is always a non-positive function of rsince it represents the mass lost rate. On the other hand, γis a negative parameter in our case. 43 Constant of Motion, Lagrangian and Hamilto- nian A constant of motion for the dynamical system (13) is a function K=K(r,v) which satisfies the partial differential equation [9] v∂K ∂r+/bracketleftbigg−Gm1 r2+l2 θ m2 2r3−m′ 2+γ m2v2/bracketrightbigg∂K ∂v= 0. (14) The general solution of this equation is given by [10] K(x,v) =F(c(r,v)), (15a) whereFis an arbitrary function of the characteristic curve c(r,v) which has the following expression c(r,v) =m2 2(r)e2γλ(r)v2+/integraldisplay/parenleftbigg2Gm1 r2−2l2 θ m2 2r3/parenrightbigg m2 2(r)e2γλ(r)dr , (15b) and the function λ(r) has been defined as λ(r) =/integraldisplaydr m2(r). (15c) During a cycle of oscillation, the function m2(r) can be different for the comet approaching the sun and for the comet moving away from the sun. L et us denote m2+(r)forthefirstcaseand m2−(r)forthesecondcase. Therefore, onehastwocases to consider in Eqs. (15a), (15b) and (15c) which will denoted by ( ±). Now, if mo 2± denotes the mass at aphelium (+) or perielium (-) of the comet, F(c) =c±/2mo 2± represents the functionality in Eq. (15a) such that for m2constant and γequal zero, this constant of motion is the usual gravitational energy. T hus, the constant of motion can be chosen as K±=c(r,v)/2m0 2±, that is, K±=m2 2±(r) 2mo 2±e2γλ±(r)v2+V± eff(r), (16a) where the effective potential Veffhas been defined as V± eff(r) =Gm1 mo 2±/integraldisplaym2 2±(r)e2γλ±(r)dr r2−l2 θ mo 2±/integraldisplaye2γλ±(r)dr r3(16b) 5This effective potential has an extreme at the point r∗defined by the relation r∗m2 2(r∗) =l2 θ Gm1(17) which is independent on the parameter γand depends on the behavior of m2(r). This extreme point is a minimum of the effective potential since one has /parenleftBigg d2V± eff dr2/parenrightBigg r=r∗>0. (18) Using the known expression [11-13] for the Lagrangian in terms of t he constant of motion, L(r,v) =v/integraldisplayK(r,v)dv v2, (19) the Lagrangian, generalized linear momentum and the Hamiltonian are given by L±=m2 2±(r) 2mo 2±e2γλ±(r)v2−V± eff(r), (20) p=m2 2±(r)v mo 2±e2γλ±(r), (21) and H±=mo 2±p2 2m2 2±(r)e−2γλ±(r)+V± eff(r). (22) The trajectories in the space ( x,v) are determined by the constant of motion (16a). Given the initial condition ( ro,vo), the constant of motion has the specific value K± o=m2 2±(ro) 2mo 2±e2γλ±(ro)v2 o+V± eff(ro), (23) and the trajectory in the space ( r,v) is given by v=±/radicalBigg 2mo 2± m2 2±(r)e−γλ±(r)/bracketleftbigg K± o−V± eff(r)/bracketrightbigg1/2 . (24) Note that one needs to specify ˙θoalso to determine Eq. (9). In addition, one normallywants toknowthetrajectoryintherealspace, thatis, t heacknowledgment 6ofr=r(θ). Since one has that v=dr/dt= (dr/dθ)˙θand Eqs. (9) and (24), it follows that θ(r) =θo+l2 θ/radicalbig2mo 2±/integraldisplayr rom2±(r)eγλ±(r)dr r2/radicalBig K±o−V± eff(r). (25) The half-time period (going from aphelion to perihelion (+), or backwa rd (-)) can be deduced from Eq. (24) as T± 1/2=1/radicalbig2mo 2±/integraldisplayr2 r1m2±(r)eγλ±(r)dr/radicalBig K±o−V± eff(r), (26) wherer1andr2are the two return points resulting from the solution of the following equation V± eff(ri) =K± oi= 1,2. (27) Ontheother hand, thetrajectory inthespace ( r,p) isdetermine by theHamiltonian (21), and given the same initial conditions, the initial poandH± oare obtained from Eqs. (21) and (22). Thus, this trajectory is given by p=±/radicalBigg 2m2 2±(r) mo 2±eγλ±(r)/bracketleftbigg H± o−V± eff(r)/bracketrightbigg1/2 . (28) It is clear just by looking the expressions (24) and (28) that the tr ajectories in the spaces (r,v) and (r,p) must be different due to complicated relation (21) between v andp(see reference [14]). 4 Mass-Variable Model and Results As a possible application, consider that a comet looses material as a r esult of the interaction with star wind in the following way (for one cycle of oscillatio n) m2±(r) =/braceleftBiggm2−(r2(i−1))/parenleftbigg 1−e−αr/parenrightbigg incoming (+)v <0 m2+(r2i−1)−b/parenleftbigg 1−e−α(r−r2i−1)/parenrightbigg outgoing (−)v >0 7(29) where the parameters b >0 andα >0 can be chosen to math the mass loss rate in the incoming and outgoing cases. The index ”i” represent the ith-se mi-cycle, being r2(i−1)andr2i−1the aphelion( ra) and perihelion( rp) points ( rois given by the initial conditions, and one has that m2−(ro) =mo). For this case, the functions λ+(r) and λ−(r) are given by λ+(r) =1 αmaln/parenleftbigg eαr−1/parenrightbigg , (30a) and λ−(r) =−1 α(b−mp)/bracketleftBigg αr+ln/parenleftbig mp−b(1−e−α(r−rp))/parenrightbig/bracketrightBigg . (30b) where we have defined ma=m2(ra) andmp=m2(rp). Using the Taylor expansion, one gets e2γλ+(r)=e2γr/ma/bracketleftbigg 1−2γ αmae−αr+1 22γ αma/parenleftbigg2γ αma−1/parenrightbigg e−2αr+.../bracketrightbigg ,(31a) and e2γλ−(r)=e−2γr (b−mp) (mp−b)2γ α(mp−b)/bracketleftbigg 1+2γ α(mp−b)e−α(r−rp) mp−b +1 22γ α(mp−b)/parenleftbigg2γ α(mp−b)−1/parenrightbigge−2α(r−rp) (mp−b)2+.../bracketrightbigg (31b) The effective potential for the incoming comet can be written as V+ eff(r) =/bracketleftbigg −Gm1ma r+l2 θ 2ma1 r2/bracketrightbigg e2γr/ma+W1(γ,α,r), (32) and for the outgoing comet as V− eff(r) =/bracketleftbigg −Gm1ma r+l2 θ 2ma1 r2/bracketrightbigge2γr (mp−b) (mp−b)2γ α(mp−b)+W2(γ,α,r),(33) whereW1andW2are given in the appendix. 8We will use the data corresponding to the sun mass (1 .9891×1030Kg) and the Halley comet [15-17], mc≈2.3×1014Kg, r p≈0.6au, r a≈35au, l θ≈10.83×1029Kg·m2/s,(34) with a mass lost of about δm≈2.8×1011Kgper cycle of oscillation. Although, the behavior of Halley comet seem to be chaotic [18], but we will neglect this fine detail here. Now, the parameters αand ”b” appearing on the mass lost model, Eq. (29), are determined by the chosen mass lost of the comet during t he approaching to the sun and during the moving away from the sun (we have assumed t he same mass lost in each half of the cycle of oscillation of the comet around the sun ). Using Eqs. (32) and (33), Eq. (24), the trajectories can be calculated in the spaces (r,v) . Fig. 1 shows these trajectories using δm= 2×1010Kg(orδm/m= 0.0087%) for γ= 0 and (continuos line), and for γ=−3Kg/m(dashed line), starting both cases from the same aphelion distance. As one can see on the minimum, dissipation causes to reduce a little bit the velocity of the comet , and the antidissipation inc reases the comet velocity, reaching a further away aphelion point. Also, when o nly mass lost is considered ( γ= 0) the comet returns to aphelion point a little further away from th e initial one during the cycle of oscillation. Something related with this eff ect is the change of period as a function of mass lost ( γ= 0). This can be see on Fig. 2, where the period is calculated starting always from the same aphelion point ( ra). Note that with a mass lost of the order 2 .8×1011Kg(Halley comet), which correspond to δm/m=.12%, the comet is well within 75 years period. The variationofthe rat io of the change of aphelion distance as a function of mass lost ( γ= 0) is shown on Fig.3. On Fig. 4, the mass lost rate is kept fixed to δm/m= 0.0087%, and the variation of the period of the comet is calculated as a function of the dissipative- antidissipative parameter γ <0 (using|γ|for convenience). As one can see, antidissipation always wins to dissipation, bringing about the increasing of the period as a fu nction of this parameter. The reason seems to be that the antidissipation ac ts on the comet when this ones is lighter than when dissipation was acting (dissipation a cts when the comet approaches to the sun, meanwhile antidissipation acts wh en the comet goes away from the sun). Since the period of Halley comets has not c hanged much during many turns, we can assume that the parameter γmust vary in the interval (−0.01,0]Kg/m. Finally, Fig. 5 shows the variation, during a cycle of oscillation, of the ratio of the new aphelion ( r′ a) to old aphelion ( ra) as a function of the parameter γ. 95 Conclusions and comments The Lagrangian, Hamiltonian and a constant of motion of the gravita tional attrac- tion of two bodies were given when one of the bodies has variable mass and the dissipative-antidissipative effect of the solar wind is considered. By c hoosing the reference system in the massive body, the system of equations is r educe to 1-D problem. Then, the constant of motion, Lagrangian and Hamiltonian were obtained consistently. A model for comet-mass-variation was given, and wit h this model, a study was made of the variation of the period of one cycle of oscillatio n of the comet when there are mass variation and dissipation-antidissipation. When mass variation is only considered, the comet trajectory is moving away from the su n, the mass lost is reduced as the comet is farther away (according to our model), a nd the period of oscillations becomes bigger. When dissipation-antidissipation is added , this former effect becomes higher as the parameter γbecomes higher. It is important to mention that if instead of loosing mass the body wou ld had winning mass, the period of oscillation of the system would decrease. One can imagine, for example, a binary stars system where one of the star is winning mass fromtheinterstellar space or fromtheother star companion. So, due to thiswinning mass, the period of the star would decrease depending on how much mass the star is absorbing. 106 Appendix Expression for W1andW2: W1=Gm2 2− mo 2+/braceleftBigg −p(p−1)e(−4+p)αr 2r+αpEi(αpr)−2αp(p−1)Ei/parenleftbig (−4+p)αr/parenrightbig +αp2(p−1) 2Ei/parenleftbig (−4+p)αr/parenrightbig +p(p−1) r/bracketleftbig e(p−3)αr+3α(1−p)rEi/parenleftbig (p−3)αr/parenrightbig/bracketrightbig +p(p+3) 2/bracketleftbigg −e(p−2)αr r+α(p−2)Ei/parenleftbig (p−2)αr/parenrightbig/bracketrightbigg +p+2 r/bracketleftbig e(p−2)αr+α(p−1)rEi/parenleftbig (p−1)αr/parenrightbig/bracketrightbig/bracerightBigg +l2 θ 2m2 2+r2/braceleftBigg p(p−1) 2e(p−2)αr−pe(p−1)αr−αp(p−1)e(p−2)αr+αp(p−1) 2epαr +α2p(p−1)r 2e(p−2)αr+pαre(p−1)αr−p2αre(p−1)αr−p2α2r2Ei/parenleftbig pαr/parenrightbig −α2(p−2)2p(p−1)r2 2Ei/parenleftbig (p−2)αr/parenrightbig +pα2r2Ei/parenleftbig (p−1)αr/parenrightbig −2α2p2r2Ei/parenleftbig (p−1)αr/parenrightbig +p3α2r2Ei/parenleftbig (p−1)αr/parenrightbig/bracerightBigg (A1) wheremais the mass of the body at the aphelion, and we have made the definitio ns p=2γ αma(A2) and the function Eiis the exponential integral, Ei(z) =/integraldisplay∞ −ze−t tdt (A3) 11W2=Gm2 2− mo 2+/braceleftBigg e(q−2)αr r/bracketleftbigg 1+q(q−1) 2(mp+αq)e2qαr+2q mp+αqeqαr/bracketrightbigg +qαEi/parenleftbig qαr/parenrightbig −q(q−1)e2qαr (mp+αq)2r/bracketleftbig e(q−3)αr−α(q−3)rEi/parenleftbig (q−3)αr/parenrightbig/bracketrightbig +qeqαr (mp+αq)r/bracketleftbig e(q−3)αr−α(q−3)rEi/parenleftbig (q−3)αr/parenrightbig/bracketrightbig −2αEi/parenleftbig (q−2)αr/parenrightbig +αqEi/parenleftbig (q−2)αr/parenrightbig −q(q−1)αe2qαr (mp+αq)2Ei/parenleftbig (q−2)αr/parenrightbig +q2(q−1)αe2qαr 2(mp+αq)2Ei/parenleftbig (q−2)αr/parenrightbig −4αeqαr mp+αqEi/parenleftbig (q−2)αr/parenrightbig +2q2αeqαr (mp+αq)rEi/parenleftbig (q−2)αr/parenrightbig +2 r/bracketleftbig e(q−1)αr−(q−1)αrEi/parenleftbig (q−1)αr/parenrightbig/bracketrightbig +qeqαr (mp+αq)r/bracketleftbig e(q−1)αr−(q−1)αrEi/parenleftbig (q−1)αr/parenrightbig/bracketrightbig/bracerightBigg +l2 θ 2m2 2+(mp+αq)q/braceleftBigg −qαeqαr r+q2α2Ei/parenleftbig qαr/parenrightbig +q(q−1)e(3q−2)αr 2(mp+αq)2r2/bracketleftbig −1+2αr−qαr+(2−q)2α2r2e(2−q)αrEi/parenleftbig (q−2)αr/parenrightbig/bracketrightbig −qe(2q−1)αr (mp+αq)r2/bracketleftbig −1+αr+qαr+(q−1)2α2r2e(1−q)αrEi/parenleftbig (q−1)αr/parenrightbig/bracketrightbig/bracerightBigg (A4) wherempis the mass of the body at the perihelion, and we have made the definit ion q=2γ α(mp−b)(A5) 127 Bibliography 0.G. 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Astrophys., 221, (1989) 146. 151/Multiply10122/Multiply10123/Multiply10124/Multiply10125/Multiply10126/Multiply1012r/LParen1m/RParen1 /Minus20000/Minus100001000020000v/LParen3m s/RParen3 Figure 1: Trajectories in the ( r,v) space with δm/m= 0.009.0 0.1 0.2 0.3 0.4 0.5 0.6 δm/m (%)100150200250300350400450500T (years) Figure 2: Period of the comet as a function of the mass lost ratio. 160 0.1 0.2 0.3 0.4 0.5 0.6 δm/m (%)012345δ ra/ra Figure 3: Ratio of aphelion distance change as a function of the mass lost rate.0 1 2 3 4 5 6 7 | γ |708090100110120130140T (years)δm=2x1010 Kg(δm/m=0.0087%) Figure 4: Period of the comet as a function of the parameter γ. 170 1 2 3 4 5 6 7 | γ |11.21.41.61.82ra'/raδm/m=0.0087% Figure 5: Ratio of the aphelion increasing as a function of the parame terγ. 18

2009-10-24

We study two-bodies gravitational problem where the mass of one of the bodies varies and suffers a damping-antidamping effect due to star wind during its motion. A constant of motion, a Lagrangian and a Hamiltonian are given for the radial motion of the system, and the period of the body is studied using the constant of motion of the system. An application to the comet motion is given, using the comet Halley as an example.

Two bodies gravitational system with variable mass and damping-antidamping effect due to star wind

0910.4684v2

Bloch oscillations in lattice potentials with controlled aperiodicity Stefan Walter,1,Dominik Schneble,1and Adam C. Durst1,y 1Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA (Dated: 25 March 2010) We numerically investigate the damping of Bloch oscillations in a one-dimensional lattice poten- tial whose translational symmetry is broken in a systematic manner, either by making the potential bichromatic or by introducing scatterers at distinct lattice sites. We nd that the damping strongly depends on the ratio of lattice constants in the bichromatic potential and that even a small concen- tration of scatterers can lead to strong damping. Moreover, collisional interparticle interactions are able to counteract aperiodicity-induced damping of Bloch oscillations. The discussed eects should readily be observable for ultracold atoms in optical lattices. PACS numbers: 03.75.-b, 67.85.Hj, 72.10.Fk I. INTRODUCTION The oscillatory motion of particles in a periodic po- tential, when subject to an external force, was predicted by Felix Bloch in 1928 [1]. Bloch oscillations were rst observed in the 1990s both in semiconductor superlat- tices [2] and in systems of laser-cooled atoms in optical lattices [3, 4]. Since then, they have also been studied in atomic quantum gases [5{8] and have found appli- cations in ultracold atomic-physics-based precision mea- surements [9{12]. Bloch oscillations are absent in solids due to fast damp- ing from scattering by defects and phonons. Their obser- vation in long-period semiconductor superlattices relies on oscillation periods that are shorter than the charac- teristic scattering lifetime, and even in those systems, damping of Bloch oscillations due to disorder is ubiq- uitous [13, 14]. In contrast, such damping is absent in optical lattice systems which are inherently defect-free, allowing for the observation of a large number of oscil- lations [6{8]. Damping can however be induced by the introduction of disorder into the lattice potential, which can in principle be done with various techniques [15{19]. Also, in the case of quantum gases, a damping of Bloch oscillations arises from mean-eld interactions between weakly interacting, Bose-condensed atoms [7, 8, 20{24]. In the presence of both interactions and disorder, a re- duction of disorder-induced damping due to screening of disorder by the mean eld has been predicted [25]. Ex- periments with ultracold atoms thus not only constitute a versatile testbed for behavior expected in solid-state systems but may also display novel eects. Recently, the damping of Bloch oscillations in a Bose- Einstein condensate [26] has been observed for an optical lattice with a superimposed randomly corrugated opti- cal eld. A related theoretical investigation of disorder- induced damping of Bloch oscillations has considered the case of Gaussian spatial noise [25]. In this paper, present address: Dept. of Physics, Univ. W urzburg, Germany ypresent address: Photon Research Associates, Port Jeerson, NYwe numerically investigate the dynamics of an atomic wave packet in a potential with a systematically degraded translational symmetry, considering two scenarios: The rst is based on the use of a weak bichromatic poten- tial [16] of a variable wavelength ratio, and the second considers scatterers (impurities) pinned at single sites of the potential [17]. We nd that the damping strongly depends on the ratio of lattice constants in the bichro- matic case and that even a small concentration of scat- terers can lead to strong damping. We also include ef- fects of the mean-eld interaction and nd that the rate at which damping of the Bloch oscillations occurs is re- duced, similar to the case of Gaussian disorder [25]. Both eects should be observable experimentally with existing ultracold-atom technology. This paper is organized as follows: After a brief discus- sion of fundamental aspects of tilted lattices in Sec. II, Section III investigates the in uence of aperiodicity on the damping dynamics of Bloch oscillations. Sec. IV ad- dresses the interplay of aperiodicity and the interaction between atoms. Conclusions are given in Sec. V. II. TILTED PERIODIC POTENTIALS The Hamiltonian for the motion of a particle in a one- dimensional periodic potential V(x) =V(x+a) with lat- tice constant apossesses a complete set of eigenfunctions that obey Bloch's theorem '(x+a) =eika'(x). The corresponding eigenvalues En(k) are periodic in momen- tum space and form energy bands, En(k+K) =En(k) (wherenis the band index, ~kthe quasimomentum, andK= 2=a the width of the rst Brillouin zone). Under the in uence of an externally applied constant force (\tilt") F, the quasimomentum evolves as ~k(t) = ~k0+Ft. Due to the periodicity of the energy bands, this results in oscillations of the particle's group veloc- ityvg;n(k) = (1=~)dEn(k)=dk. These oscillations occur with a period TB= 2=!B=h=(aF) and a maximum displacement 2 AB;n=
n=Fin coordinate space, where
n=jEn(K=2)En(0)jis the width of the n-th band. In the following, we consider particles that are con- ned to the lowest Bloch band, n=1, corresponding toarXiv:0911.1108v3 [cond-mat.quant-gas] 11 Apr 20102 1 0 FIG. 1. Evolution of the momentum-space density j (k;t)j2 (a) in a tilted periodic potential (b) in the presence of an additional periodic potential, and (c) in the presence of two localized scatterers as discussed in the text. Additional mo- mentum components emerge in (b) and (c), broadening the momentum distribution. The parameters in (b) are = 0:01 and= 1=p 5; the scatterers in (c) are spaced 11 sites apart. A detailed explanation is given in the text. The density of j (k;t)j2is normalized to 1; a corresponding color scale is shown on the right. suciently deep potentials and small enough tilts such that Zener tunneling [27] at the band edges is negligible. Using the split-operator method [28], we perform numer- ical simulations of the dynamics of Bloch oscillations of a Gaussian wave packet (x;t= 0) =1 (22)1=4exp x2 (2)2 (1) that evolves according to the Hamiltonian H=~2@2 x 2m+V0cos(Kx) +~V(x) +Fx (2) for motion in a tilted periodic potential V(x) = V0cos(Kx) +Fxthat is modied by a weak additional potential ~V(x). III. EFFECTS OF MODIFIED PERIODICITY This section investigates the in uence of the additional potential ~V(x), which either is a weak additional peri- odic potential with variable lattice constant, or arises from the local interaction with scatterers pinned at sin- gle sites of the tilted periodic potential. In the absence of~V(x), the energies of neighboring sites dier by a xed amountFa=~!B, corresponding to a spatially homoge- neous phase dierence
(t) =!Bt. In the presence of ~V(x),
generally becomes position-dependent, leading to global dephasing and thus to a broadening of the wave packet in momentum space, as illustrated in Fig. 1. 02468101214161820/Minus0.7/Minus0.6/Minus0.5/Minus0.4/Minus0.3/Minus0.2/Minus0.10.0 t/LParen1units of TB/RParen1/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1/LParen1a/RParen1 0246810121416182001234 t/LParen1units of TB/RParen1/LAngleBracket1Σ/MinusΣ 0/RAngleBracket1/LParen1units of 10/Minus3a/RParen1/LParen1b/RParen1FIG. 2. Damped Bloch oscillations in a bichromatic lattice ( = 0:005,= 0:4). The collapse of the oscillations in coordinate space (a) is accompanied by breathing-mode exci- tations of the wave packet (b), and vice versa (see also text). The dashed line in (a) is the envelope Aexp(t2) +B. A. Bichromatic potentials A tunable bichromatic potential is generated by the addition of ~V(x) = V0cos(Kx ); (3) with variable relative amplitude 1 and lattice- constant ratio . Ifis a rational number =p=q, the total potential has a periodicity  = aq. If further- more  exceeds the spatial range probed by the wave packet, the potential can be considered disordered. The evolution of the wave packet (Eq. 1) typically ex- hibits collapses and revivals of center-of-mass oscillations that are coupled to breathing-mode excitations at twice the Bloch frequency as is shown in Fig. 2 (visible after a transient phase immediately following the switch-on of ~V). The presence of a collapse and a revival of the Bloch oscillations is a consequence of the absence of dissipation in our model. To characterize the decay of Bloch oscil- lations, the numerical data for their initial decay, for a given and, are tted with the function f(t) =Aexp(t2) cos (!Bt) +B: (4) Results for the Gaussian-decay constant are shown in Fig. 3. Clearly an increase in the perturbation amplitude leads to an overall increase in the damping of the oscil- lations. However, the dependence of the damping on the ratiois less trivial. For integer values of , the poten- tial retains its original periodicity and no dephasing of Bloch oscillations occurs. It is also suppressed for half- integer values of (i.e. forq= 2), which correspond to a doubling of the lattice constant, and correspondingly a halving of the Brillouin zone, and of the corresponding Bloch period. For this case, the dynamics of the wave packet in the modied band structure can easily be visu- alized. 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Damping constant as a function of the lattice- constant ratio , for three dierent depth ratios: = 0:005 (squares), = 0:01 (diamonds) and = 0:013 (circles). The inset shows the behavior of the decay constant in the vicinity of= 1:5. 0246810121416182000.0050.010.015 t/LParen1units of TB/RParen1/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1 /VertBar1/LParen1a/RParen1 1 43 2 1 42 3 /MinusΠ/Slash12 0 Π/Slash12 k/LParen1units of 1 /Slash1a/RParen1E/LParen1k/RParen1/LParen1b/RParen1 /LBrace1 /RBrace1 FIG. 4. Wave packet in a bichromatic potential with = 1=2; = 0:005. (a) Overlap jh (x;t= 0)j (x;t)ij, exhibiting the original periodicity of TB, with small, growing, contri- butions at odd multiples of TB=2. (b) Band structure of the potential. The inset shows the small gap at the Brillouin zone boundaries, with a calculated width ~
= 9:5105Er. The arrows with the corresponding numbers indicate the motion of the particle in kspace. The wave function completes one Bloch cycle after TB, going from 1!2!3!4, with most of the wave function tunneling across the gap. forming a closed loop, with a small splitting ~
/ at the new zone boundaries (cf. Fig. 4). In one Bloch cy- cle, most of the wave function tunnels through the tiny gap from the lower to the upper portion at one boundary and then back to the lower portion at the other bound- ary (Bloch-Zener tunneling [29, 30]). The time needed for one such cycle is TB, that is the Bloch period for the unperturbed potential V(x). This behavior can directly be seen in the time dependence of the overlap of the wave function with the initial wave packet jh (x;0)j (x;t)ij, which has a periodicity of TB; contributions at odd mul- tiples ofTB=2 only grow very slowly over a large number of cycles.The dynamics of Bloch oscillations in Figs. 2 and 3 are for a wave packet with width 0= 20ain a lattice with depthV0= 10Er[whereEr(~K=2)2=2mis the recoil energy], and a small external force F= 0:011V0=a. In coordinate space the amplitude of the Bloch oscillations isAB= 0:34a. The additional lattice ~V(x) is chosen to have a small relative amplitude = 0:005, which is sucient to cause noticeable damping already after a few Bloch cycles, depending on the lattice constant ratio . Experimentally, such a bichromatic potential is straightforward to realize in the context of optical lat- tices, using two laser beams of dierent wavelength. With present-day tunable-laser technology, a large fraction of the range of shown in Fig. 3 can be accessed. B. Scatterers on distinct sites For scatterers pinned at a set of distinct sites fngof the lattice, we model the potential ~Vas a sum of Gaussians, ~V(x) =X fng~Aexp (xxn)2=(2~2) : (5) For the simulation, the parameters of the optical lattice and the tilt are chosen such that the size of the wave packet and its oscillation amplitude ABin the unper- turbed titled potential each cover a large number of lat- tice sites. A small number of scatterers are then ran- domly placed within the range of the wave packet's mo- tion. The amplitude ~Aand width ~of the scatterers are chosen such that the valleys of the unperturbed poten- tialV0cos(Kx) are eectively lled up where the scat- terers are located. Their eect on the Bloch oscillations is shown in Fig. 5. The general trend with an increas- ing number of scatterers is a more rapid damping of the Bloch oscillations, with the details of the dynamics de- pending on their spatial arrangement. Clearly, already a small number of scatterers can lead to a rapid damping of Bloch oscillations. The results shown in Fig. 5 are for a comparatively shallow lattice ( V0= 1:4Er) and a tilting force F= 0:022Er=a, resulting in a large amplitude AB= 16a for undamped Bloch oscillations. This condition for the scatterers is well fullled by setting ~ =a=20 and ~A= Er. In the context of ultracold atoms, the placement of scatterers pinned at single lattice sites can be achieved, for example by using atoms with two internal states in conjunction with a state-dependent lattice depth [31], or by using two atomic species in a species-dependent opti- cal lattice [32]. IV. EFFECT OF INTERACTIONS The discussion so far has been restricted to noninter- acting particles. We now consider the case of a Bose-4 0 1 2 3 4/Minus30/Minus20/Minus100/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1/LParen1a/RParen1 10 1 2 3 4/Minus30/Minus20/Minus100 2 0 1 2 3 4/Minus30/Minus20/Minus100/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1 3 0 1 2 3 4/Minus30/Minus20/Minus100 4 0 1 2 3 4/Minus30/Minus20/Minus100 t/LParen1units of TB/RParen1/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1 5 0 1 2 3 4/Minus30/Minus20/Minus100 t/LParen1units of TB/RParen110 0 1 2 3 400.250.50.751/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1/VertBar1/LParen1b/RParen1 1 0 1 2 3 400.250.50.7512 0 1 2 3 400.250.50.751/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1/VertBar13 0 1 2 3 400.250.50.7514 0 1 2 3 400.250.50.751 t/LParen1units of TB/RParen1/VertBar1/LAngleBracket1Ψ/LParen1x,0/RParen1/VertBar1Ψ/LParen1x,t/RParen1/RAngleBracket1/VertBar15 0 1 2 3 400.250.50.751 t/LParen1units of TB/RParen110 FIG. 5. (a) Evolution of the packet position hx(t)iin the presence of randomly distributed scatterers. Increasing the number of scatterers (as indicated) increases the damping of Bloch oscillations. Each curve in a plot represents a dier- ent spatial congurations. (b) Overlap with the initial wave function for the case of randomly distributed scatterers. The parameters for (a) and (b) are given in the text. Einstein condensate of Natoms with repulsive interpar- ticle interaction U(xixj) =g(xixj). The evolution of the condensate wave function (x;t) =p N (x;t) in the bichromatic potential is then determined by the mean-eld Hamiltonian H=H0+ V0cos(Kx ) +gNj(x;t)j2: (6) The simulation results (Fig. 6) show that an increase in the coupling strength gleads to a reduction of thedamping constant , up to a characteristic value gc, be- yond which an increase in gleads to an increase in . We interpret the reduction of as a partial screening of the potential corrugations by the mean eld [25] as g increases, which is eventually overcompensated by mean- eld-induced dephasing [21, 22]. 0 1 2 3 4 5 6 7 8/Minus0.7/Minus0.6/Minus0.5/Minus0.4/Minus0.3/Minus0.2/Minus0.10.0 t/LParen1units of TB/RParen1/LAngleBracket1x/RAngleBracket1/LParen1units of a /RParen1/LParen1a/RParen1 0.0 0.15 0.30 0.45 0.600.0000.0050.0100.0150.020 g/LParen1units of Era/RParen1Damping constant Η/LParen1units of TB/Minus2/RParen1/LParen1b/RParen1 FIG. 6. Interplay between aperiodicity and mean-eld inter- action, showing (a) evolution for g= 0 (solid), g= 0:1Era (dotted) and g= 0:4Era(dashed), and (b) dependence ofon the coupling constant g, reaching a minimum at gc= 0:33Era . The parameters for the simulation are the same as those in Sec. 3.1, leading to an associated coupling strengthgc= 0:33Era. This value should be compared with an eective one-dimensional interaction parameter g=g3D=(2a2 ?) for a trapped atomic Bose-Einstein con- densate, where g3D= 4~2asN=m (with atomic s-wave scattering length asand atomic mass m), anda?R, whereRis the Thomas-Fermi radius. Already for a small condensate of87Rb atoms with N= 1104atoms and R= 5:3m in an isotropic 50 Hz trap [33], and an op- tical lattice with a= 532 nm, we obtain g0:42Era, which is in the vicinity of gc. Hence, a signicant mod- ication of the damping rate in a bichromatic potential due to mean-eld eects can be expected; the coupling gdepends, for example on the atom number Nin the condensate, which is variable. Alternatively, an inves- tigation of the interplay is possible for species in which the mean-eld interaction can be tuned via a Feshbach resonance. In this context we mention that Bloch oscil- lations with widely controllable mean-eld interactions in a (monochromatic) optical lattice have recently been demonstrated with cesium condensates [7, 8]. V. CONCLUSIONS We have numerically investigated the damping of Bloch oscillations resulting from a controlled breakdown of the periodicity of the lattice potential. The eects dis- cussed here, including the eects of the mean-eld inter- action on disorder-induced dephasing, should be readily observable in experiments with ultracold atoms in optical lattices.5 ACKNOWLEDGMENTS We thank D. Pertot and B. Gadway for a critical read- ing of the manuscript. This work is supported by NSFGrant Nos DMR-0605919 (S.W. and A.C.D) and PHY- 0855643 (D.S.) as well as a DAAD scholarship (S.W.). [1] F. Bloch, Z. Phys. A 52, 555 (1929). [2] C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. K ohler, Phys. Rev. Lett. 70, 3319 (1993). [3] S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. 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2009-11-05

We numerically investigate the damping of Bloch oscillations in a one-dimensional lattice potential whose translational symmetry is broken in a systematic manner, either by making the potential bichromatic or by introducing scatterers at distinct lattice sites. We find that the damping strongly depends on the ratio of lattice constants in the bichromatic potential, and that even a small concentration of scatterers can lead to strong damping. Moreover, mean-field interactions are able to counteract aperiodicity-induced damping of Bloch oscillations.

Bloch oscillations in lattice potentials with controlled aperiodicity

0911.1108v3

arXiv:0911.4628v1 [cond-mat.mes-hall] 24 Nov 2009Origin of adiabatic and non-adiabatic spin transfer torque s in current-driven magnetic domain wall motion Jun-ichiro Kishine Department of Basic Sciences, Kyushu Institute of Technolo gy, Kitakyushu, 804-8550, Japan A. S. Ovchinnikov Department of Physics, Ural State University, Ekaterinbur g, 620083 Russia (Dated: 24 November 2009) Aconsistent theorytodescribe thecorrelated dynamics ofq uantummechanical itinerantspins and semiclassical local magnetization is given. We consider th e itinerant spins as quantum mechanical operators, whereas local moments are considered within cla ssical Lagrangian formalism. By appro- priately treating fluctuation space spanned by basis functi ons, including a zero-mode wave function, we construct coupled equations of motion for the collective coordinate of the center-of-mass motion and the localized zero-mode coordinate perpendicular to th e domain wall plane. By solving them, we demonstrate that the correlated dynamics is understood t hrough a hierarchy of two time scales: Boltzmann relaxation time τel, when a non-adiabatic part of the spin-transfer torque appe ars, and Gilbert damping time τDW, when adiabatic part comes up. Spin torque transfer (STT) process is expected to rev- olutionize the performance of memory device due to non- volatility and low-power consumption. To promote this technology, it is essential to make clear the nature of the current-driven domain wall (DW) motion[1, 2]. Recent theoretical [3, 4, 5, 6, 7, 8, 9, 10] and experimental[11] studies have disclosed that the STT consists of two vec- tors perpendicular to the local magnetization m(x) and can be written in general as N=c1∂xm+c2m× ∂xm[3]. The c1andc2-terms respectively come from adiabatic[1, 4] and non-adiabatic [5] processes between conduction electrons and local magnetization, and the terminal velocity of a DW is controlled by not c1but smallc2term. The origin of the c2term is ascribed to the spatial mistracking of spins between conduction elec- trons and local magnetization[5]. Behind appearance of thec2term is the so called transverse spin accumula- tion (TSA) of itinerant electrons generated by the elec- tric current[6, 7]. Now, any consistent theory should ex- plain how the adiabatic and non-adiabatic STT come up starting with microscopic model. In particular, it should be made clear how the TSA caused by the non-adiabatic STTeventuallyleadstotranslationalmotionofthewhole DW. In this letter, to solve this highly debatable prob- lem, we propose a consistent theory to describe the cor- related dynamics of quantum mechanical itinerant spins and semiclassical local magnetization. We consider a single head-to-head N´ eel DW through a magnetic nanowire with an easy xaxis and a hard zaxis. Fee electrons travel along the DW axis ( x- axis). We describe a local spin by a semiclassical vec- torS=Sn=S(sinθcosϕ,sinθsinϕ,cosθ) whereS=|S| and the polar coordinates θandϕare assumed to be slowly varying functions of one-dimensional coordinate x [Fig.1(a)]. The DW formation is described by the Hamil- tonian (energy per unit area) in the continuum limit, HDW=JS2 2a/integraldisplay∞ −∞dx/bracketleftBig (∂xn)2−λ−2ˆn2 x+κ−2ˆn2 z/bracketrightBig ,(1) whereais the cubic lattice constant, Jis the ferromag- netic exchange strength, λ=/radicalbig J/Kandκ=/radicalbig J/K⊥respectively represent the single-ion easy and hard axis anisotropies measured in the length dimension. The stationary N´ eel wall ( θ0=π/2) is described by n0= (cosϕ0,sinϕ0,0) withϕ0(z) = 2arctan( ex/λ). In the in- finite continuum system, the DW configuration has con- tinuous degeneracy labeled by the center of mass posi- tion,X, of the DW. This degeneracy apparently leads to rigidtranslationoftheDW,i.e., n0(x)→n0(x−X)[12]. As explicitly shown below, however, the translation in off-equilibrium accompanies internal deformation of the DW. The creation operator of a conduction electron is writ- ten in a spinor form as c†(x) = (c† ↑(x),c† ↓(x)). By per- forming the local gauge transformation c(x) =ˆU(x)¯c(x) with the unitary operator ˆU(x) =eiˆσzϕ0(x)/2(ˆσzis a (a) (b) zx−y− 0n xy TSAT1 T2 OPZAy xz0nx−y− X 0 00 T2TT ns'x '0 ȟx FIG. 1: (a) Stationary configuration of local spins ( n0) asso- ciated with a single N´ eel wall. Labotatory frame x, y, zand local frame ¯ x,¯y, zare indicated. (b) Schematic view of the transverse spin accumulation (TSA) of itinerant spin sand the out-of-plane ( θ) zero-mode accumulation (OPZA) of local spinn. These magnetic accumulations respectively cause the non-adiabatic torque T2and adiabatic torque T1.2 Pauli matrix) the quantization axis becomes parallel to the local spin located at x. Assuming |a∂xϕ0(x)| ≃ a/λ≪1, i.e. wall thickness is much larger than atomic lattice constant, this procedure leads to the single- particle Hamiltonian, Hel=/planckover2pi12 2m∗a/integraldisplay∞ −∞dx/bracketleftbigg1 2|∂x¯c|2+i(∂x¯c†)ˆAz¯c/bracketrightbigg +c.c,(2) where the effective mass of the conduction electron is m∗. The SU(2) gauge field[8, 13] is introduced as ˆAz≡ i−1ˆU−1∂xˆU=−(∂xϕ0)ˆσz/2. The conduction electrons are assumed to interact with the local spins by a s-d coupling represented in the form, Hsd=−Jsd a3/integraldisplay∞ −∞dxˆs(x)·S(x−X),(3) whereˆsandS=Snare respectively the spins of itin- erant and localized electrons .We treat ˆs(x) =1 2c†ˆσcas fully quantum mechanical operator, while nis a semi- classical vector. Boltzmann relaxation : let switch on the electric field Eatt= 0. We introduce the Boltzmann relaxation timeτeland the number density of the conduction elec- tronsfkσin the state k,σ. We assume that the devia- tionfromequilibriumFermi-Diracdistribution f0(εkσ) = [exp[(εkσ−µ)/kBT]+1]−1issmall, where εkσ(σ=↑,↓) is the single-particle energy, µis the chemical potential. Using standard Boltzmann kinetic equation with relax- ation time approximation[6], the distribution function is written as fkσ≃f0(εkσ)+eEτelvkσ∂f0(εkσ) ∂εkσ,(4) where the electron charge is −eand the spin-dependent velocity is vkσ≡/planckover2pi1−1∂εkσ/∂k. The spin-dependence of εkσoriginates from the SU(2) gauge fields ( ˆAz)↑↑and (ˆAz)↓↓. In the process of approaching to stationary cur- rentflowingstate aroundthe time t∼τel, aswewillshow explicitly, the statistical average of the conduction elec- tron’s spin component perpendicular to the local quan- tization axis accumulates and acquires finite value. As schematically depicted in Fig.1(b), this process is ex- actly the TSA. The TSA causes an additional magnetic field actingon the localspins and exertthe non-adiabatic torque on the local spins. Local spin dynamics : next we formulate dynamics of the local spins coupled with the conduction electrons. We introduce the δθ(x,t) (out-of-plane) and δϕ(x,t) (in-plane) fluctuations of the local spins around the sta- tionary DW configuration n0(x).We say “out-of-plane” and “in-plane” with respect to the DW plane. The fluctuations are spanned by the orthogonal basis func- tionsvqanduqasϕ(x) =ϕ0(x−X)+δϕ(x−X) and θ(x) =π/2+δθ(x−X),where δϕ(x) =/integraldisplay∞ −∞dq ηq(t)vq(x), δθ(x)=/integraldisplay∞ −∞dq ξq(t)uq(x). (5) At this stage, Xis not a dynamical variable, but just a parameter. The basis functions obey the Schr¨ odingerequations,/parenleftbig JS2/2/parenrightbig (−∂2 x−2λ−2sin2ϕ0+λ−2)vq(x) = εϕ quq(x) and/parenleftbig JS2/2/parenrightbig (−∂2 x−2λ−2sin2ϕ0+λ−2+κ−2) uq(x) =εθ quq(x).Bothθandϕmodes consist of a sin- gle bound state ( zero mode ) and continuum states ( spin- wave modes ). The dimensionless zero mode wave func- tions are given by u0(x) =v0(x) = Φ0(x), where Φ0(x)≡/radicalbigg aλ 2∂xϕ0(x) =/radicalbigga 2λ1 cosh(x/λ),(6) with the corresponding energies respectively given by εθ 0=JS2/(2κ2) andεϕ 0= 0. The normalization is given by a−1/integraltext∞ −∞dx[Φ0(x)]2= 1.Although to excite the out-of-plane ( θ) zero mode costs finite energy gap εθ 0coming from the hard-axis anisotropy, we still call this “zero mode.” The spin-wave states have energy dispersions given by εθ q=1 2JS2/parenleftbig q2+λ−2+κ−2/parenrightbig and εϕ q=1 2JS2/parenleftbig q2+λ−2/parenrightbig . Because the zero mode and the spin-wave states are orthogonal to each other and sepa- rated by the anisotropy gaps, the spin-wave modes are totally irrelevant to a low energy effective theory. There- fore, we ignore the spin-wave modes from now on. Out-of-plane zero-mode(OPZ) coordinate ξ0: in order to obtain the correct form of the dynamical Hamiltonian, one has to regard the variable Xas a dynamical variable X(t) and replace the zero mode coordinate η0withX(t). Following this idea, the zero-mode fluctuations should be given by, ϕ(x,t) =ϕ0[x−X(t)], (7) θ(x,t) =π/2+ξ0(t)Φ0[x−X(t)].(8) Eq. (8) is a key ingredient of this letter, which has neverbeen explicitly treated sofar[14]. That is to say, we naturally include the out-of-plane(OPZ) zero-mode, in addition to the in-plane ( ϕ) zero-mode replaced by X(t). The zero-mode wave function Φ 0[x−X(t)] serves as the basis function of the θ-fluctuations localized around the center of the DW and ξ0(t) is the OPZ coordinate . Now, our effective theory is fully described by two dynamical variables X(t) andξ0(t) which naturally give physical coordinates along the Hilbert space of orthogonal θand ϕfluctuations. As we will see, we have ξ0(t)/ne}ationslash= 0 only for inequilibrium current flowing state under E/ne}ationslash= 0 [Fig. 2(a)]. It is here important to note an essential difference be- tween Tatara and Khono’s approach[8] and ours. Tatara and Khono used X(t) and the weighted average, θ0(t) =/integraltext∞ −∞dx θ(x,t)sin2ϕ[x−X(t)], as dynamical vari- ables. Later, they systematically used complex coordi- nateξ=eiϕtan(θ/2) and described the fluctuations in the form ξ=e−u(x,t)+iϕ0+η[x−X(t)][9](their notation is reproduced by putting θ→π/2−θ,ϕ→ϕin our nota- tion). Inourunderstanding, thesedescriptionsinevitably cause redundant coupling between uandvmodes in Eq. (5). Actually, our natural choice of the dynamical vari- ables is essential to appropriately derive relaxational dy- namics described by the following equations of motion given by (12a) and (12b). Equations of motion of the DW : now, we construct an effective Lagrangian L=LDW+Lsdto describe the3 xπ 0 Xπ/2(a) j (b) (c)0 ȟ'0 j OPZAȥX. 0, ˢk ˢkJsd ȥ0ȍ0 FIG. 2: (a) Spatial profile of the polar angles ϕ(x,t) = ϕ0[x−X(t)] andθ(x,t) =π/2 +ξ0(t)Φ0[x−X(t)] in the current flowing state. (b) Linear dependence of ξ0and˙X(t) on thecurrentdensity j. (c) Single- particle propagation (rep- resented by solid line) with spin flip process by the s-d inter - action (represented by wavy line) which leads to the STT. DW motion and resultant equations of motion (EOM). Using (7) and (8), the local spin counterpart is given by LDW=/planckover2pi1S a3/integraltext∞ −∞dx(cosθ−1) ˙ϕ−HDWexplicitly written as LDW=/planckover2pi1S a3/parenleftBigg/radicalbigg 2a λξ0+π/parenrightBigg ˙X−JS2 2κ2ξ2 0.(9) To understand the effect of the s-d coupling, it is use- ful to note n[θ0+δθ,ϕ0+δϕ]≃n0−ezδθ−n0δθ2/2, where we dropped δϕbecause this degree of freedom is eliminated by the global gauge fixing[14]. We have thus s-d Lagrangian, Lsd=a−3JsdS/parenleftbig F0−S/bardblξ02/2/parenrightbig , (10) where,F0[X(t)]≡/integraltext∞ −∞dxˆn0[x−X(t)]· /an}bracketle{ts(x,t)/an}bracketri}ht andS/bardbl[X(t)]≡/integraltext∞ −∞dx{Φ0[x−X(t)]}2n0[x−X(t)]· /an}bracketle{ts(x,t)/an}bracketri}ht. Finally, to take account of dissipative dynamics, we use the Rayleigh dissipation function WRayleigh=α 2/planckover2pi1S a3/integraltext∞ −∞dx˙n2explicitly written as WRayleigh=α 2/planckover2pi1S a3/parenleftbigg a˙ξ2 0+2 λ˙X2/parenrightbigg ,(11) whereαis the Gilbert damping parameter. It is simple to write down the Euler-Lagrange-Rayleigh equations, d(∂L/∂˙qi)/dt−∂L/∂q i=−∂W/∂˙qi,for the dynami- cal variables q1=Xandq2=ξ0. We obtain the EOMs which contain the dynamical variables in linear order, /planckover2pi1/radicalbigg 2a λ˙ξ0+JsdT⊥=−2α/planckover2pi1 λ˙X, (12a) −/planckover2pi1/radicalbigg 2a λ˙X+/parenleftbigga3JS κ2+JsdS/bardbl/parenrightbigg ξ0=−α/planckover2pi1a˙ξ0,(12b) where the quantities T⊥≡ −∂F0 ∂X=/integraldisplay∞ −∞dx ∂xϕ0[x−X(t)]/an}bracketle{t¯sy(x)/an}bracketri}ht,(13a) S/bardbl≡/integraldisplay∞ −∞dxΦ2 0[x−X(t)]/an}bracketle{t¯sx(x)/an}bracketri}ht, (13b)respectively give the non-adiabatic STT and longitudinal spin accumulation[7]. The statistical average of the con- duction electron’s spin component is denoted by /an}bracketle{t···/an}bracketri}ht. The gauge-transformed spin variables are introduced by ¯s(x) =ˆU−1[x−X(t)]ˆs(x)ˆU[x−X(t)] which has lo- cal quantization axis tied to the local spin at the po- sition of x−X(t). To obtain Eq.(13a), we used rela- tions∂xn0[x−X(t)] =−∂xϕ0(x)ez×n0[x−X(t)] and /an}bracketle{t¯sy/an}bracketri}ht=−/an}bracketle{tˆsx/an}bracketri}htsinϕ0+/an}bracketle{tˆsy/an}bracketri}htcosϕ0.The relation (13a) im- plies that the translation of the DW ( x→x−X) natu- rally gives rise to the TSA, /an}bracketle{t¯sy/an}bracketri}ht, along the local ¯ yaxis. The appearance of /an}bracketle{t¯sy/an}bracketri}htcauses local magnetic moment which triggers the local spins to precess around the local ¯yaxis and consequently produce finite deviation of the polar angle δθ=θ−θ0.It is seen that upon switching the external electric field, the deviation δθrelaxes to finite magnitude in the stationary current-flowing state, i.e., the OPZ coordinate ξ0(t) accumulates and reaches finite terminal value ξ∗ 0. We call this process out-of-plane zero- mode accumulation(OPZA)as schematically depicted in Fig.1(b). This effect is physically interpreted as appear- ance of demagnetization field phenomenologically intro- duced by D¨ oring, Kittel, Becker[15], and Slonczewski[1]. It is also to be noted that we ignored the term ∂S/bardbl/∂X. This simplification is legitimate for the case of of small sd-coupling. Gilbert relaxation : coupled equations of motion (12a) and(12b)arereadilysolvedtogiverelaxationalsolutions, ξ0=ξ∗ 0(1−e−t/τDW), V≡˙X=V∗(1−e−t/τDW),(14) where the OPZA reaches the terminal value, ξ∗ 0=−1 α/radicalbig aλ/2JsdT⊥/parenleftbig a3JSκ−2+JsdS/bardbl/parenrightbig≃ −α−1/radicalbigg λ 2a/parenleftBigκ a/parenrightBig2Jsd JST⊥, (15) and correspondingly the terminal velocity of the DW reachesV∗=−λ 2α/planckover2pi1JsdT⊥.The relaxation time of the DW magnetization, τDW, is given by τDW=/planckover2pi1aα−1+α κ−2a3JS+JsdS/bardbl≃α−1/parenleftBigκ a/parenrightBig2/planckover2pi1 JS.(16) This result clearly shows that the DW magnetization try to relax through the Gilbert damping toward the direc- tion of the newly established precession axis. We stress that without the OPZ coordinate ξ0in Eqs. (12a) and (12b), onlytheterminalvelocityisavailableandthetran- sient relaxational dynamics is totally lost. As depicted in Fig.2(a), the OPZA[Eq. (15)] gives rise to finite out-of-plane ( z) component of the local spin, nz(x,t) = cosθ≃1 2α/parenleftBigκ a/parenrightBig2Jsd JS1 cosh[(x−X(t))/λ]T⊥. (17) The resultant local spin S⊥=Seznz(x,t) gives the de- magnetization field phenomenologically treated by Slon- czewski and gives rise to the adiabatic torque T1= c1∂xn(x) =c1(∂xϕ0)(−sinϕ0,cosϕ0,0). At the in- terface of the DW boundary, ϕ0=π/2 andT1= c1(∂xϕ0)(−1,0,0), i.e., the adiabatic torque rotate the local spin to counterclockwise direction when the electric4 current flows in the (1 ,0,0)-direction. As is clear from the above discussion, this adiabatic torque is established afterthe stationary current-flowing [ j= (ne2τel/m∗)E] state establishes the non-adiabatic torque, T⊥. Around the time scale of t≃τel+τDW, the whole system (in- cluding conduction electrons and DW) reaches non- equilibrium but stationary state. In this state, the DW magnetizations continuously feel the OPZA and macro- scopically rotate around it. This process exactly corre- sponds to stationary translation of the DW. Computation of T⊥: the final step is to compute an explicit form of T⊥. By taking Fourier transform ¯ ckσ(t) =1√ L/summationtext keikx¯cσ(x,t), and retaining only the momentum conserving process, we have T⊥=1 2/integraldisplayπ/a −π/adkReG< k↑,k↓(t,t), (18a) S/bardbl=a 2π/integraldisplayπ/a −π/adkImG< k↑,k↓(t,t).(18b) Here, the expectation values are computed by using the lesser component of the path-oriented Green function G< kσ,k′σ′(t,t′) =i/an}bracketle{t¯c† k′σ′(t′)¯ckσ(t)/an}bracketri}ht, wheret(t′) is defined on the upper (lower) branch of Keldysh contour. Since S/bardbldoes not play an essential role, we pay attention to an essential quantity T⊥. To evaluate the Green functions, we perturbatively treat the s-d coupling and write down the Dyson equation. Then, we truncate the Dyson equa- tion by using the Born approximation including the s-d coupling in linear order which causes a single spin flip process [Fig.2(c)] and gives rise to off-diagonal compo- nent in spin space, G< k↑,k↓(t,t) =−iJsd 2fk↑−fk↓ εk↑−εk↓−i0.(19) To obtain the explicit form of εkσ, we write the single- particle Hamiltonian (2) in Fourier space and obtain Hel=H0+Hgauge, whereH0represents free conduction andHgaugecomes form the second term in Eq. (2). By retaining only momentum conserving process, we have Hel=/summationtext k,σεkσ¯c† kσ¯ckσ,whereεk↑,↓=/planckover2pi12(k∓δk)2/2m∗, where the shift of the Fermi wave numbers due to the background DW is given by δk=π/(2a). Using Eqs. (4), (19), and (18a), we finally obtain the STT which points in the z-direction, T1=T⊥ez, where its magnitude is given in a form, T⊥=1 4Jsd kBT1 cosh2[(ε0−µ)/2kBT]j j0,(20) wherej0= 4ne/planckover2pi1/(πam∗) andε0=/planckover2pi12π2/(8m∗a2) corre- spondstothechemicalpotentialathalf-filling. Wehavea master formula which gives relation between the current density and the terminal velocity of the DW, V∗=−1 8αλJsd /planckover2pi1Jsd kBT1 cosh2[(ε0−µ)/2kBT]j j0.(21) As shown in Fig.2(b), we see there is no threshold for the velocity, which is consistent with the result obtained byThiaville et al.[10]. Standard choice of parameters, j0≃ 1016[A·m−2],λ= 10−8[m],α= 10−2,j≃1011[A·m−2] give a rough estimate V∗≃ −100(Jsd/kBT)2[m/s].Of course, to pursuit more quantitative result needs numer- ical estimation of T⊥taking account of real band struc- ture. It is essential that the Gilbert damping coefficient, α, entersEq.(21). Therelaxationprocessofthe DWdynam- ics is governed by the Boltzmann relaxation followed by the Gilbert damping in hierarchical manner. As summa- rized in Figs.2(a) and (b), in our treatment, it is crucial to recognize that the OPZ coordinate ξ0acquires finite value (i.e., accumulation) only for the current flowing state which is non-equilibrium but stationary. This is the case where dynamical relaxation leads to finite accumula- tion of physical quantities which are zero in equilibrium. Although essential role of the sliding mode to describe localized spin dynamics was pointed out before[8, 12] and importance of out-of-plane canting of the local spins was stressed[1, 8], the OPZA presented in this letter has not been discussed before. For example, the sliding mo- tion in Ref.[12] does not contain internal deformation of the DW. The OPZA is an outcome of time-reversal- symmetry breaking by electric current. This interpre- tation seems natural because current-flowing state is off equilibrium. J. K. acknowledges Grant-in-Aid for Scientific Re- search (C) (No. 19540371) from the Ministry of Educa- tion, Culture, Sports, Science and Technology, Japan. [1] J.C. Slonczewski, J. Magn. Magn. Mat. 159L1 (1996). [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] M. D. Stiles and A. Zangwill, Phys. Rev. B66, 014407(2002). [4] Ya.B. Bazaliy, B.A. Jones, and S.-C. Zhang, Phys. Rev. B57, R3213 (1998). [5] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [6] J. Xiao, A. Zangwill and M. D. Stiles, Phys. Rev. B73, 054428 (2006). [7] S. Zhang, P.M. Levy and A. Fert, Phys. Rev. Lett. 88, 236601 (2002). [8] G. Tatara, H.Kohno, Phys.Rev.Lett. 92, 086601 (2004). [9] G. Tatara, H. Kohno and J. Shibata, Phys. Rep. 468, 213 (2008). [10] A. Thiaville, et al.,Europhys. Lett. 69, 990 (2005). [11] S. Petit, et al.,Phys. Rev. Lett. 98, 077203 (2007); Z. Li,et al.,Phys. Rev. Lett. 100, 246602 (2008). [12] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). [13] G. E. Volovik, J. Phys. Condens. Matter 20, 83(1987). [14] The out-of-plane zero-mode was discussed in the con- text of chiral helimagnet by the present authors: I. G. Bostrem, J. Kishine, and A. S. Ovchinnikov, Phys. Rev.B77, 132405 (2008); Phys. Rev. B78, 064425 (2008); I.G. Bostrem, J. Kishine, R. V. Lavrov, A.S. Ovchinnikov, Phys. Lett. A 373, 558(2009). [15] W. D¨ oring, Zeits. f. Naturforschung 3a, 374 (1948); R. Becker, Proceedings of the Grenoble Conference, July (1950); C. Kittel, Phys. Rev. 80, 918 (1950).

2009-11-24

A consistent theory to describe the correlated dynamics of quantum mechanical itinerant spins and semiclassical local magnetization is given. We consider the itinerant spins as quantum mechanical operators, whereas local moments are considered within classical Lagrangian formalism. By appropriately treating fluctuation space spanned by basis functions, including a zero-mode wave function, we construct coupled equations of motion for the collective coordinate of the center-of-mass motion and the localized zero-mode coordinate perpendicular to the domain wall plane. By solving them, we demonstrate that the correlated dynamics is understood through a hierarchy of two time scales: Boltzmann relaxation time when a non-adiabatic part of the spin-transfer torque appears, and Gilbert damping time when adiabatic part comes up.

Origin of adiabatic and non-adiabatic spin transfer torques in current-driven magnetic domain wall motion

0911.4628v1

APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS A. C. GILBERT, Y. LI, E. PORAT, AND M. J. STRAUSS Abstract. Anapproximate sparse recovery system consists of parameters k;N, anm-by-Nmea- surement matrix ,, and a decoding algorithm, D. Given a vector, x, the system approximates x bybx=D(x), which must satisfy kbxxk2Ckxxkk2, where xkdenotes the optimal k-term approximation to x. For each vector x, the system must succeed with probability at least 3/4. Among the goals in designing such systems are minimizing the number mof measurements and the runtime of the decoding algorithm, D. In this paper, we give a system with m=O(klog(N=k)) measurements|matching a lower bound, up to a constant factor|and decoding time O(klogcN), matching a lower bound up to log(N) factors. We also consider the encode time ( i.e., the time to multiply byx), the time to update measurements ( i.e., the time to multiply by a 1-sparse x), and the robustness and stability of the algorithm (adding noise before and after the measurements). Our encode and update times are optimal up to log( N) factors. The columns of have at most O(log2(k) log(N=k)) non-zeros, each of which can be found in constant time. If xis an exact k-sparse signal and 1and2are arbitrary vectors (regarded as noise), then, setting bx=D((x+1) +2), we get kbxxk22k1k2+ log(k)k2k2 kk2 2; wherekk2 2is a natural scaling factor that makes our result comparable with previous results. (The log(k) factor above, improvable to log1=2+o(1)k, makes our result (slightly) suboptimal when 26= 0.) We also extend our recovery system to an FPRAS. 1.Introduction Tracking heavy hitters in high-volume, high-speed data streams [4], monitoring changes in data streams [5], designing pooling schemes for biological tests [10] (e.g., high throughput sequencing, testing for genetic markers), localizing sources in sensor networks [15, 14] are all quite dierent technological challenges, yet they can all be expressed in the same mathematical formulation. We have a signal xof lengthNthat is sparse or highly compressible; i.e., it consists of ksignicant entries (\heavy hitters") which we denote by xkwhile the rest of the entries are essentially negligible. We wish to acquire a small amount information (commensurate with the sparsity) about this signal in a linear, non-adaptive fashion and then use that information to quickly recover the signicant entries. In a data stream setting, our signal is the distribution of items seen, while in biological group testing, the signal is proportional to the binding anity of each drug compound (or the expression level of a gene in a particular organism). We want to recover the identities and values only of the heavy hitters which we denote by xk, as the rest of the signal is not of interest. Mathematically, we have a signal xand anm-by-Nmeasurement matrix with which we acquire measurements y=x, and, from these measurements y, we wish to recover bx, withO(k) entries, such that kxbxk2Ckxxkk2: Gilbert is with the Department of Mathematics, The University of Michigan at Ann Arbor. E-mail: annacg@umich. edu. Li is with the Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor. E-mail: leeyi@umich.edu . Porat is with the Department of Computer Science, Bar-Ilan University. E- mail: porately@cs.biu.ac.il . Strauss is with the Department of Mathematics and the Department of Electrical Engineering and Computer Science, The University of Michigan at Ann Arbor. E-mail: martinjs@umich.edu . 1arXiv:0912.0229v1 [cs.DS] 1 Dec 20092 GILBERT, LI, PORAT, STRAUSS Paper No. Measurements Encode time Column sparsity/ Decode time Approx. error Update time [8, 3] klog(N=k)Nklog(N=k)klog(N=k)N`2(1=p k)`1 [4, 7] klogcN NlogcN logcN klogcN`2C`2 [6] klogcN NlogcN logcN klogcN`1C`1 This paper klog(N=k)NlogcN logcN klogcN`2C`2 Figure 1. Summary of the best previous results and the result obtained in this paper. Our goal, which we achieve up to constant or log factors in the various criteria, is to design the measurement matrix and the decoding algorithm in an optimal fashion: (i) we take as few measurements as possible m=O(klog(N=k)), (ii) the decoding algorithm runs in sublin- eartimeO(klog(N=k)), and (iii) the encoding and update times are optimal O(Nlog(N=k)) and O(klog(N=k)), respectively. In order to achieve this, our algorithm is a randomized algorithm; i.e., we specify a distribution on the measurement matrix and we guarantee that, for each signal, the algorithm recovers a good approximation with high probability over the choice of matrix. In the above applications, it is important both to take as few measurements as possible and to recover the heavy hitters extremely eciently. Measurements correspond to physical resources (e.g., memory in data stream monitoring devices, number of screens in biological applications) and reducing the number of necessary measurements is critical these problems. In addition, these applications require ecient recovery of the heavy hitters|we test many biological compounds at once, we want to quickly identify the positions of entities in a sensor network, and we cannot aord to spend computation time proportional to the size of the distribution in a data stream application. Furthermore, Do Ba, et al. [2] give a lower bound on the number of measurements for sparse recovery ( klog(N=k)). There are polynomial time algorithms [13, 3, 12] meet this lower bound, both with high probability for each signal and the stronger setting, with high probability for all signals1. Previous sublinear time algorithms, whether in the \for each" model [4, 7] or in the \for all" model [11], however, used several additional factors of log( N) measurements. We summarize the previous sublinear algorithms in the \for each" signal model in Figure 1. The column sparsity denotes how many 1s there are per column of the measurement matrix and determines both the decoding and measurement update time and, for readability, we suppress O(
). The approximation error signies the metric we use to evaluate the output; either the `2or`1metric. In this paper, we focus on the `2metric. We give a joint distribution over measurement matrices and sublinear time recovery algorithms that meet this lower bound (up to constant factors) in terms of the number of measurements and are within log( k) factors of optimal in the running time and the sparsity of the measurement matrix. Theorem 1. There is a joint distribution on matrices and algorithms, with suitable instantiations of anonymous constant factors, such that, given measurements x=y, the algorithm returns bx and approximation error kxbxk22k1k2 with probability 3=4. The algorithm runs in time O(klogc(N))andhasO(klog(N=k))rows. Furthermore, our algorithm is a fully polynomial randomized approximation scheme. Theorem 2. There is a joint distribution on matrices and algorithms, with suitable instantiations of anonymous constant factors (that may depend on , such that, given measurements x=y, the algorithm returns bxand approximation error kxbxk2(1 +)k1k2 1albeit with dierent error guarantees and dierent column sparsity depending on the error metric.APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 3 with probability 3=4. The algorithm runs in time O((k=) logc(N))andhasO((k=) log(N=k)) rows. Finally, our result is robust to corruption of the measurements by an arbitrary noise vector 2, which is an important feature for such applications as high throughput screening and other physical measurement systems. (It is less critical for digital measurement systems that monitor data streams in which measurement corruption is less likely.) When 26= 0, our error dependence is on 2is suboptimal by the factor log( k) (improvable to log1=2+o(1)k). Equivalently, we can use log( k) times more measurements to restore optimality. Theorem 3. There is a joint distribution on matrices and algorithms, with suitable instantiations of anonymous constant factors (that may depend on ), such that, given measurements x+2= y+2, the algorithm returns bxand approximation error kxbxk2(1 +)k1k2+log(k)k2k2 kk2 2 with probability 3=4. The algorithm runs in time O((k=) logc(N))andhasO(k=log(N=k)) rows. Previous sublinear algorithms begin with the observation that if a signal consists of a single heavy hitter, then the trivial encoding of the positions 1 through Nwith log(N) bits, referred to as a bit tester, can identify the position of the heavy hitter. The second observation is that a number of hash functions drawn at random from a hash family are sucient to isolate enough of the heavy hitters, which can then be identied by the bit tester. Depending on the type of error metric desired, the hashing matrix is pre-multiplied by random 1 vectors (for the `2metric) in order to estimate the signal values. In this case, the measurements are referred to as the Count Sketch in the data stream literature [4] and, without the premultiplication, the measurements are referred to as Count Median [6, 7] and give `1C`1error guarantees. In addition, the sublinear algorithms are typically greedy, iterative algorithms that recover portions of the heavy hitters with each iteration or that recover portions of the `2(or`1) energy of the residual signal. We build upon the Count Sketch design but incorporate the following algorithmic innovations to ensure an optimal number of measurements: With a random assignment of Nsignal positions to O(k) measurements, we need to encode onlyO(N=k) positions, rather than Nas in the previous approaches. So, we can reduce the domain size which we encode. We use a good error-correcting code (rather than the trivial identity code of the bit tester). Our algorithm is an iterative algorithm but maintains a compound invariant: the number of un-discovered heavy hitters decreases at each iteration while, simultaneously, the required error tolerance and failure probability become more stringent. Because there are fewer heavy hitters to nd at each stage, we can use more measurements to meet more stringent guarantees. In Section 2 we detail the matrix algebra we use to describe the measurement matrix distribution which we cover in Section 3, along with the decoding algorithm. In Section 4, we analyze the foregoing recovery system. 2.Preliminaries 2.1.Vectors. Letxdenote a vector of length N. For eachkN, letxkdenote either the usual k'th component of xor the signal of length Nconsisting of the jlargest-magnitude terms in x; it will be clear from context. The signal xkis the best k-term representation of x. The energy of a signal xiskxk2 2=PN i=1jxij2.4 GILBERT, LI, PORAT, STRAUSS operator name input output dimensions and construction rrow direct sum A:r1NM: (r1+r2)N B:r2NMi;j=( Ai;j; 1ir1 Bir1;j;1 +r1ir2 element-wise product A:rNM:rN B:rNMi;j=Ai;jBi;j nrsemi-direct product A:r1NM: (r1r2)N B:r2hMi+(k1)r2;`=( 0;Ak;`= 0 Ak;`Bi;j;Ak;`=jth nonzero in row ` Figure 2. Matrix algebra used in constructing an overall measurement matrix. The last column contains both the output dimensions of the matrix operation and its construction formula. 2.2.Matrices. In order to construct the overall measurement matrix, we form a number of dier- ent types of combinations of constituent matrices and to facilitate our description, we summarize our matrix operations in Table 2. The matrices that result from all of our matrix operations have Ncolumns and, with the exception of the semi-direct product of two matrices nr, all operations are performed on matrices AandBwithNcolumns. A full description can be found in the Appendix. 3.Sparse recovery system In this section, we specify the measurement matrix and detail the decoding algorithm. 3.1.Measurement matrix. The overall measurement matrix, , is a multi-layered matrix with entries inf1;0;+1g. At the highest level, consists of a random permutation matrix Pleft- multiplying the row direct sum of (lg( k)) summands, (j), each of which is used in a separate iteration of the decoding algorithm. Each summand (j)is the row direct sum of two separate matrices, an identication matrix,D(j), and an estimation matrix,E(j). =P2 6664(1) (2) ... (lg(k))3 7775where (j)=E(j)rD(j): In iteration j, the identication matrix D(j)consists of the row direct sum of O(j) matrices, all chosen independently from the same distribution. We construct the distribution ( C(j)nrB(j))S(j) as follows: Forj= 1;2;:::; lg(k), the matrix B(j)is a Bernoulli matrix with dimensions kcj-by-N, wherecis an appropriate constant 1 =2<c< 1. Each entry is 1 with probability  1=(kcj)
and zero otherwise. Each row is a pairwise independent family and the set of row seeds is fully independent. The matrixC(j)is an encoding of positions by an error-correcting code with constant rate and relative distance. That is, x an error-correcting code and encoding/decoding algorithm that encodes messages of (log log N) bits into longer codewords, also of length (log logN), and can correct a constant fraction of errors. The i'th column of C(j)is the direct sum of (log log N) copies of 1 with the direct sum of E(i1);E(i2);:::, where i1;i2;::: are blocks of O(log logN) bits each whose concatenation is the binary expansion ofiandE(
) is the encoding function for the error-correcting code. The number of columns inC(j)matches the maximum number of non-zeros in B(j), which is approximately theAPPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 5 expected number,  cjN=k
, wherec<1. The number of rows in C(j)is the logarithm of the number of columns, since the process of breaking the binary expansion of index iinto blocks has rate 1 and encoding by E(
) has constant rate. The matrixS(j)is a pseudorandom sign- ip matrix. Each row is a pairwise independent family of uniform 1-valued random variables. The sequence of seeds for the rows is a fully independent family. The size of S(j)matches the size of C(j)nrB(j). Note that error correcting encoding often is accomplished by a matrix-vector product, but we are notencoding a linear error-correcting code by the usual generator matrix process. Rather, our matrix explicitly lists all the codewords. The code may be non-linear. The identication matrix at iteration jis of the form D(j)=2 664h (C(j)nrB(j))S(j)i 1 :::h (C(j)nrB(j))S(j)i O(j)3 775: In iteration j, the estimation matrix E(j)consists of the direct sum of O(j) matrices, all chosen independently from the same distribution, B0(j)S0(j), so that the estimation matrix at iteration jis of the form E(j)=2 664h B0(j)S0(j)i 1 :::h B0(j)S0(j)i O(j)3 775: The construction of the distribution is similar to that of the identication matrix, but omits the error-correcting code and uses dierent constant factors, etc., for the number of rows compared with the analogues in the identication matrix. The matrixB0(j)is Bernoulli with dimensions O(kcj)-by-N, for appropriate c, 1=2<c< 1. Each entry is 1 with probability  1=(kcj)
and zero otherwise. Each row is a pairwise independent family and the set of seeds is fully independent. The matrixS0(j)is a pseudorandom sign- ip matrix of the same dimension as B0(j). Each row ofS0(j)is a pairwise independent family of uniform 1-valued random variables. The sequence of seeds for the rows is a fully independent family. 3.2.Measurements. The overall form of the measurements mirrors the structure of the measure- ment matrices. We do not, however, use all of the measurements in the same fashion. In iteration jof the algorithm, we use the measurements y(j)=(j)x. As the matrix (j)=E(j)rD(j), we have a portion of the measurements w(j)=D(j)xthat we use for identication and a portion z(j)=E(j)xthat we use for estimation. The w(j)portion is further decomposed into measurements [v(j);u(j)] corresponding to the run of O(log logN) 1's inC(j)and measurements corresponding to each of the blocks in the error-correcting code. There are O(j) i.i.d. repetitions of everything at iterationj. 3.3.Decoding. The decoding algorithm is shown in Figure 3 in the Appendix. 4.Analysis In this section we analyze the decoding algorithm for correctness and eciency.6 GILBERT, LI, PORAT, STRAUSS 4.1.Correctness. Letx=xk+1where we assume xis normalized so that k1k2= 1 and xkis the vector xwith all but the largest-magnitude kentries zeroed out. Our goal is to guarantee an approximation bxwith approximation error kxbxk2(1 +)k1k2+k2k2. But observe that 2 is a dierent type of object from xorbx;2is added to x. For the main theorem to make sense, therefore, we need to normalize . We discuss this now. Observe that the matrix can be scaled up by an arbitrary constant factor c>1 which can be undone by the decoding algorithm: Let D0be a new decoding algorithm that calls the old decoding algorithm Das follows: D0(y) =D1 cy
, so that D0(cx+2) =D x+1 c2
. Thus we can reduce the eect of 2by an arbitrary factor cand so citing performance in terms of k2kalone is not sensible. Note also that 2andxare dierent types of objects; , as an operator, takes an object of the type of xand produces an object of the type of 2. We will stipulate that the appropriate norm of be bounded by 1, in order to make our results quantitatively comparable with others. Our error guarantee is in `2norm, so we should use a 2-operator norm; i.e.,, max kxk2overx withkxk2= 1. But our algorithm's guarantee is in the \for each" signal model, so we need to modify the norm slightly. Denition 4. Thekk2 2norm of a randomly-constructed matrix ismaxxEhkxk2 xi . the smallestMsuch that, for all xwithkxk2= 1, we havekxk2<M except with probability 1/4. Now we boundkk2 2. Each row of a Bernoulli( p) matrix with sign ips, BS, satises E[jxj2] =pkxk2 2. So 1=psuch rows satisfy k(BS)xk2 2O kxk2 2 . Our matrix repeats the abovejtimes in the j'th iteration, jlog2(k), and combines it with an error-correcting code matrix of (log( N=k)) dense rows. It follows that kk2 2 2=O(log2(k) log(N=k)): We are ready to state the main theorem. Theorem 3 Consider the matrices in Section 3.1 and the algorithms in Section 3.3 (that share randomness with the matrices). The joint distribution on those matrices and algorithms, with suitable instantiations of anonymous constant factors (that may depend on ), are such that, given measurements x+2=y+2, the algorithm returns bxwith approximation error kxbxk2(1 +)k1k2+log(k)k2k2 kk2 2 with probability 3=4. The algorithm runs in time klogcNandhasO(klog(N=k))rows. In this extended abstract, we give the proof only for = 1. Our results generalize in a straightfor- ward way for general >0 (roughly, by replacing kwithk=at the appropriate places in the proof) and the number of measurements is essentially optimal in . Because our approach builds upon theCount Sketch approach in [4], we omit the proof of intermediary steps that have appeared earlier in the literature. We maintain the following invariant. At the beginning of iteration j, the residual signal has the form (Loop Invariant )r(j)=x(j)+(j) 1with x(j) 0k 2j, and (j) 1 223 4j except with probability1 4(1(1 2)j), wherek
k0is the number of non-zero entries. The vector x(j) consists of residual elements of xk. Clearly, maintaining the invariant is sucient to prove the overall result. In order to show that the algorithm maintains the loop invariant, we demonstrate the following claim. Claim 1. Letb(j)be the vector we recover at iteration j.APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 7 The vectorb(j)contains all but at most1 4k 2jresidual elements of x(j) k, with \good" estimates. The vectorb(j)contains at most1 4k 2jresidual elements of xkwith \bad" estimates. The total sum square error over all \good" estimates is at most " 23 4j+1# " 23 4j# =1 43 4j : Proof. To simplify notation, let Tbe the set of un-recovered elements of xkat iteration j; i.e., the support of x(j). We know thatjTjk=2j. The proof proceeds in three steps. Step 1. Isolate heavy hitters with little noise. Consider the action of a Bernoulli sign- ip matrixBSwithO(k=2j) rows. From previous work [4, 1], it follows that, if constant factors parametrizing the matrices are chosen properly, Lemma 5. For each row ofB, the following holds with probability (1) : There is exactly one element tofT\hashed" byB; i.e., there is exactly one t2Twith t= 1. There areO(N
2j=k)total positions (out of N) hashed byB. The dot product (S)r(j)isStr(j) tO 2j k (j) 1 2 . Proof. (Sketch.) For intuition, note that the estimator St(S)r(j)is a random variable with mean r(j) tand variance (j) 1 2 2. Then the third claim and the rst two claims assert that the expected behavior happens with probability (1).  In our matrix B(j), the number of rows is not k=2jbutkcjfor somec, 1=2< c < 1. Take c= 2=3. We obtain a stronger conclusion to the lemma. The dot product ( S)r(j)is Str(j) tO1 k(2=3)j (j) 1 2 =Str(j) t1 8 (3=4)j2j k (j) 1 2 ; provided constants are chosen properly. Our lone hashed heavy hitter twill dominate the dot product provided r(j) t1 8 (3=4)j2j k (j) 1 2 : We show in the remaining steps that we can likely recover such heavy hitters; i.e., Identify identies them and Estimate returns a good estimate of their values. There are at most ( k=2j) heavy hitters of magnitude less than1 8 (3=4)j2j k (j) 1 2 which we will not be able to identify nor to estimate but they contribute a total of1 8 (3=4)j (j) 1 2 noise energy to the residual for the next round (which still meets our invariant). Step 2. Identify heavy hitters with little noise. Next, we show how to identify t. Since there areN=k(1)positions hashed by B(j), we need to learn the O(log(N=k)) bits describing tin this context. Previous sublinear algorithms [7, 11] used a trivial error correcting code, in which the t'th column was simply the binary expansion of tin direct sum with a single 1. Thus, if the signal consists of xtin thet'th position and zeros elsewhere, we would learn xtandxttimes the binary expansion of t(the latter interpreted as a string of 0's and 1's as real numbers). These algorithms require strict control on the failure probability of each measurement in order to use such a trivial encoding. In our case, each measurement succeeds only with probability (1) and, generally, fails with probability (1). So we need to use a more powerful error correcting code and a more reliable estimate ofjxtj.8 GILBERT, LI, PORAT, STRAUSS To get a reliable estimate of jxtj, we use the b= (log log N)-parallel repetition code of all 1s. That is, we get bindependent measurements of jxtjand we decode by taking the median. Let p denote the success probability of each individual measurement. Then we expect the fraction pto be approximately correct estimates of jxtj, we achieve close to the expectation, and we can arrange thatp>1=2. It follows that the median is approximately correct. We use this value to threshold the subsequent measurements (i.e., the bits in the encoding) to 0 =1 values. Now, let us consider these bit estimates. In a single error-correcting code block of b= (log log N) measurements, we will get close to the expected number, bp, of successful measurements, except with probability 1 =log(N), using the Cherno bound. In the favorable case, we get a number of failures less than the (properly chosen) distance of the error-correcting code and we can recover the block using standard nearest-neighbor decoding. The number of error-correcting code blocks associated with tisO(log(N=k)=log logN)O(logN), so we can take a union bound over all blocks and conclude that we recover twith probability (1). The invariant requires that the failure probability decrease with j. Because the algorithm takes O(j) parallel independent repetitions, we guarantee that the failure probability decreases with jby taking the union over the repetitions. We summarize these discussions in the following lemma. We refer to these heavy hitters in the list  as the j-large heavy hitters. Lemma 6. Identify returns a set of signal positions that contains at least 3=4of the heavy hitters inT,jTjk=2j, that have magnitude at least1 8 (3=4)j2j k (j) 1 2 . We also observe that our analysis is consistent with the bounds we give on the additional mea- surement noise 2. The permutation matrix Pinis applied before 2is added and then P1is applied after 2by the decoding algorithm. It follows that we can assume 2is permuted at random and, therefore, by Markov's inequality, each measurement gets at most an amount of noise energy proportional to its fair share of k2k2 2. Thus, If there are m= (klogN=k) measurements, each measurement getsk2k2 2 mnoise energy and identication succeeds anyway provided the lone heavy hittertin that bucket has square magnitude at leastk2k2 2 m, so the at most ksmaller heavy hitters, that we may miss, together contribute energykk2k2 2 m=Ok2k2 2 log(N=k) . If we recall the denition and value ofkk2 2, we see that this error meets our bound. Step 2. Estimate heavy hitters. Many of the details in this step are similar to those in Lemma 5 (as well as to previous work as the function Estimate is essentially the same as Count Sketch ), so we give only a brief summary. First, we discuss the failure probability of the Estimate procedure. Each estimate is a complete failure with probability 1 (1) and the total number of identied positions is O jk(2=3)j . Because we perform jparallel repetitions in estimation, we can easily arrange to lower that failure probability, so we assume that the failure probability is at most  (3=4)j , and that we get approximately the expected number of (nearly) correct estimates. There are k(2=3)jheavy hitters in , so the expected number of failures is (1 =4)(k=2j). These, along with the at most 1 =4(k=2j) missedj-large heavy hitters, will form x(j+1), the at-most- k=2j+1residual heavy hitters at the next iteration. In iteration j,Identity returns a list  with k(2=3)jheavy hitter position identied. A group of k(2=3)jmeasurements in E(j)yields estimates for the positions in  with aggregate `2errorO(1), additively. An additional O (4=3)j times more measurements, O(k(8=9)j) in all, improves the estimation error to (1 =8) (3=4)j, additively. These errors, together with the omitted heavy hitters that are not j-large and(j)form the new noise vector at the next iteration, (j+1).APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 9 Finally, consider the eect of 2. We would like to argue that, as in the identication step, the noise vector 2is permuted at random and each measurement is corrupted byk2k2 2 m, wherem= (klog(N=k)) is the number of measurements, approximately its fair share of k2k2 2. Unfortunately, the contributions of 2to the various measurements are not independent as 2is permuted, so we cannot use such a simple analysis. Nevertheless, they are negatively correlated and we can achieve the result we want using [9]. The total `2squared error of the corruption over all O(k) estimates isk2k2 2=log(N=k), which will meet our bound. That is, since kk2 2 2=O(log2(k) log(N=k)), the 2contribution to the error is O k2k2p logN=k! =Olog(k)k2k2 kk2 2 ; as claimed, whence we read o the factor, log( k) (improvable to log1=2+o(1)k), which is directly comparable to other results that scale properly.  4.2.Eciency. 4.2.1. Number of Measurements. The analysis of isolation and estimation matrices are similar; the number of measurements in isolation dominates. The number of measurements in iteration jis computed as follows. There are O(j) parallel repetitions in iteration j. They each consist of k(2=3)jmeasurements arising out of B(j)for identication times O(log(N=k)) measurements for the error correcting code, plus k(2=3)jtimes O((4=3)j) for estimation. This gives  jk2 3j log(N=k) +jk8 9j! =klog(N=k)8 9+o(1)j : Thus we have a sequence bounded by a geometric sequence with ratio less than 1. The sum, over allj, isO(klog(N=k)). 4.2.2. Encoding and Update Time. The encoding time is bounded by Ntimes the number of non- zeros in each column of the measurement matrix. This was analyzed above in Section 4.1; there are log2(k) log(N=k) non-zeros per column, which is suboptimal by the factor log2(k). By comparison, some proposed methods use dense matrices, which are suboptimal by the exponentially-larger factor k. This can be improved slightly, as follows. Recall that we used jparallel repetitions in iteration j;j < log(k), to make the failure probability at iteration be; e.g., 2j, so the sum over jis bounded. We could instead use failure probability 1 =j2, so that the sum is still bounded, but the number of parallel repetitions will be log( j), forjlog(k). This results in log( k) log log(k) log(N=k) non-zeros per column and 2contribution to the noise equal top log(k) log log(k)k2k2 kk2 2. We can use a pseudorandom number generator such as i7!b(ai+bmodd)=Bcfor random a andb, whereBis the number of buckets. Then we can, in time O(1), determine into which bucket anyiis mapped and determined the i'th element in any bucket. Another issue is the time to nd and to encode (and to decode) the error-correcting code. Observe that the length of the code is O(log logN). We can aord time exponential in the length, i.e., time logO(1)N, for nding and decoding the code. These tasks are straightforward in that much time. 4.2.3. Decoding Time. As noted above, we can quickly map positions to buckets and nd the i'th element in any bucket, and we can quickly decode the error-correcting code. The rest of the claimed runtime is straightforward.10 GILBERT, LI, PORAT, STRAUSS 5.Conclusion In this paper, we construct an approximate sparse recovery system that is essentially optimal: the recovery algorithm is a sublinear algorithm (with near optimal running time), the number of measurements meets a lower bound, and the update time, encode time, and column sparsity are each within log factors of the lower bounds. We conjecture that with a few modications to the distribution on measurement matrices, we can extend this result to the `1C`1error metric guarantee. We do not, however, think that this approach can be extended to the \for all" signal model (all current sublinear algorithms use at least one factor O(logN) additional measurements) and leave open the problem of designing a sublinear time recovery algorithm and a measurement matrix with an optimal number of rows for this setting. References [1] N. Alon, Y. Matias, and M. Szegedy. The Space Complexity of Approximating the Frequency Moments. J. Comput. System Sci. , 58(1):137{147, 1999. [2] K. Do Ba, P. Indyk, E. Price, and D. Woodru. Lower bounds for sparse recovery. In ACM SODA , page to appear, 2010. [3] E. J. Cand es, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. , 59(8):1208{1223, 2006. [4] M. Charikar, K. Chen, and M. Farach-Colton. Finding frequent items in data streams. ICALP , 2002. [5] G. Cormode and S. Muthukrishnan. What's hot and what's not: Tracking most frequent items dynamically. In Proc. ACM Principles of Database Systems , pages 296{306, 2003. [6] G. Cormode and S. Muthukrishnan. Improved data stream summaries: The count-min sketch and its applications. FSTTCS , 2004. [7] G. Cormode and S. Muthukrishnan. Combinatorial algorithms for Compressed Sensing. In Proc. 40th Ann. Conf. Information Sciences and Systems , Princeton, Mar. 2006. [8] D. L. Donoho. Compressed Sensing. IEEE Trans. Info. Theory , 52(4):1289{1306, Apr. 2006. [9] Devdatt Dubhashi and Volker Priebe Desh Ranjan. Negative dependence through the fkg inequality. In Research Report MPI-I-96-1-020, Max-Planck-Institut fur Informatik, Saarbrucken , 1996. [10] Yaniv Erlich, Kenneth Chang, Assaf Gordon, Roy Ronen, Oron Navon, Michelle Rooks, and Gregory J. Han- non. Dna sudoku|harnessing high-throughput sequencing for multiplexed specimen analysis. Genome Research , 19:1243|1253, 2009. [11] A. C. Gilbert, M. J. Strauss, J. A. Tropp, and R. Vershynin. One sketch for all: fast algorithms for compressed sensing. In ACM STOC 2007 , pages 237{246, 2007. [12] P. Indyk and M. Ruzic. Near-optimal sparse recovery in the l1norm. FOCS , 2008. [13] D. Needell and J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comp. Harmonic Anal. , 2008. To appear. [14] Y. H. Zheng, N. P. Pitsianis, and D. J. Brady. Nonadaptive group testing based ber sensor deployment for multiperson tracking. IEEE Sensors Journal , 6(2):490{494, 2006. [15] Y.H. Zheng, D. J. Brady, M. E. Sullivan, and B. D. Guenther. Fiber-optic localization by geometric space coding with a two-dimensional gray code. Applied Optics , 44(20):4306{4314, 2005. 6.Appendix We have a full description of the matrix algebra dened in Table 2. Row direct sum. The row direct sum ArBis a matrix with Ncolumns that is the vertical concatenation of AandB. Element-wise product. IfAandBare bothrNmatrices, then ABis also anrN matrix whose ( i;j) entry is given by the product of the ( i;j) entries inAandB. Semi-direct product. SupposeAis a matrix of r1rows (andNcolumns) in which each row has exactly hnon-zeros and Bis a matrix of r2rows andhcolumns. Then BnrAis the matrix with r1r2rows, in which each non-zero entry aofAis replaced by atimes the j'th column of B, whereais thej'th non-zero in its row. This matrix construction has the following interpretation. Consider ( BnrA)xwhereAconsists of a single row, , with hnon-zeros and xis a vector of length N. Lety=xbe the element-wise product of APPROXIMATE SPARSE RECOVERY: OPTIMIZING TIME AND MEASUREMENTS 11 andx. Ifis 0/1-valued, ypicks out a subset of x. We then remove all the positions in ycorresponding to zeros in , leaving a vector y0of lengthh. Finally, (BnrA)xis simply the matrix-vector product By0, which, in turn, can be interpreted as selecting subsets of y, and summing them up. Note that we can modify this denition when Ahas fewer than hnon-zeros per row in a straightforward fashion.12 GILBERT, LI, PORAT, STRAUSS Recover (;y) Output:bx=approximate representation of x y=P1y a(0)= 0 Forj= 0 toO(logk)f y=yP1a(j) splity(j)=w(j)rz(j)  =Identify (D(j);w(j)) b(j)=Estimate (E(j);z(j);) a(j+1)=a(j)+b(j) g bx=a(j) Identify (D(j);w(j)) Output:  = list of positions  =; Dividew(j)into sections [v;u]of sizeO(log(cj(N=k))) For each section f u= median(jv`j) For each` // threshold measurements u`= (u`u=2) //(u) = 1 ifu>0,(u) = 0 otherwise Divideuinto blocks biof sizeO(log logN) For eachbi i=Decode (bi) // using error-correcting code =Integer (1;2;:::) // integer rep'ed by bits 1;2;:::  = [fg g Estimate (E(j);z(j);) Output:b=vector of positions and values b=; For each2 b= median`s:t:B(j) `;=1(z(j) `S(j) `;) Figure 3. Pseudocode for the overall decoding algorithm.

2009-12-01

An approximate sparse recovery system consists of parameters $k,N$, an $m$-by-$N$ measurement matrix, $\Phi$, and a decoding algorithm, $\mathcal{D}$. Given a vector, $x$, the system approximates $x$ by $\widehat x =\mathcal{D}(\Phi x)$, which must satisfy $\| \widehat x - x\|_2\le C \|x - x_k\|_2$, where $x_k$ denotes the optimal $k$-term approximation to $x$. For each vector $x$, the system must succeed with probability at least 3/4. Among the goals in designing such systems are minimizing the number $m$ of measurements and the runtime of the decoding algorithm, $\mathcal{D}$. In this paper, we give a system with $m=O(k \log(N/k))$ measurements--matching a lower bound, up to a constant factor--and decoding time $O(k\log^c N)$, matching a lower bound up to $\log(N)$ factors. We also consider the encode time (i.e., the time to multiply $\Phi$ by $x$), the time to update measurements (i.e., the time to multiply $\Phi$ by a 1-sparse $x$), and the robustness and stability of the algorithm (adding noise before and after the measurements). Our encode and update times are optimal up to $\log(N)$ factors.

Approximate Sparse Recovery: Optimizing Time and Measurements

0912.0229v1

arXiv:0912.3125v1 [gr-qc] 16 Dec 2009Toward a dynamical shift condition for unequal mass black hole binary simulations Doreen M¨ uller, Bernd Br¨ ugmann Theoretical Physics Institute, University of Jena, 07743 Jena, G ermany Abstract. Movingpuncturesimulationsofblackholebinariesrelyonaspecificgau ge choice that leads to approximately stationary coordinates near ea ch black hole. Part of the shift condition is a damping parameter, which has to be proper ly chosen for stable evolutions. However, a constant damping parameter does n ot account for the difference in mass in unequal mass binaries. We introduce a position de pendent shift damping that addresses this problem. Although the coordinates ch ange, the changes in the extracted gravitational waves are small. PACS numbers: 04.25.D-, 04.25.dg,04.30.Db 1. Introduction Earlynumericalrelativitysimulationsusinga3+1splitoftheEinsteineq uationssuffered from so-called slice stretching, an effect which occurs when using sin gularity avoiding slicing together with a vanishing shift. The slices become highly distort ed when time marches on in the outer regions of the grid but slows down in the vicinit y of the black hole. It became clear that a non-vanishing, outwards pointing shift vector would be required in order to redistribute grid points and also to prevent grid points from falling into the black hole. Inspired by Balakrishna et al.[1], Alcubierre et al.[2, 3] combined the1+log slicing conditionwith a dynamical shift condition called gamma- driver. These gauge conditions successfully prevented slice stretching in black ho le simulations using excision. It turned out that such gauge conditions could be used als o for fixed punctures with slight modifications to keep the puncture from evolving [4]. The fix ed-puncture modification was removed in [5, 6] when the moving puncture method was introduced. 1+log slicing with gamma-driver shift succeeds in moving the puncture freely through the grid while simultaneously avoiding slice-stretching. The basic reas on for the success ofthisgaugeconditionisthatwhentheslicesstarttostretch, the shiftvectorcounteracts by pulling out grid points from the region near the black hole. In this paper we focus on the dissipation or damping parameter in the gamma- driver shift condition, which plays an important role in the success of this gauge. In order to reduce oscillations in the shift vector, the authors of [3] n oticed the necessity of a damping term in the shift condition. Adjusting the strength of t he damping via a dampingparameterwasfoundtoallowfreezingoftheevolutionatlat etimesin[4]andtoToward a dynamical shift condition for unequal mass black ho le binary simulations 2 avoid drifts in metric variables in [7]. Additionally, the value of the damp ing coefficient was found to affect the coordinate location of the apparent horizo n and therefore the resolution ofthe black holeonthe numerical grid [7, 8]. The right cho ice ofthe damping value is therefore important if one wants to resolve the black hole pr operly while still driving the coordinates to a frame where they are stationary when the physical situation is stationary and hence obtain a stable evolution. The specific value of the damping parameter has to be adapted to th e black hole mass in order to obtain long term stable evolutions. If the damping pa rameter is either too small or too large, there are unwanted oscillations or a co ordinate instability, respectively. In binary simulations, a typical choice is a constant va lue of roughly 2 /M, whereMis the total mass of the system. However, using a constant dampin g parameter for black hole binaries with unequal masses leads to a fundamental p roblem. With a constant damping parameter, the effective damping near each blac k hole is asymmetric for unequal black hole masses since the damping parameter has dime nsions 1/M. For largemassratios, thisasymmetryinthegridcanbesolargethatsimu lationsfailbecause the damping may become too large for one of the black holes. This is on e of the reasons why the highest mass ratio that has been successfully simulated up t o now is 10 : 1 [9]. Advantageous would be a position-dependent damping parameter t hat adapts to the local mass, in particular such that in the vicinity of the ithpuncture with mass Miits value approaches 1 /Mi. It was noticed before [4, 9] that a damping coefficient adapted to the various parameters of the simulation would be benefi cial. In [10] a position-dependent formula was introduced for head-on collisions o f black holes, which to our knowledge was only used in one other publication [11], prior to th e moving puncture framework. In this paper, we take first steps towards a position-dependent damping parameter for moving punctures. As a consequence, the local coordinates change compared to standard simulations, but this does not signific antly affect gauge invariant quantities like the extracted waves as we discuss below. 2. Dynamical damping in the shift equation 2.1. Numerical setup We focus on the gauge condition used in the 3+1 splitting of the Einste in equations, in particular on the condition for the shift vector. The slices are determined by the 1+log slicing condition [12] for the laps e function α, ∂0α=−2αK, (1) whereKis the trace of the extrinsic curvature. The coordinates of a given slice are governed by the gamma-driver shift condition introduced in [4] as ∂2 0βi=3 4∂0˜Γi−ηs∂0βi, (2)Toward a dynamical shift condition for unequal mass black ho le binary simulations 3 where˜Γiare the contracted Christoffel symbols of the conformal metric ˜ γij,βiis the shift vector and ηsis the damping coefficient we will discuss in this publication. In Eqs. (1) and (2), ∂0is defined as ∂0=∂t−βi∂ias suggested by [13, 14, 15]. Examining the physical dimensions, we see that [ βi] = 1 and [∂0] = 1/M, whereM is the mass (e.g. the total mass of the spacetime under considerat ion). For this reason, the second term on the right hand side of equation (2) requires the damping parameter to carry units, [ηs] =1 M. (3) In simulations of a single Schwarzschild puncture of mass M1, we typically choose a damping parameter of ηs≈1/M1for obtaining enough damping in the shift without producing instabilities. In numerical experiments for a Schwarzsch ild puncture (to be discussed elsewhere), we find that 0 ≤ηs/lessorapproxeql3.5/M1is necessary for a stable and convergent numerical evolution. Some minimal amount of damping is im portant to suppress noise in the gauge when a puncture is moving. On the other hand, ifηsis too largethen there aregaugeinstabilities, leading toa loss of converge nce and toinstability of the entire numerical evolution. Furthermore, early simulations f or fixed punctures also found that ηsshould take values around 1 /M, whereMis the total mass, to avoid long-term coordinate drifts at the outer boundary [4]. In simulations of black hole binaries with total mass M=M1+M2, we usually set ηs= 2/Mwhich has been found to work well in equal mass binaries simulations. ( For equal masses, M1=M2=M/2, so near one of the punctures the value of ηsdiscussed above for Schwarzschild becomes ηs= 1/M1= 2/M.) For unequal mass binaries, the different black holes tolerate different ranges of ηsaccording to the above statement about single black holes. Ideally, ηsshould be ≈1/Mi, which cannot be accomplished simultaneously for unequal masses using a constant value of ηs= 2/M. In fact, for the mass ratio 1:10 in [9], the choice ηs= 2/Mfailed, but a smaller value for ηswas chosen such thatηs/lessorapproxeql3.5/Mifor bothi= 1 andi= 2. To overcome the conflicts between punctures with different masse s in evolutions of two or more black holes, we suggest to construct a non-constant , position-dependent damping parameter which knows about the position and mass of each puncture and takes a suitable value at every grid point. 2.2. Using ψ−2to determine the position of the punctures We thus desire a definition of ηswhich respects the unit requirements found in Eq. (3) and which asymptotes to specifiable values at the location of the pun ctures and at infinity. Typical values are ηs= 1/Miat theithblack hole and ηs= 2/Mat large distances. Thiscanbeachievedbydetermining ηsthroughapositiondependent function defined on the whole grid instead of using a constant as before. We d esire a smooth ηs which avoids modes which travel at superluminal speeds. Since we us e the Baumgarte– Shapiro–Shibata–Nakamura (BSSN) system of Einstein’s equations [16, 17], we wantToward a dynamical shift condition for unequal mass black ho le binary simulations 4 the form of ηsto depend only on the BSSN variables in a way that does not change th e principal part of the differential operators. In this paper, we choose to use the conformal factor ψ, which contains information aboutthelocationsandmassesofthepunctures. Theformulawew illusefordetermining the damping coefficient ηs(/vector r) is ηs(/vector r) =ˆR0/radicalbig ˜γij∂iψ−2∂jψ−2 (1−ψ−2)2, (4) with ˜γijthe inverse of the conformal 3–metric and ˆR0a dimensionless constant. While ψ, ˜γij, andˆR0are dimensionless, the partial derivative introduces the appropria te dependence on the mass since [ ∂i] = 1/Mand hence [ ηs(/vector r)] = 1/M. For a single Schwarzschild puncture of mass Mlocated atr= 0 the behavior of Eq. (4) near the puncture and near infinity is as follows. According t o [18], for small radiir(near the puncture) the conformal factor asymptotically equals ψ−2≃p1r (5) for a known constant p1. The next to leading order behavior is less simple [19]. The pointr= 0 corresponds to a sphere with finite areal radius R0, R0= lim r→0ψ2r=1 p1=ˆR0M. (6) Numerically, ˆR0≈1.31. The inverse of the conformal metric behaves like ˜γij≃δij. (7) Therefore, we find for small r /radicalbig ˜γij∂iψ−2∂jψ−2≃p1=1 ˆR0M(8) and (1−ψ−2)2≃(1−p1r)2≃1 (9) when keeping only leading order terms in r. Equations (8) and (9) combine according to (4) to give ηs(r= 0) = 1/M. (10) For largerwe can expand the conformal factor in powers of 1 /r, ψ−2≃/parenleftbigg 1+M 2r/parenrightbigg−2 ≃1−M r, (11) resulting in /radicalbig ˜γij∂iψ−2∂jψ−2=M r2. (12) and ηs(r→ ∞)≃ˆR0M/r2 (M/r)2=ˆR0 M. (13)Toward a dynamical shift condition for unequal mass black ho le binary simulations 5 In summary, Eq. (4) leads to ηs(r)→/braceleftBigg 1/M, r →0 ˆR0/M, r→ ∞(14) for a single puncture at r= 0. Note that using Eq. (4) in Eq. (2) does not affect the principal part of (2). Therefore, the system remains strongly hy perbolic, same as for ηs= const. according to [14, 13]. 3. Results Our Eq. (4) analytically gives the desired 1 /Mbehavior near the puncture and near infinity for a single, non–spinning and non–moving puncture. Now it re mains to be tested whether these properties persist in actual numerical simu lations, especially for unequal mass binaries. Simulations are performed with the BAM code described in [7, 20]. The c ode uses the BSSN formulation of Einstein’s equations and employs the mo ving puncture framework [5, 6]. Spatial derivatives are 6thorder accurate and time integration is performed using the 4thorder Runge–Kutta scheme. The numerical grid is composed of nested boxes with increasing resolution, where the boxes of high est resolution are centered around the black holes. These boxes are advanced in time with Berger–Oligar time stepping [21]. We are using puncture initial data with Bowen–York extrinsic curvature and solve the Hamiltonian constraint using a pseudospec tral collocation method described in [22]. The momentum parameter in the Bowen–Yo rk extrinsic curvature is chosen such that we obtain quasi–circular orbits in our binary simulations using the method of [23]. For binary simulations with unequal masses, we will use the mass ratio q=M2/M1 to denote the runs, Mibeing the bare mass of the ithpuncture. The physical masses of the punctures (obtained after solving the constraints) differ b y less than 10% from the bare masses for the orbits considered here, so the ηsvalues derived for a single puncture should remain valid. When comparing simulations run with ηs= 2/Mand ηs(/vector r) following Eq. (4) we will refer to them as “standard” and “new” or “ dynamical” gauge, respectively, throughout this paper. 3.1. Single Schwarzschild Black Hole In order to test the 1 /M–behavior of (4) near the puncture and infinity, we first performed a series of evolutions for a time of 100 Mof a single, non–spinning puncture while varying its mass. We then measured the value of ηs(/vector r) near the puncture and at the outer boundary of the grid and compared these values to th e limits (14). The data points in Fig. 1 correspond to these measurements while the line s are fits to the numerical data. The values of ηs(/vector r) near the puncture as a function of total mass M are fitted to ηs(M) = 1.05/Mwhich agrees well with the analytical limit r→0 of (14). Fitting toηsmeasured near the physical boundary of the grid reveals ηs(M) = 1.311/MToward a dynamical shift condition for unequal mass black ho le binary simulations 6 /SolidCircle /SolidCircle /SolidCircle /SolidCircle/SolidSquare /SolidSquare /SolidSquare /SolidSquare 0 1 2 3 40.51.01.52.02.53.03.5 MΗs/LParen1r0/RParen1r0/EquΑl125 M r0/RArrow0 Figure 1. Numerical test of the analytical limits (14) of ηs(/vector r) using single, non– spinning punctures with different masses. Shown are the values of ηs(r) near the puncture (gray squares) and at the outer boundary (black dots ). Note that here Mis identifiedwith adimensionlessnumber, so ηs(r) isdimensionlessaswell. Thefits tothe data points are consistent with the analytic prediction. Numerically, ηs(M) = 1.05/M (grayline) nearthe puncture and ηs(M) = 1.311/M(blackline) at the outerboundary. 0 2 4 6 8 100.000.050.100.150.20 r/LBracket1M/RBracket1ΒxΗs/LParen1r/OverRVector/RParen1 Ηs/EquΑl2/Slash1M 0 2 4 6 8 101.001.051.101.151.201.25 r/LBracket1M/RBracket1Γ/OverTilde xxΗs/LParen1r/OverRVector/RParen1 Ηs/EquΑl2/Slash1M Figure 2. x-component of the shift vector (upper panel) and xx-component of the conformal 3-metric (lower panel) of a single, non-spinning punctur e at time t= 100M, where the simulations have reached a stationary state. The dashe d black curves use dynamical damping, Eq. (4), the gray ones use ηs= 2.0/Min the shift condition Eq. (2). and therefore fulfills the limit r→ ∞of (14) even though the outer boundary is situated only at 130 M. Using a modified shift condition, the shift itself will, of course, change . We compare thex-component of the shift vector for using ηs= 2.0/Mandηs(/vector r) in Fig. 2. A change in the shift implies a change of the coordinates and therefore, coor dinate dependent quantities willchange, too. Asanexample, the xx-component oftheconformal3-metric, ˜γxx, is compared for ηs= 2/Mandηs(/vector r) in the lower panel of Fig. 2. The comparisons are made at time t= 100M, when the simulations have reached a stationary state. The changes in the shift should only affect the coordinates and coor dinate independent quantities should not change. This can be examined by lo oking at a scalar as a function of another scalar, e.g. the lapse αas a function of extrinsic curvature K, α=α(K). Both scalars should see the same coordinate drifts and therefo re, no changesToward a dynamical shift condition for unequal mass black ho le binary simulations 7 0.00 0.05 0.10 0.15 0.20 0.25 0.300.00.20.40.60.8 K/LBracket1M/Minus1/RBracket1Α0123450.00.20.40.60.8 r/LBracket1M/RBracket1Α 0123450.000.050.100.150.200.250.30 r/LBracket1M/RBracket1K/LBracket1M/Minus1/RBracket1 Ηs/LParen1r/OverRVector/RParen1 Ηs/EquΑl2/Slash1M Figure 3. The lapse function αas a function of extrinsic curvature Kfor a single, non-spinning and non-movingpuncture after a time t= 50.M.We compare using ηs(/vector r) (black, dashed line) and ηs= 2.0/M(gay line). The two curves lie perfectly on top of each other and are therefore indistinguishable. The insets show lapse (upper panel) and extrinsic curvature (lower panel) as functions of distance fro m the puncture. are expected in α(K). Figure 3 confirms this expectation. The two curves α(K) for ηs= 2.0/Mandηs(/vector r) are lying perfectly on top of each other. We therefore believe that using the dynamical damping introduces only coordinate chang es in our puncture simulations. 3.2. Black hole binary with equal masses While Eq. (4) has been introduced in order to allow for numerical simula tions of two black holes with highly different masses, we first apply it to equal mass simulations in order to perform several consistency checks. The (first order) coordinate independent quantity to look at in bina ry simulations is the Newman–Penrose scalar Ψ 4. We use Ψ 4for the extraction of gravitational waves (see [7] for details of the wave extraction algorithm), decomposed into modes using spin-weighted spherical harmonics Y−2 lm. Since Ψ 4is only first-order gauge invariant and we furthermore extract waves at a finite, fixed coordinate radius , it is a priori an open question how much the changes in the shift affect the wave forms. As the most dominant mode of Ψ 4in an equal mass simulation is the l=|m|= 2 mode, its real part multiplied by the extraction radius ( rex= 90Min this case) is displayed in Fig. 4. We look at amplitude and phase of this mode in Figs. 5 a nd 6. The initial separation was chosen to be D= 7M. The black holes complete about 3 orbits. Three different resolutions are used corresponding to the three d ifferent colors in Figs. 4, 5 and 6. We use the number of grid points in the inner boxes (centere d around the black holes) to denote the different resolutions. The grid configurations , in the terminology of [7], areφ[5×56 : 5×112 : 6],φ[5×64 : 5×128 : 6], and φ[5×72 : 5×144 : 6]Toward a dynamical shift condition for unequal mass black ho le binary simulations 8 260 280 300 320 340 360/Minus0.06/Minus0.04/Minus0.020.000.020.040.06 t/LBracket1M/RBracket1Re/LBrace1/CΑpPsi422/RBrace1rexN/EquΑl56 N/EquΑl64 N/EquΑl72 Figure 4. Real part of the 22-mode of Ψ 4times extraction radius rexfor an equal mass binary with initial separation D= 7Musingηs= 2.0/M(solid lines) and ηs(/vector r) following Eq. (4) (dashed lines) in three different resolutions (blue, r ed, green lines) according to the grid configurations described in the text. which corresponds to resolutions on the finest grids of 3 M/112, 3M/128 andM/48, respectively. These are the grid configurations used in [20]. In Fig. 5, we compare the amplitude A22in the standard gauge, ηs= 2.0/M, displayed as solid lines, to the new one, Eq. (4), plotted as dashed line s. We find that the differences between standard and new gauge for a given grid re solution are much smaller than differences due to using different resolutions. This stre ngthens the belief that we only introduced coordinate changes to the system when us ing Eq. (4). The maximum relative deviation between the amplitudes A22of old and new gauge amounts to about 3% for the lowest resolution ( N= 56) and decreases with increasing resolution, which can be seen in the inset of Fig. 5. For the phase, the absolute d ifferences are not visible by eye and therefore, we only plot the relative deviations betw eenφ22in the standard and new gauge for the three resolutions. For the lowest resolution ( N= 56), the maximum deviation is only 0 .35%. As for the amplitude, this deviation decreases with increases in resolution. This shows how the differences in the wav eforms disappear with increasing resolution. The fact that there actually aredifferences visible in the waves, though very small ones, is not surprising when considering the way we extract gravita tional waves. We fix a certain extraction radius and compute the Newman–Penrose sca lar on a sphere of this radius. The radius itself is coordinate dependent and we are compar ing Ψ4extracted at slightly different radii in the standard and new gauges. In future work we plan to compare wave forms extrapolated in radius to infinity, although it is w orth noting how small the deviations are without additional processing. 3.3. Black hole binary with mass ratio 4:1 After having examined the influence of using a dynamical damping coe fficientηs(/vector r) for an equal mass binary, the next step is to look at its behavior for une qual masses. The following results are obtained from a simulation of two black holes with m ass ratioq= 4 and initial separation D= 7M. We used the grid configurations φ[5×N: 7×2N: 6]Toward a dynamical shift condition for unequal mass black ho le binary simulations 9 260 280 300 320 340 3600.000.010.020.030.040.050.060.07 t/LBracket1M/RBracket1A22 260 280 300 320 340 360/Minus0.02/Minus0.010.000.010.020.03 t/LBracket1M/RBracket1/CΑpDeltΑ/CΑpAlpΗΑ22/Slash1/CΑpAlpΗΑ22N/EquΑl56 N/EquΑl64 N/EquΑl72 Figure 5. Amplitude of the 22-mode of Ψ 4for the same binary as in Fig. (4) using ηs= 2.0/M(solid lines) and ηs(/vector r) (dashed lines) in three different resolutions (blue, red, green lines) according to the grid configurations described in t he text. The inset shows the relative deviation ∆ A22/A22between the amplitude in the standard and in the new gauge, again for the three different resolutions. 260 280 300 320 340 360/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.50.0 t/LBracket1M/RBracket1/CΑpDeltΑΦ22/Slash1Φ22/LBracket1/Multiply10/Minus3/RBracket1N/EquΑl56 N/EquΑl64 N/EquΑl72 Figure 6. Relative phase difference of the 22-mode of Ψ 4for the same binary as in Fig. (4) using ηs= 2/Mandηs(/vector r). Compared are three different resolutions (blue, red and green lines) according to the grid configurations described in th e text. withN= 72,80 which have also been used in [24] for mass ratio 4:1. An interesting question in this context is how Eq. (4) behaves for a simulation with tw o punctures with different masses. The analytical behavior (14) was deduced fo r a single, non- moving, stationary puncture but now we are using it for two moving p unctures, which are during most of the simulations far from having reached a station ary state globally, but approximately stationary locally at the punctures. Figure 7 illustrates the distribution of ηs(/vector r) between the two punctures. For convenience, the conformal factor φ= lnψis also plotted in order to indicate the positions of the punctures via its maxima (the divergences are not r esolved). The snapshot is taken at a time during the simulation when the punctures are still well separated. Similar to the simulations of a single puncture, according to (14) we expect to findηs(/vector r)≃2 near the puncture with mass M1= 0.5 andηs(/vector r)≃0.5 in the vicinity of the second puncture with M2= 2.0. Near the outer boundary, ηsis supposed to take the value 1 .31/(M1+M2) = 0.52. Here the Miare chosen to be dimensionless, so ηs(/vector r) is dimensionless as well. Figure 7 confirms that we do obtain the expec ted values, although they are not reached exactly. The latter is not a problem a s simulations workToward a dynamical shift condition for unequal mass black ho le binary simulations 10 /Minus15/Minus10/Minus5 0 5 100.00.51.01.52.0 d/LBracket1M/RBracket1Ηs,Φ Figure 7. ηs(/vector r) (black line) for two punctures of bare masses M1= 0.5 andM2= 2.0 and the conformal factor φ(gray line) whose maxima show the current positions of the punctures. In this plot the Miare dimensionless, so ηs(/vector r) is dimensionless as well. The snapshot is taken right after the beginning of the q= 4 simulation with initial separation D= 5Mdescribed in the text. The smaller black hole is at position d=−3.6M, the larger one at position d= 1.2M. 100 110 120 130 140 150 160/Minus0.03/Minus0.02/Minus0.010.000.010.020.03 t/LBracket1M/RBracket1Re/LBrace1/CΑpPsi422/RBrace1rexN/EquΑl72 N/EquΑl80 Figure 8. Real part of the 22-mode of Ψ 4multiplied by the extraction radius rex forq= 4 and initial separation D= 5Mruns. Compared are results for employing ηs= 2.0/M(solid lines) and ηs(/vector r) (dashed lines) in two different resolutions (red and green lines) according to the grid configurations described in the te xt. nicely as long as ηs(/vector r) is in the right range for each black hole. For this reason, Eq. (4) also seems to work rather nicely for two punctures with unequal ma sses. As we did in the equal mass case in section 3.2, we compare the 22-mod e of Ψ 4 in the new gauge with the standard gauge. To see how small the differ ences using ηs orηs(/vector r) are, we plot its real part using two different resolutions which corr espond to the two different colors in Fig. 8. Figure 9 shows the amplitude for the two different resolutions. The inset gives the relative differences between amplitu des in the standard and new gauge. The maximum relative deviation appears for the lower resolution and amounts to about 3%. The high resolution gives 0 .5% relative difference. The phases in standard and new gauge are compared in Fig. 10. Again, we show on ly the relative deviations as the absolute ones are too small to be seen. We find rela tive differences of up to 0.4% for the lower resolution and only 0 .1% for the high one. This confirms that we are changing only the coordinates, as we found before in Section s 3.1 and 3.2. While the invariance of the waveforms is the most important feature of the newToward a dynamical shift condition for unequal mass black ho le binary simulations 11 100 110 120 130 140 150 160 1700.0000.0050.0100.0150.0200.0250.0300.035 t/LBracket1M/RBracket1A22 80 100 120 140 160/Minus0.020.000.020.04 t/LBracket1M/RBracket1/CΑpDeltΑ/CΑpAlpΗΑ22/Slash1/CΑpAlpΗΑ22N/EquΑl72 N/EquΑl80 Figure 9. Amplitude of the 22-mode of Ψ 4for the same runs as in Fig. 8. We compare results using ηs= 2.0/M(solid lines) and ηs(/vector r) (dashed lines) in two different resolutions (red and green lines) according to the grid configuratio ns described in the text. The inset shows the relative deviation ∆ A22/A22between the amplitude in the standard and in the new gauge, again for the same two resolutions. 80 100 120 140 160/Minus4/Minus3/Minus2/Minus1012 t/LBracket1M/RBracket1/CΑpDeltΑΦ22/Slash1Φ22/LBracket1/Multiply10/Minus3/RBracket1N/EquΑl72 N/EquΑl80 Figure 10. Relative phase difference of the 22-mode of Ψ 4for the same run as in Fig. 8ηs= 2.0/Mandηs(/vector r). Compared are two different resolutions according to the grid configurations described in the text. gaugeηs(/vector r), it is illuminating to examine how the black holes are represented on th e numerical grid. To this end, the apparent horizons (AH) are compu ted for both gauges in Fig. 11. We show the result in the ( x,y)–plane, in which the orbital plane lies. For clarity, the slices through the apparent horizons are only shown at 4 different times. In the beginning of the simulations, the AH pertaining to the same black h ole are lying on top of each other. With time, they separate as the coordinates become more and more different in the two simulations. Two observations can be made. First, the ratio between the coordinate area ofthe AH of thelarger black hole andt he oneof the smaller black hole is larger in the simulation using ηs= 2.0/M. This means the black holes are represented more equally on the grid in the simulation using Eq. (4). T his fact can be seen even more clearly in Fig. 12 where we plot the coordinate area of the apparent horizons comparing the standard gauge (red lines) and the new one (black lines). While the coordinate sizes of the smaller black hole (dashed lines) are near ly equal in both gauges, the sizes of the larger black hole (solid lines) differ by roughly 2M2. Second, the shape of the horizon of the smaller black hole is more and more dist orted in the ηs= 2.0/M-simulation when the black holes come closer together. This deforma tion isToward a dynamical shift condition for unequal mass black ho le binary simulations 12 -4-2 0 2 4 -4 -2 0 2 4PSfrag replacements x[M]y[M] t = 5.7M-4-2 0 2 4 -4 -2 0 2 4PSfrag replacements x[M]y[M] t = 32 .8M -4-2 0 2 4 -4 -2 0 2 4PSfrag replacements x[M]y[M] t = 54 .4M-4-2 0 2 4 -4 -2 0 2 4PSfrag replacements x[M]y[M] t = 70 M Figure 11. Comparison of apparent horizons in the orbital plane using ηs= 2.0/M (red lines) and ηs(/vector r) (black lines) at different times during the evolution for a q= 4 run with initial separation D= 5M. 0 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60PSfrag replacements AAH[M2] t [M]0 1 2 3 4 5 6 7 8 9 Figure 12. Comparison of the coordinate area of the apparent horizons using ηs= 2.0/M(red lines) and ηs(/vector r) (black lines) over evolution time for a q= 4 run with initial separation D= 5Muntil shortly before a common apparent horizon appears. The dashed lines belong to the smaller black hole whereas the solid lines r epresent the larger black hole. not visible in the new coordinates. The progressive stretching of th e apparent horizon shapeandthereforethedistortionofthecoordinatesnear theb lack holescanbeasource of instabilities, e.g. [25]. Using Eq. (4) seems to be profitable in this reg ard. 3.4. Behavior of ηs(/vector r)and influence on the shift vector Despite the encouraging results we have seen so far, there is a non –negligible concern using Eq. (4) in the gamma-driver condition (2). Although we do not d etermine the damping coefficient via a wave equation, we see wavy features in ηs(/vector r) traveling outwards. These distortions even leave remnants on the grid, esp ecially when they pass through a refinement boundary. The form of ηs(x) after different evolution times canToward a dynamical shift condition for unequal mass black ho le binary simulations 13 0 50 100 150 200 250 300 350 400 450 800700600500400300200100 0 0 0.5 1 1.5 2 2.5 PSfrag replacements x[M]t[M]ηsηs Figure 13. Form of ηs(/vector r) inx-direction at different times during an equal mass binary simulation. Noise travels outwards and leaves strong distort ions on the grid. 0 100 200 300 400 5000.0000.0020.0040.0060.0080.010 x,y/LBracket1M/RBracket1Βx,ΒyΗs/LParen1r/OverRVector/RParen1 Ηs/EquΑl2/Slash1M Figure 14. x-component of the shift vector in x-direction (solid curves) and y- component of the shift vector in y-direction (dashed curves) after the merger of two equal mass black holes at time t= 500Mwhen using either the standard gauge ηs= 2.0/M(gray lines) or the dynamical one, ηs(/vector r), (black lines). be seen in Fig. 13 for the equal mass binary described in Sec. 3.2. The result is similar in theq= 4 simulation and even in the Schwarzschild simulation, an outward tra veling pulse is present, which however does not leave visible distortions on t he grid and the relative amplitude of which decreases for higher mass. The effort we made before in order to achieve the correct value of ηs(/vector r) near the outer boundary seem to be canceled out by the disturbed shape we find now. As the peaks travel to a re gion of the grid where we have no punctures, we might take the point of view that th e exact value of ηs(/vector r) and therefore the distortions are of no importance for our simula tions. Indeed the oscillations do not translate to oscillations in the shift vector as one m ight think. In the shift, we find no gauge “waves” related to the ones in ηs(/vector r). Nevertheless, there is an unusual behavior. After merger, when going away from the punct ures the shift does not fall off to zero as fast as it does when using ηs= const. but keeps a shoulder (compare Fig. 14) which might lead to an unphysical and unwanted drift of the c oordinate system. We are planning to investigate these issues in more detail in the futur e.Toward a dynamical shift condition for unequal mass black ho le binary simulations 14 4. Discussion We presented a new approach to determine the coordinates in slices of spacetime for binary black hole simulations where we take the distribution of mass ov er the grid into account. We have shown that our approach of determining the dam ping parameter in the gamma-driver conditiondynamically via Eq. (4) gives stable evolut ions and doesnot significantly change the gravitational waves extracted from binar y systems of equal or unequal masses. Furthermore, the use of Eq. (4) in an unequal m ass simulation resulted in a more regular shape of the apparent horizon of the smaller black h ole as the binary merges. Thecoordinatesizeoftheapparenthorizonsbecamemor euniformwiththenew damping coefficient which is a first step towards representing and re solving black holes with different masses equally and hence removing the large asymmetr y which usually distorts the numerical grid in unequal mass simulations. We found ga uge waves in our damping coefficient which might affect the stability in very long-term sim ulations and lead to coordinate drifts after the merger of the binary. We will add ress these issues in a future publication [26]. Acknowledgments It is a pleasure to thank Jason Grigsby for discussions and for his va luable comments on this publication. We also thanks David Hilditch for discussions on the hyperbolicity of the BSSN system. This work was supported in part by DFG grant S FB/Transregio 7 “Gravitational Wave Astronomy” and the DLR (Deutsches Zentru m f¨ ur Luft und Raumfahrt). Doreen M¨ uller was additionally supported by the DFG R esearch Training Group 1523 “Quantum and Gravitational Fields”. Computations wer e performed on the HLRB2 at LRZ Munich. References [1] J. Balakrishna, G. Daues, E. Seidel, W.-M. Suen, M. Tobias, and E. Wang. Coordinate conditions in three-dimensional numerical relativity. Class. Quantum Grav. , 13:L135–142, 1996. [2] Miguel Alcubierre and Bernd Br¨ ugmann. Simple excision of a black h ole in 3+1 numerical relativity. Phys. Rev. D , 63:104006, 2001. [3] Miguel Alcubierre, Bernd Br¨ ugmann, Denis Pollney, Edward Seide l, and Ryoji Takahashi. Black hole excision for dynamic black holes. Phys. Rev. D , 64:061501(R), 2001. [4] Miguel Alcubierre, Bernd Br¨ ugmann, Peter Diener, Michael Kop pitz, Denis Pollney, Edward Seidel, and Ryoji Takahashi. Gauge conditions for long-term numer ical black hole evolutions without excision. Phys. Rev. D , 67:084023, 2003. [5] Manuela Campanelli, Carlos O. Lousto, Pedro Marronetti, and Yos ef Zlochower. Accurate evolutions of orbiting black-hole binaries without excision. Phys. Rev. Lett. , 96:111101, 2006. [6] JohnG.Baker, JoanCentrella, Dae-IlChoi, MichaelKoppitz, an dJamesvanMeter. Gravitational wave extraction from an inspiraling configuration of merging black ho les.Phys. Rev. Lett. , 96:111102, 2006. [7] Bernd Br¨ ugmann, Jos´ e A. Gonz´ alez, Mark Hannam, Sascha H usa, Ulrich Sperhake, and Wolfgang Tichy. Calibration of moving puncture simulations. Phys. Rev. D , 77:024027, 2008.Toward a dynamical shift condition for unequal mass black ho le binary simulations 15 [8] Frank Herrmann, Deirdre Shoemaker, and Pablo Laguna. Unequ al-mass binary black hole inspirals. 2006. [9] Jos´ e A. Gonz´ alez, Ulrich Sperhake, and Bernd Br¨ ugmann. Bla ck-hole binary simulations: the mass ratio 10:1. Phys. Rev. , D79:124006, 2009. [10] Miguel Alcubierre, Bernd Br¨ ugmann, Peter Diener, Frank Her rmann, Denis Pollney, Edward Seidel, andRyojiTakahashi. Testing excisiontechniques fordynam ical3Dblackholeevolutions. 2004. [11] Y. Zlochower, J. G. Baker, M. Campanelli, and C. O. Lousto. Acc urate black hole evolutions by fourth-order numerical relativity. Phys. Rev. D , 72:024021, 2005. gr-qc/0505055. [12] C. Bona, J. Mass´ o, E. Seidel, and J. Stela. New Formalism for Nu merical Relativity. Phys. Rev. Lett., 75:600–603, July 1995. [13] Horst Beyer and Olivier Sarbach. On the well posedness of the B aumgarte-Shapiro- Shibata- Nakamura formulation of Einstein’s field equations. Phys. Rev. D , 70:104004, 2004. [14] Carsten Gundlach and Jose M. Martin-Garcia. Well-posedness o f formulations of the Einstein equations with dynamical lapse and shift conditions. Phys. Rev. D , 74:024016, 2006. [15] James R. van Meter, John G. Baker, Michael Koppitz, and Dae- Il Choi. How to move a black hole without excision: gauge conditions for the numerical evolution of a m oving puncture. Phys. Rev. D, 73:124011, 2006. [16] M. Shibata and T. Nakamura. Evolution of three-dimensional gr avitational waves: Harmonic slicing case. Phys. Rev. D , 52:5428–5444, 1995. [17] T. W. Baumgarte and S. L. Shapiro. On the Numerical integratio n of Einstein’s field equations. Phys. Rev. D , 59:024007, 1998. [18] MarkHannam, SaschaHusa, Denis Pollney, Bernd Br¨ ugmann, a nd Niall ´O Murchadha. Geometry and regularity of moving punctures. Phys. Rev. Lett. , 99:241102, 2007. [19] Bernd Br¨ ugmann. Schwarzschild black hole as moving puncture in isotropic coordinates. Gen. Rel. Grav. , 41:2131–2151, 2009. [20] Sascha Husa, Jos´ eA. Gonz´ alez, Mark Hannam, Bernd Br¨ ug mann, and Ulrich Sperhake. Reducing phase error in long numerical binary black hole evolutions with sixth or der finite differencing. Class. Quantum Grav. , 25:105006, 2008. [21] Marsha J. Berger and Joseph Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. , 53:484–512, 1984. [22] Marcus Ansorg, Bernd Br¨ ugmann, and Wolfgang Tichy. A single -domain spectral method for black hole puncture data. Phys. Rev. D , 70:064011, 2004. [23] Benny Walther, Bernd Br¨ ugmann, and Doreen M¨ uller. Numeric al black hole initial data with low eccentricity based on post-Newtonian orbital parameters. Phys. Rev. , D79:124040, 2009. [24] Thibault Damour, Alessandro Nagar, Mark Hannam, Sascha Hus a, and Bernd Br¨ ugmann. Accurate effective-one-body waveforms of inspiralling and coalesc ing black-hole binaries. Phys. Rev., D78:044039, 2008. [25] BerndBr¨ ugmann, WolfgangTichy, and Nina Jansen. Numerical simulationof orbitingblack holes. Phys. Rev. Lett. , 92:211101, 2004. [26] Doreen M¨ uller, Jason Grigsby, and Bernd Br¨ ugmann. Dynamic al damping for unequal mass black hole binary simulations. 2009. in preparation.

2009-12-16

Moving puncture simulations of black hole binaries rely on a specific gauge choice that leads to approximately stationary coordinates near each black hole. Part of the shift condition is a damping parameter, which has to be properly chosen for stable evolutions. However, a constant damping parameter does not account for the difference in mass in unequal mass binaries. We introduce a position dependent shift damping that addresses this problem. Although the coordinates change, the changes in the extracted gravitational waves are small.

Toward a dynamical shift condition for unequal mass black hole binary simulations

0912.3125v1

Page1 Strategies and tolerances of spin transfer torque switching Dmitri E. Nikonov, George I. Bourianoff Intel Corp., Components Research, Santa Clara, California, 95052 Graham Rowlands, Ilya N. Krivorotov Department of Physics and Astronomy, Univers ity of California–Irvine, Irvine, California 92697, USA dmitri.e.nikonov@intel.com Abstract Schemes of switching nanomagnetic memories via the effect of spin torque with various polarizations of injected electrons are studied. Simulations based on macrospin and micromagnetic theories are performed and compared. We demonstrate that switching with perpendicularly polarized current by short pulses and free precession requires smaller time and energy than spin torque switching with collinear in plane spin polarization; it is also found to be superior to other ki nds of memories. We study the tolerances of switching to the magnitude of current and pulse duration. An increased Gilbert damping is found to improve tolera nces of perpendicular switching without increasing the threshold current , unlike in plane switching. Page2 1. Introduction Research in spintronics resulted in huge technological impact via development of extremely high capacity hard drives and magnetic RAM1. The basic paradigm of such devices is a stack of two ferromagnetic (FM) layers separated by a non-magnetic layer, see Fig. 1a. Giant magnetoresistance2,3 (GMR) – dependence of resi stance of such a stack on the relative direction of magne tization in the FM layers – provides the crucial path to interfacing magnetic and electroni c states in the device. If the non-magnetic layer is a dielectric, this effect is calle d tunneling magnetoresistance (TMR)4,5,6,7. A better way of switching ma gnetization in such devices was first theoretically predicted8,9 followed by the demonstration10,11,12,13,14 of spin transfer torque effect. This effect consists of precession of magnetization of one of the FM layers as current flows across the stack. As this happens, the angul ar momentum of spin-polarized current originating in one FM layer is transferred to the magnetization of another FM layer, see Fig 1a. Since this original work, a tremendous amount of research has been conducted in the field (see review15). Good values of the performance metrics have been achieved: the memory cell size has been reduced to a few square microns, the switching time to a few nanoseconds, and the switching current to a few milliamps. Spin transfer torque random access memory (STTRAM) has been prototyped16 and is close to commercialization. Still to be competitive with the incumbent memory technologies, such as SRAM, DRAM, and flash, STTRAM has to surpass them in the majority of a set of metrics (size, speed, Page3 energy). The fact that STTRAM is non-volatile is an important advantage, but would not alone ensure commercial success. For this r eason it is necessary to devise ways to decrease STTRAM’s threshold switching curren t and time, and thus the overall switching energy. One of the possible pathways to th is end is conducting sw itching by short pulses rather than quasi-steady currents. A previous theoretical treatment17 of pulsed currents envisioned two pulses of opposite polarity without a gap between them. Experi mental demonstration of switching with pulses was performed with single pulses 18,19 or double pulses 20 of constant polarity. In the present work we attempt to give a comprehensive view of the pulsed switching of nanomagnetic memories. We consider a variet y of cases of spin polarization of the current injected from the fixed FM layer to the free FM layer. In considering various dynamical switching strategies, we first define the energy landscape of the system and represent a specific switching strategy as a specific path across th e topological surface which defines the energy landscape The anal ysis contained here identifies the optimal switching strategy to be along the path of steep est accent (unlike the tr aditional, in-plane collinear polarization switching which pro ceeds along the path of minimum accent). We perform simulations both in the macrospi n and micromagnetic approximations and compare them side by side in order to make conclusions on applicabil ity of these methods in each of the cases. In these simulations we se parate the contribution of spin transfer and field-like torques 21,22,23 to draw a conclusion on how they affect switching in each of the cases. Finally we perform multiple simulations over a range of switching parameters in order to predict the tolerances of the memori es relative to the va riations in both the Page4 magnitude of current and the pulse duration. Su ch variations of curre nt and pulse duration are bound to exist in elec tronic circuits due to fabricati on variability and the temperature drift. Counter-intuitively, we find that Gilbert damping is be neficial for the tolerance of switching and increases neither the thres hold current nor the switching duration. The paper is organized as follows. Section 2 co ntains the description of the mathematical models of the macrospin approximation and micromagnetic simulator OOMMF used in this paper. Section 3 analyzes the energy dependence on the direction of magnetization and various strategies for switching based on it. Time dependence of magnetization in various strategies of switching is also ex emplified. Results of multiple simulation runs, presented in Section 4, establish tolerances of switching relative to current magnitude and pulse duration. In Section 5 we compare th e results of macrospin and micromagnetic simulations side by side. The conclusions of this work ar e summarized in Section 6. 2. Mathematical models The mathematical model of macrospin dynami cs is based on the Landau-Lifshitz-Gilbert (LLG) equation (see the review24). In addition it assumes that spatial variation of magnetization can be neglected, and the w hole magnetic moment of the nanomagnet (“macrospin”) can be represented by a single average vector of magnetization M, or dimensionless magnetization /sM=mM , where the saturation magnetization of the material is sM. Thus the LLG equation, containing the spin torque terms Γ, is Page5 0 effdd dt dtγµ α⎡⎤⎡⎤=− + +⎣⎦ ⎢⎥⎣⎦mmm×H m× Γ, (1) where the gyromagnetic constant is Bgµγ==, the Bohr magneton is Bµ, and the Lande factor is g, the permeability of vacuum is 0µ, and the Gilbert damping factor is α. The effective magnetic field originates from all contributions to the energy E per unit volume of the nanomagnet: 01 effEδ µδ=−HM (2) The energy of the nanomagnet includes the demagnetization term – coming from the interaction of magnetic dipoles between them selves, the material anisotropy, and the Zeeman energy due to an external magnetic fiel d. In this article we disregard the latter two contributions for the free FM layer, and focus on the former. The demagnetization term (synonymous with the shape an isotropy of the nanomagnet) is 0 2Eµ= MNM , (3) The demagnetization tens or is diagonal and =1xx yy zzNNN++ , if the coordinate axes coincide with the principal axes of the nanomagnet: Page6 00 0000xx yy zzN N N⎛⎞ ⎜⎟=⎜⎟⎜⎟⎝⎠N . (4) For the shape of typical nanomagnets, ellipti cal cylinder, the demagnetization tensor is calculated according to Ref. 25. A heuristic rule for demagneti zation energy is that it is lowest when magnetization points along the l ongest axis of a nanomagnet and highest when it points along the shortest axis. For exam ple, in the case considered here of the nanomagnet with dimensions of 120nm*60nm*3nm, the demagnetization tensor elements are =0.0279, 0.0731, =0.8990xx yy zzNN N = . The spin torque contribution is described by th e spin transfer (Slonczewski) and field like terms (in the brackets of the following equation) [][]() ' sJ Me tγεε=+ ⎡⎤⎣⎦Γ m× p×m p×m=, (5) where t is the thickness of the nanomagnet, e is the absolute value of the electron charge, J is the density of current perpendicu lar to the plane of the nanomagnet, p is the unit vector of polarizati on of electrons , and P is the degree of polari zation of these electrons. In this paper we disregard the angular depende nce in the prefactor fo r simplicity, so that 2Pε= (6) The mathematical model of micromagnetics is realized in a widely used simulator OOMMF 26. It is based on the same LLG equation (1). The main difference from the Page7 macrospin model is that now magnetization M is considered a function of spatial coordinates (discretized on a gr id) and its spatial variation plays an important role. The exchange interaction of between spin s is included as the energy density ( )2 22 xyz EAm m m= ∇ +∇ +∇ (7) and the demagnetization energy is calculat ed by explicit summation of dipole-dipole interactions between the parts of the nanomagnet, see Ref. 26. In this paper we will use the following set of typical parameters a nd hope that the reader agrees that the conclusions of the paper do not depend on this particular choice of numerical values: saturation magnetization 1 /sM MA m= , Lande factor 2g=, spin polarization 0.8P= , and the exchange constant 1121 0 /A Jm−=⋅ . In the cases when we include the field-like torque, we set its constant '0 . 3εε= to be in approximate agr eement with the results of Refs. 21 and 22. 3. Energy profile and strategies for switching We are applying the above mathematical model to treat various cases of spin transfer torque switching, Fig. 1. In these schemes of nanopillars, the upper blue layer designates a free nanomagnet. We introduce th e coordinate axes as follow s: x along the long axis of the nanomagnet, in plane of the chip, y perp endicular to x in plane of the chip, and z perpendicular to the plane of the chip. We also introduce the angles to specify the magnetization direction: θ, the angle from the x-axis, and φ, the angle of the projection on the yz-plane from the y axis. The free nanomagnet has two stable (lowest energy) Page8 states when magnetization points al ong +x or –x directions, i.e. 0, θπ= . The goal of memory engineering is to switch magnetiza tion between these two states. The bottom blue layer designates the fixed nanomagnet. Even though the spin torque acts on this layer as well, one keeps its magnetization fr om switching by coupling it to an adjacent (“pinning”) antiferromagnetic la yer (not shown in the pictur e). The magnetization of the fixed nanomagnet can be set in various direc tions by fabricating it with the right shape and magnetocrystalline anisotropy. One cas e is when the fixed magnetization is approximately along the x-axis, Fig. 1a. If both magnetizations were exactly along the x- axis, the spin torque acting on the free laye r would be zero, and sw itching would not take place. In fact, thermal fluctuations cause the angles of both free and fixed nanomagnets to have random values around 0θ=. Therefore in the macrospi n simulations, we formally set the initial angle of the free nanomagnet to 0.1θ= radians, and set the angle of the fixed layer to 0θ=. Another case, Fig 1b, is that of th e fixed magnetization in plane of the chip, with various θ and 0φ=. Finally, the case in Fig. 1c is that of perpendicular magnetization of the fixed nanomagnet /2φπ=± . We will see further on that the directions of magnetization of the fixed layer, which we consider identical to directions of spin polarization of the el ectrons injected into the free layer, correspond to different strategies of switching. In order to gain an intuitive understandi ng of the process of switching, one needs to visualize the “energy landscape” – the patter n of demagnetization energy in the phase space of magnetization angles, according to Eq. (3). The map of this sphere of angles on the plane is shown in Fig. 2. For a different sh ape of the nanomagnet, or in the presence of material anisotropy and exte rnal field, this dependence wi ll be quantitatively different, Page9 but the same qualitative approach applies. The salient features of the energy landscape are: a) the stretched ellipse-shaped “basins” close to the poles – th e stable equilibrium states; b) the two “valleys” stretching from one pole to the other – the states with in-plane magnetization; c) they have “mount ain passes”, or saddle points at /2θπ= and 0,φπ= d) two peaks corresponding to magnetiza tion perpendicular to the plane, at /2θπ= and /2φπ=± . The iso-energy lines are shown in the contour plot, Fig. 2. In the absence of damping and spin torque, they would coincide with closed orbits of magnetization. There are orbits of low energy precessing around one of the poles, and orbits of high energy oscillating between the two poles. In the presence of damping, the nanomagnet will evolve to one of the basins and eventually to the pole inside it. Spin torque can cause the nanomagnet to gain or lose energy, under some conditions moving between the basins, i.e. switching. At this point, let us agree on the definitions for switching time. We will consider switching under the action of rectan gular pulse currents of magnitude I and duration puτ (marked in the following plots). These valu es are relevant for the switching charge puIτand energy swp uE UIτ= dissipated in switching under the voltage bias U. Therefore the pulse duration puτ is the time important in the technological sense for optimizing the energy per writing a bit. From simulations we obtain the switching time, which characterizes how fast the nanomagnet responds to the current pulse. We customarily define the switching time swτ as the time over which the magne tization is switched from 10% to 90% of its limiting values. In our part icular case it is the interval between the first time the projection of magnetization xm goes below 0.8 till the last time it is over - Page 10 0.8. The 10-90% time gives the lower bound of switching time. Its importance is to characterize the switchin g time pertinent to the strategy, ra ther than influence of initial conditions. The reason that we us e it instead of 0-100% time is that, in the collinear in- plane case, the latter strongly depends on the choice of the initial angle of magnetization (see discussion below). The total write time needs to be longer than the largest of the two time measures. In the following, we will plot the switching times swτ resulting from the simulation of magnetization dynamics over certain time interv als, typically 1, 2.5, 3, or 4ns. When the switching time reaches this constant value, it really means that switching does not occur over the simulation time, and, with high probabi lity, even after any duration of evolution. Thus these limiting constant values on th e plot are just tokens for “no switching occurring”. Collinear polarization spin torque switching is done in a configuration of Fig 1a. Since the injected spin polarization is in plane close to 0θ=, the resulting spin transfer torque pushes the magnetization to rotate in plane, along the slope of sl owest accent in energy, which might seem at first glance like the optimal strategy of switching. An example of switching dynamics for this case is shown in Fig. 3. From the time evolution plot we see that magnetization performs an oscillatory mo tion close to in plan e position with slowly increasing amplitude. Torque increases with the angle from the x-axis, and this reinforces the growth of this angle. At some point the projections on the x-axis abruptly switches to a negative value and then the amplit ude of the oscillations is damped27,28. From the trajectory in phase space of magnetization direction, we see that the switching happens by Page 11 moving along one of the valleys and crossing over a saddle point. The duration of the pulse is sufficient if it is l onger than the time necessary to go to the other side of this saddle point. Another strategy is the in th e configuration of Fig. 1b. It is similar to the previous strategy, with a few modifications. In case of th e spin polarization 90 degree in plane, i.e. /2θπ= , see Fig. 4, the torque is maximal in the initial instant when the magnetization is at 0θ=. At very large current values, when th e magnetization reaches the saddle point, the torque turns to zero and the nanomagnet dwells in an unstable equilibrium until the end of the current pulse. At this point it falls towards one of the equilibriums; the choice of which is governed by the randomness of its position at that moment. Overall, this looks like an unreliable method of switching. Also it requires a much higher current than collinear polarization switching a nd the switching time is longer29. The situation is drastically improved for the case of injected spin polarization at a different angle, e.g. 135 degree in plane, 3/ 4θπ= , shown in Fig. 5. The path of switching still goes along the energy valley and over the saddle point. But in that case, torque is not zero at the initial instant, and it does not turn to zero at the position of th e energy saddle point. Switching time proves to be shorter. But one im portant similarity is that the pulse time needs to be relatively long in order to cross over the saddle point. The strategy of switching with perpendicular spin polarization, as shown in Fig 1c, turns out to be very different from collinear polari zation switching. It stems from the fact that the spin transfer torque acts in the direction perpe ndicular to the sample plane as well. In a counter-intuitive manner, it pushes the nanomagne t along the path of steepest accent. The way to take advantage of this situation is to use a very short pulse, which will supply Page 12 sufficient energy to the nanomagnet. The adva ntage of a short pulse is that it requires smaller switching energy supplied by the curren t. After the initial short current pulse, the nanomagnet precesses due to the torque from the shape anisotropy at zero spin torque, see Fig 6 for an example of such evolu tion. Under the condition of a small Gilbert damping, its trajectory will be close to the iso-energy line. A necessary condition for switching is that the nanomagnet has sufficient energy to be on the trajectory that crosses to the other basin, even with the account of loss to damping. The sufficient condition of successful switching is that by the time magne tization reaches the other basin, it loses enough energy so that it cannot cross back to the original basin. If this condition is satisfied, the nanomagnet slowly loses energy to damping and approaches the equilibrium. Due to the small value of damping the switching time turns out to be long. A variation on this strategy is to apply a nother pulse of curren t after the nanomagnet crosses to the other basin, Fig. 7. Th is pulse can have the same duration puτ and start after a delay time, gτ, after the trailing edge of the fi rst pulse but must have the opposite polarity of the current. Such a tw o-pulse sequence with the total time 2tp u gτττ=+ , will efficiently decrease the energy and avoid the process of slow energy damping. As a result the switching time swτ becomes very short, ~0.2ns, comparable topuτ. The downside of this strategy is that two pulses of course require twice the energy of one pulse with the same current magnitude and duration. Also it re quires a more complicated circuit to time the pulses of opposite polarity. We note that th e difference of the strategy considered here from the one of Kent et al.17 is that in their approach the two pulses of the opposite polarity did not have a gap between them, so the stage of free precession was absent. Page 13 We make the following approximate estimate of the pulse parameters for the pulsed switching. The energy that the nanomagnet gains in the first pulse must be larger than the energy necessary to cros s over the saddle point. 2 zz z y y x xmN N N=− (8) On the other hand, the out of plane projection at the end of the pulse is obtained from Eq. 2Bp u z sgI PmMeVµτ= . (9) For the parameters used in this paper, it amounts to the switching charge of 81puIf Cτ= per bit. This is much smalle r than the best achieved values for the collinear polarization switching, ~5pC per bit. Moreover, if we compare the switching time and energy projected here with incumbent type s of memory, see Table 3 in Ref. 30, we see that STTRAM with pulsed switching is superior to all other types of memory. From prototypes of STTRAM 16 we also know that its density can be comparable to that of DRAM. Therefore the proposed improvement gives it the crucial performance boost to potentially be the universal memory and to replace all other kinds. 4. Tolerances of switching and in fluence of field-like torque Spin transfer torque memories operate in el ectronic circuits. The ci rcuits naturally have variability originating from fabrication imperfections. Also the state of the circuit is subject to electronic noise and temperature drif t. Obviously it is not possible to guarantee precise values of switching current and time fo r elements of memories. Therefore it is Page 14 especially important to study the tolerances of memory operation relative to external parameters. We believe such studies have not been conducted up to now. Here we run multiple simulations over a wide set of parameters to draw some conclusions about these tolerances. At the same time we study the effect of the field-like torque (FLT) contribution. Experiments on separate measurements of the spin transfer (Slonczewski) and field like torques have been conducted 21,22. But the implications of th ese contributions to current induced magnetization switching have not been sufficiently clarified. The experiments show that FLT increases with increasing appl ied voltage. To account for this we consider two cases – no field-like torque '0ε= and large field-like torque '0 . 3εε= . The contour plots of switching time vs. current and pulse duration for collinear polarization switching are shown in Fig. 8. It is common to think of spin torque switching as having a threshold (or critical) current cI. However from this plot we see that, for sufficiently short pulse du ration, the switching current 1/cp uIIτ−∼ is much larger than the critical current. Therefore it is the threshold charge, 1.2puI pCτ≈ , that determines whether the memory state is sw itched. Above this threshold, switching can be quite fast, ~0.2ns, but the total write time is limited by the pulse duration instead. One can notice that the shapes of the swit ching time dependence with and without FLT are remarkably similar, but they appear to be shifted. For th at reason, one needs to be cautious of the fact that for a specific values of current and pul se duration, the dynami cs may be different with and without FLT. The reason for similarity is that for this cas e FLT plays a role of an effective magnetic field in the z-direction, in addition to a large effective field from Page 15 demagnetization. There are curious geometri cal features of the switching threshold border. Even though they are persistent in simula tion, we believe that they are artifacts of the choice of initial magnetization and of os cillations (“ringing”) of magnetization after switching. In reality, thermal fluctuation wi ll vary the initial magnetization, and the features on the plot will be washed away for the thermally averaged switching time. Overall, this strategy of switching gives an excellent tolerance when the switching current and pulse duration are se t high enough above the threshold. The switching time diagrams for the 90 degree in plane polarization ar e shown in Fig. 9. In this case, the threshold current is actua lly a good criterion of switching; and this threshold turns out to be very high, ~10mA. Without FLT, even above threshold there are tightly interlaced regions of successful and uns uccessful switching. This attests to the unstable nature of such switchi ng. It cannot be used for a pr actical device. The situation is different with FLT. There are large regions of small sw itching time, and therefore good tolerance to parameters, above th e threshold. The reason for this stabilization is that even though the Slonczewski torque vanishes at /2θπ= , FLT is still finite and it succeeds in pushing the nanomagnet over the energy saddl e point. However at excessively high current we encounter the regions of unsucce ssful switching that memory designers need to avoid. The switching diagrams for 135degree in plan e polarization are shown in Fig 10. The threshold behavior is in be tween the collinear polarizati on and 90degree in plane cases. The threshold current is not c onstant, and it is ~2-4mA, which is lower than that for 90 degrees. But it is not inversely proportional to the pulse duration either. The threshold Page 16 charges are in the range 0.6 1.6puI pC τ≈÷ . Overall it has the same excellent tolerance to current and pulse variation as the collinear polarization switc hing, but in the absence of FLT, regions of unsuccessful switchi ng are observed at higher current. The switching diagrams for out of plane sp in polarization and one switching pulse are shown in Fig. 11. The areas of successful sw itching are seen as na rrow strips across the plot. They are interlaced with stripes of unsuccessful switching. The reason for such behavior is the precession character of magnetization dynamics for this switching strategy. If too much energy is transferre d from the current to the nanomagnet, it overshoots and returns to the basin around the initial ma gnetization state. The set of successful switching correspond to 0.5, 1.5, 2.5 etc. full turns of magnetization. The lowest of these stripes corresponds to the threshol d of switching, 100puIf C τ≈ . Though the threshold is only approximately given by the product of the current and the pulse duration. This is in a very good agreemen t with the analytical estimate (9). This is the first case when we encounter the problem of tolerances in earnest. From the left plot in Fig. 11 for Gilbert damping of 0. 01, we estimate the tolerances to be 1ps and 0.1mA. This is likely too tight for a realistic memory circuit. The right plot is calculated for Gilbert damping 0.03. We see that, c ontrary to what is known about collinear polarization switching, the thre shold is almost unchanged at a higher Gilbert damping. At the same time, the tolerances are much rela xed, to 5ps and 0.7mA. The switching stripes became wider, and the next order of switc hing with 1.5 turns is moved to a higher switching charge. This seems to be the fi rst occasion when increasing damping is beneficial for the device performance. We note that these simulations are done with Page 17 inclusion of FLT. The results with zero F LT (not included in this paper) are almost indistinguishable from those. The reason for this is that the current pulses act when magnetization is close to the poles, and FLT has projections mostly in-plane of the nanomagnet, which contribute only negligibly to the precessional type of switching. The switching diagrams for out of plane polariz ation and two pulses are shown in Fig. 12. The total pulse time is fixed at 0.2t nsτ= , while the pulse time puτ is given on the horizontal axis of the plot. Their overall character is sim ilar to those for a single pulse. The threshold condition is approximately the sa me. The first stripe is very narrow, with the tolerances 0.3ps and 30uA. Surprisingly the second stripe corresponding to 1.5 turns, has a much larger and acceptable value of to lerance of 4ps and 0.4mA. We do not have an intuitive explanation for this difference between the two switching cases. For smaller Gilbert damping 0.01 the switching time is quite short, ~0.2ps. This is due to the fact that the second pulse eliminates ringing of magnetizat ion, as it was discussed in the previous section. With increase of Gilbert damping to 0.03, the tolerances in the first stripe improve to 2ps and 0.2mA. This is manifested as appearances of several satellite strips around the first one, meaning that at highe r damping the condition of timing the gap between the pulses becomes less crucial fo r successful switchi ng. Conversely, the switching time gets slower, ~0.5ps. This is because the magnetization oscillations are not eliminated as efficiently is the pulses are not finely timed. Like for the single pulse, the inclusion of FLT changes the results very insignificantly. Page 18 5. Comparison of macrospin an d micromagnetic simulations The simulation of micromagnetic dynamics is a more rigorous model and presumably gives a better approximation to reality that the macrospin approximation. However it is also much more computationally demanding. Fo r this reason it is us eful to compare the results of macrospin and micromagnetic m odels. We compare the switching time at various pulse durations but fixe d values of current. Microma gnetic simulations start with an initial state obtained by rela xing its energy to a minimum. It has the general direction of 0θ=, and a “leaf state” pattern, i.e. magnetization is closer to being parallel to the longer sides of the ellipse. The comparison for the collinear polarization switching is shown in Fig. 13. Here we set the direction of the sp in polarization of the fixed layer to be 0.1θ= . This value is an estimate of the r.m.s deviation of the angle of magnetization due to thermal distribution of energy. We make this choice of the initia l angle only in the case of collinear in-plane switching, because spin torque would vanish for 0θ=. For other cases the spin torque does not vanish at the zero initial angle. The agreement is surprisingly good. Both models give approximately the same value of threshold charge. This may be because we are focusing on the evolu tion starting from angle 0.1θ= . The evolution around 0θ= occurs under a much smaller torque and carries more un certainty. Macrospin model exhibits more oscillations close to the thre shold. Micromagnetic model predicts shorter switching time above threshold, probably due to more efficient damping of higher modes of magnetization. Page 19 The comparison in the case of 90degree in plan e polarization, Fig. 14, shows essentially the same lower envelope of switching time. The micromagnetic model shows fewer areas of failed switching. We speculate that this is due to stronger effect of FLT on non- uniform magnetization. The disagreement is more pronounced in the cas e of single pulse out of plane polarization, Fig. 15. The similarities are the same positi on on the time scale of successful switching stripes for 0.5 and 3.5 turns. The regions of unsuccessful switching for 1 and 3 turns are absent in the micromagnetic model, probabl y because the barrier for crossing into the other basin is increased for a non-uniform ma gnetization pattern. The switching times are generally predicted to be shor ter in the micromagnetic model. A similar situation is observed in out of plane pol arization switching with two pulses, see Fig. 16. The positions of successful switching on the time scale are shifted. This is probably due to the fact that the nanomagnet energy is di fferent with the account of exchange and dipole-dipole interaction, and thus the time of free precession of magnetization is different too. As before, micromagnetics pr edict shorter switching times, as well as giving better tolerance for 0.5 turn switching. In fact, micromagnetic simulations for one and two pulses look quite si milar. This points to higher significance of damping than of the second pulse in bri nging the magnetization to its final state. This paper includes just a few examples of co mparison. A more complete set of data is available 31. All of this supports the conclusion th at macrospin simulati ons give generally the same qualitative dependence of switchi ng time on the current a nd pulse duration as Page 20 OOMMF simulations. The former can be used to infer general trends . The latter should be saved for obtaining quanti tatively precise results. 6. Conclusions We have compared various strategies of spin torque switching. The tolerances of switching, performance limits of out-of plane pol arization devices, and the effect of field- like torque have been comprehensively studied for the first time. In summary, switching with short pulses of out of plane polarization is the prefer red strategy. It has a much lower threshold charge than other strategies of switching, and suffers less from low tolerance to current magnitude and pulse dura tion. This problem of low tolerance can be resolved by increasing Gilbert damping in the nanomagnet. Field-like torque significantly changes the results for non-co llinear in-plane switching a nd happens to produce a minor effect for other switching strategies. Implem entation of this strategy would put STTRAM in a position of a technological leadership among memories. We find that macrospin gives good qualitative prediction of the dyna mics, though micromagnetic models should be used to get better quantitative precision. 8. Acknowledgements G. R. and I. N. K. gratefully acknowle dge the support of DARPA, NSF (grants DMR- 0748810 and ECCS-0701458) and the Nanoelectr onic Research Ini tiative through the Western Institute of Nanoelectronics. Co mputations supporting this paper were performed on the BDUC Compute Cluster donated by Broadcom, co-run by the UCI OIT and the Bren School of Inform ation and Computer Sciences. Page 21 References 1 A. Fert, “Nobel Lecture: Origin, developmen t, and future of spintronics”, Rev. Mod. Phys. 80, 1517-30 (2008). 2 M. N. Baibich, J. M. Broto, A. Fert, F. N. Vandau, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, J. Chazelas, “Giant Magneotres istance of (100) Fe / (100) Cr magnetic superlattices”, Phys. Rev. Lett. 61, 2472 (1988). 3 G. Binasch, P. Grunberg, F. Saurenbach , W. Zinn, “Enhanced magnetoresistance in layered magnetic structures with antiferromagne tic interlayer exchange”, Phys. Rev. B 39, 4828 (1989). 4 J. S. Moodera, L. R. Kinder, T. M. Wong, R. Meservey, “Large magnetoresistance at room temperature in ferromagnetic thin-film tunnel junctions”, Phys. Rev. Lett. 74, 3273 (1995). 5 T. Miyazaki, T. Yaoi, and S. Ishio, “L arge magnetoresistance effect in 82Ni-Fe/Al- Al2O3/Co magnetic tunnel junction”, J. Magn. Magn. Mater. 98, L7 (1991). 6 S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Sament, and S. Yang, Nature Mater. 3, 862 (2004). 7 S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, K. Ando, “Giant room-temperature magnetoresistance in single-crystal Fe/MgO/F e magnetic tunnel junctions”, Nature Mater. 3, 868 (2004). 8 Slonczewski, J. C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1–L7 (1996). Page 22 9 Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353–9358 (1996). 10 Tsoi, M. et al. Excitation of a magnetic multilayer by an electric current. Phys. Rev. Lett. 80, 4281–4284 (1998). 11 Myers, E. B., Ralph, D. C., Katine, J. A ., Louie, R. N. & Buhrman, R. A. “Current- induced switching of domains in magnetic multilayer devices”, Science 285, 867–870 (1999). 12 J.A. Katine, F.J. Albert, R.A. Buhrman, E. B. Myers, and D.C. Ralph, “Current-driven magnetization reversal and spin -wave excitations in Co/Cu/Co pillars”, Phys. Rev. Lett. 84, 3149 (2000). 13 Sun, J. Z. Current-driven magnetic switchi ng in manganite trilaye r junctions. J. Magn. 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Sousa, L. Buda-Prejbeanu, and B. Dieny, “ 100 ps precessional spin-transfer switching of a planar magnetic random access memory cell wi th perpendicular spin polarizer”, Appl. Phys. Lett. v. 95, 072506 (2009). 20 S. Garzon, L. Ye, R. A. Webb, T. M. Crawford, M, Covington and S. Kaka, “Coherent control of nanomagnet dynamics via ultrafast spin torque pulses”, Phys. Rev. B 78, 180401R (2008). 21 J. C. Sankey, Y.-T.Cui, J. Sun, J. C. Sl onczewski, R. A. Buhrma n, and D. C. Ralph, “Measurement of the spin-transfer-torque ve ctor in magnetic tunnel junctions”, Nature Physics 4, 67 (2008). 22 H. Kubota et al., “Quantitative measuremen t of voltage dependence of spin-transfer torque in MgO-based magnetic tunnel j unctions”, Nature Physics 4, 37 (2008). 23 I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, and W. H. Butler, “Anomalous Bias Dependence of Spin Torque in Magneti c Tunnel Junctions” , Phys. Rev. Lett. 97, 237205 (2006). 24 D.V. Berkov and J. Miltat, “Spin- torque driven magnetization dynamics: Micromagnetic modeling”, Journal of Ma gnetism and Magnetic Materials 320, 1238– 1259 (2008). Page 24 25 M Beleggia1, M De Graef, Y T Millev, D A Goode, and G Rowlands, “Demagnetization factors for elliptic Cylinders”, J. P hys. D: Appl. Phys. 38, 3333–3342 (2005). 26 M. J. Donahue and D. G. Porter, “OOMMF User’s Guide, Version 1.0,” NISTIR 6376, National Institute of Standards and T echnology, Gaithersburg, MD (Sept 1999). 27 I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, R. A. Buhrman, “Time-Resolved Measurements of Nanoma gnet Dynamics Driven by Spin-Transfer Torques”, Science 307, 228 (2005). 28 T. Devolder, J. Hayakawa, K. Ito, H. Taka hashi, S. Ikeda, P. Crozat, N. Zerounian, Joo-Von Kim, C. Chappert, and H. Ohno, “S ingle-Shot Time-Resolved Measurements of Nanosecond-Scale Spin-Transfer Induced Switc hing: Stochastic Versus Deterministic Aspects” , Phys. Rev. Lett. 100, 057206 (2008). 29 F. B. Mancoff, R. W. Dave, N. D. Rizzo, T. C. Eschrich, B. N. Engel, S. Tehrani, “Angular dependence of spin-tra nsfer switching in a magnetic nanostructure”, Appl. Phys. Lett. 83, 1596 (2003). 30 J. Gallagher and S. S. P. Parkin, “Devel opment of the magnetic tunnel junction MRAM at IBM: From first junctions to a 16-Mb MRAM demonstrator chip”, IBM J. Res. & Dev. 50, 5-23 (2006). 31 D. E. Nikonov, G. I. Bour ianoff, G. Rowlands, I. N. Krivorotov, ”Comparisons of macrospin and OOMMF simulations”, online www.nanohub.org (2010). Page 25 Figure 1. The geometry of the spin tor que memory nanopillars. Red layer – tunneling barrier oxide, orange – non-ma gnetic electrodes, blue with double-sided arrow – free layer with in-plane magnetization (easy x-axis), blue with one sided arrow – fixed layer. Polarizations: a) in plane collin ear with x-axis (left), b) in plane at an angle to x-axis (middle), c) perpendicula r to the plane (right). a b c Page 26 Figure 2. Map of the demagnetization ener gy of the nanomagnet (normalized) in the macrospin model. θ – angle of magnetization from the x-axis (easy axis), φ – angle of projection of magnetization within the yz-plane (hard plane). Page 27 Figure 3. Magnetization projections vs. time and the trajectory of magnetization for collinear polarization switching (0 degree in plane polarization). Page 28 Figure 4. Magnetization projections vs. time a nd the trajectory of magnetization for 90 degree in plane polarization. Page 29 Figure 5. Magnetization projection vs. time a nd the trajectory of magnetization for 135 degree in plane polarization. Page 30 Figure 6. Magnetization projections vs. time and the trajectory of magnetization for perpendicular out of plane polarization, one pulse. Page 31 Figure 7. Magnetization projections vs. time and the trajectory of magnetization for perpendicular out of plane polarization, two pulses with a total time of 0.2ns. Page 32 Figure 8. Contour maps of switching time, left without the field like torque and right with 0.3 factor of field like tor que, for collinear polarization switching (0 degree in plane polarization). Simulation time 2ns. Page 33 Figure 9. Contour maps of switching time, left without the field like torque and right with 0.3 factor of field like torque, for 90 degree in plane polarization. Simulation time 4ns. Page 34 Figure 10. Contour maps of switching time, le ft without the field like torque and right with 0.3 factor of field like torque, for 135 degree in plan e polarization. Simulation time 4ns. Page 35 Figure 11. Contour maps of switching time, left α=0.01 and right α =0.03, with 0.3 factor of field like torque, for perpendicular out of pl ane polarization, one pulse. Simulation time 1ns. Page 36 Figure 12. Contour maps of switching time, left α=0.01 and right α =0.03, with 0.3 factor of field like torque, fo r perpendicular out of plane pol arization, two pulses with a total time of 0.2ns. Simulation time 1 ns. Page 37 Figure 13. Comparison of the switching time vs. pulse length simulated by a macrospin model (left) and OOMMF (right ) for collinear polarization switching at I=2.5mA, field- like torque 0.3, and polarization angle in plane of 10 degrees. Simulation time 3ns. Page 38 Figure 14. Same as Fig. 13 at I=20mA, field- like torque 0.3, and polarization angle in plane of 90 degrees. Simulation time 2.4ns. Page 39 Figure 15. Same as Fig. 13, single puls e at I=4mA, field-like torque 0.3, α=0.01, and polarization angle perpendi cular to plane of 90 degrees. Simulation time 1.8ns, Page 40 Figure 16. Same as Fig. 13 for two pulses with total duration of 0. 2ns, at I=4mA, field- like torque 0.3, α=0.01, and polarization angle perp endicular to plane of 90 degrees. Simulation time 1.8ns.

2010-01-26

Schemes of switching nanomagnetic memories via the effect of spin torque with various polarizations of injected electrons are studied. Simulations based on macrospin and micromagnetic theories are performed and compared. We demonstrate that switching with perpendicularly polarized current by short pulses and free precession requires smaller time and energy than spin torque switching with collinear in plane spin polarization; it is also found to be superior to other kinds of memories. We study the tolerances of switching to the magnitude of current and pulse duration. An increased Gilbert damping is found to improve tolerances of perpendicular switching without increasing the threshold current, unlike in plane switching.

Strategies and tolerances of spin transfer torque switching

1001.4578v1

arXiv:1003.1868v1 [cond-mat.mes-hall] 9 Mar 2010Damping of Nanomechanical Resonators Quirin P. Unterreithmeier,1,∗Thomas Faust,1and J¨ org P. Kotthaus1 1Fakult¨ at f¨ ur Physik and Center for NanoScience (CeNS), Ludwig-Maximilians-Universit¨ at, Geschwister-Scholl- Platz 1, D-80539 M¨ unchen, Germany (Dated: May 28, 2018) We study the transverse oscillatory modes of nanomechanica l silicon nitride strings under high tensile stress as a function of geometry and mode index m≤9. Reproducing all observed resonance frequencies with classical elastic theory we extract the re levant elastic constants. Based on the oscillatory local strain we successfully predict the obser ved mode-dependent damping with a single frequency independent fit parameter. Our model clarifies the role of tensile stress on damping and hints at the underlying microscopic mechanisms. The resonant motion of nanoelectromechanical sys- tems receives a lot of recent attention. Their large fre- quencies, low damping i.e. high mechanical quality fac- tors, and small masses make them equally important as sensors[1–4] and for fundamental studies[3–9]. In either case, low damping of the resonant motion is very desir- able. Despite significant experimental progress[10, 11], a satisfactory understanding of the microscopic causes of damping is not yet achieved. Here we present a system- atic study of the damping of doubly-clamped resonators fabricated out of prestressed silicon nitride leading to highmechanicalqualityfactors[10]. Reproducingtheob- served mode frequencies applying continuum mechanics, we are able to quantitatively model their quality factors by assuming that damping is caused by the local strain induced by the resonator’s displacement. Considering various microscopic mechanisms, we conclude that the observed damping is most likely dominated by dissipa- tion via localized defects uniformly distributed through- out the resonator. We study the oscillatory response of nanomechanical beams fabricated from high stress silicon nitride (SiN). A released doubly-clamped beam of such a material is therefore under high tensile stress, which leads to high mechanical stability[12] and high mechanical quality fac- tors[10]. Such resonators are therefore widely used in recent experiments[6, 9]. Our sample material consists of a silicon substrate covered with 400nm thick silicon dioxide serving as sacrificial layer and a h= 100nm thick SiN device layer. Using standard electron beam lithography and a sequence of reactive ion etch and wet- etch steps, we fabricate a series of resonators having lengths of 35 /nµm,n={1,...,7}and a cross section of 100·200nm as displayed in Fig.1a and b. Since the respective resonance frequency is dominated by the large tensile stress[10, 13], this configuration leads to reso- nances of the fundamental modes that are approximately equally spaced in frequency. Suitably biased gold elec- trodes processed beneath the released SiN strings actu- ate the resonators via dielectric gradient forces to per- form out-of-plane oscillations, as explained in greater de- tail elsewhere [12]. The length and location of the gold xzxz y(a) (b) (c) 5□µm FIG.1.Setup and Geometrya SEM-pictureofoursample; the lengths of the investigated nanomechanical silicon nit ride strings are 35 /nµm,n={1,...,7}; their widths and heights are 200nm and 100nm, respectively. bZoom-in of a: the res- onator (highlighted in green) is dielectrically actuated b y the nearby gold electrodes (yellow); its displacement is recor ded with an interferometric setup. cMode profile and absolute value of the resulting strain distribution (color-coded) o f the longest beam’s 4th harmonic as calculated by elastic theory . electrodesis properlychosen to be able to also excite sev- eral higher order modes of the beams. The experiment is carried out at room temperature in a vacuum below 10−3mbar to avoid gas friction. The displacement is measured using an interferometric setup that records the oscillatory component of the re- flected light intensity with a photodetector and network analyser[12, 14]. The measured mechanical response around each resonance can be fitted using a Lorentzian lineshape as exemplarily seen in the inset of Fig.2. The thereby obtained values for the resonance frequency f and quality factor Qfor all studied resonators and ob- served modes are displayed in Fig.2 (filled circles). In order to reproduce the measured frequency spectrum, we apply standard beam theory (see e.g.[15]). With- outdamping, the differentialequationdescribingthespa- tial dependence of the displacement for a specific mode mof beam n un,m[x] at frequency fn,mwrites (with ρ= 2800kg /m3being the material density[16]; E1,σ02 50 40 30 20 10 0Signal Amplitude[□µV] 24.136 24.132 24.128 Frequency[MHz]160 140 120 100 80 60 40 20 0Quality□Factor□[103] 80 70 60 50 40 30 20 10 0 Frequency□[MHz]Beam□lenght□[µm] 35 35/2 35/5 35/3 35/6 35/4 35/7 FIG. 2. Resonance Frequency and Mechanical Qual- ity Factor The harmonics of the nanomechanical resonator show a Lorentzian response (exemplary in the inset). Fittin g yields the respective frequency and mechanical quality fac tor. The main figure displays these values for several harmonics (same color) ofdifferentbeams as indicted bythecolor. Tore - produce the resonance frequencies, we fit a continuum model to the measured frequencies. We thereby retrieve the elas- tic constants of our (processed) material, namely the built -in stressσ0= 830MPa and Young’s modulus E= 160GPa. From the displacement-induced, mode-dependent strain dis - tribution, we calculate (except for an overall scaling) the me- chanical quality factors. Calculated frequencies and qual ity factors are shown as hollow squares, the responses of the dif - ferent harmonics of the same string are connected. are the (unknown) real Young’s modulus and built in stress, respectively) 1 12E1h2∂4 ∂x4un,m[x]−σ0∂2 ∂x2un,m[x]−ρ(2πfn,m)2un,m[x] = 0 (1) Solutions of this equation have to satisfy the bound- ary conditions of a doubly-clamped beam (displacement and its slope vanish at the supports ( un,m[±l/(2n)] = (∂/∂x)un,m[±l/(2n)] = 0,l/n: beam length). These con- ditions lead to a transcendental equation that is numeri- cally solved to obtain the frequencies fn,mof the different modes. The results arefitted to excellently reproducethe mea- sured frequencies, as seen in Fig.2 (hollow squares). One thereby obtains as fit parameters the elastic con- stants of the micro-processed material E1= 160GPa, σ0= 830MPa, in good agreement with previously pub- lished measurements [13]. For each harmonic, we now are able to calculate the strain distribution within the resonator induced by the displacement u[x] and exemplarily shown in Fig.1c. The measured dissipation is closely connected to this induced strainǫ[x,z,t] =ǫ[x,z]exp[i2πft]. As in the case of a Zener model[17] we now assume that the displacement- induced strain and the accompanying oscillating stressσ[x,z,t] =σ[x,z]exp[i2πft] are not perfectly in phase; this can be expressed by a Young’s modulus E=E1+ iE2having an imaginary part. The relation reads again σ[x,z] = (E1+iE2)ǫ[x,z]. During one cycle of oscillation T= 1/f, a small volume δVof length sand cross section Athereby dissipates the energy ∆ UδV=AsπE2ǫ2. The total loss is obtained by integrating over the volume of the resonator. ∆Un,m=/integraldisplay VdV∆UδV=πE2/integraldisplay VdVǫn,m[x,z]2(2) The strain variation and its accompanying energy loss can be separated into contributions arising from overall elongation of the beam and its local bending. It turns out that here the former is negligible, de- spite the fact that the elastic energy is dominated by the elongation of the string, as discussed below. To very high accuracy we obtain for the dissipated energy ∆Un,m≈π/12E2wh3/integraltext ldx(∂2/(∂x)2un,m)2. A more rig- orous derivation can be found in the Supplementary In- formation. The total energy depends on the spatial mode (through ǫn,m, see exemplary Fig.1c) and therefore strongly dif- fers for the various resonances. To obtain the qual- ity factor, one has to calculate the stored energy e. g. by integrating the kinetic energy Un,m=/integraltext ldxAρ(2πfn,m)2un,m[x]2. The overall mechanical qual- ityfactoris Q= 2πUn,m/∆Un,m. Amoredetailed deriva- tion can be found in the Supplementary Information. Assuming that the unknown value of the imaginary partE2of the elastic modulus is independent of res- onatorlength andharmonicmode, weareleft with onefit parameter E2to reproduce all measured quality factors and find excellent agreement(Fig.2, hollowsquares). We therefore successfully model the damping of our nanores- onators by postulating a frequency independent mecha- nism caused by local strain variation. Allowing E2to depend on frequency, the accordance gets even better, as discussed in detail in the Supplement. We now discuss the possible implications of our find- ings, considering at first the cause ofthe high quality fac- tors in overall pre-stressed resonators and then the com- patibility of our model with different microscopic damp- ing mechanisms. In a relaxed beam, the elastic energy is stored in the flexural deformation and becomes for a small test volume UδV= 1/2AsE1ǫ2. In the framework of a Zener model, as employed here, this result is pro- portional to the energy loss (see eq.3) and thus yields a frequency-independent quality factor Q=E1/E2for the unstressed beam. In accordance with this finding, Ref.[10] reports a much weaker dependence of quality factor on resonance frequency, in strong contrast with the behavior of their stressed beams. Similar as in the damping model, the total stored elas- tic energy in a beam can be very accurately separated3 into a part connected to the bending and a part coming from the overall elongation. The latter is proportional to the pre-stress σ0and vanishes for relaxed beams, refer to the Supplement for details. Assuming a constant E= E1+iE2, Fig.3 displays the calculation of the elastic en- ergy and the quality factor for the fundamental mode of ourlongest( l= 35µm)beamasafunctionofoverallbuilt in stress σ0. The total elastic energy is increasingly dom- inated by the displacement-induced elongation Uelong= 1/2σ0wh/integraltext ldx(∂/(∂x)u[x])2. In contrast the bending en- ergyUbend= 1/24E1wh3/integraltext ldx(∂2/(∂x)2u[x])2, which in our model is proportional to the energy loss, is found to increase much slower with σ0. Thus one expects Qto in- creasewith σ0, afindingalreadydiscussedbySchmidand Hierold for micromechanical beams[18]. However, their model assumes the simplified mode profile of a stretched string and can not explain the larger quality factors of higher harmonics when compared to a fundamental res- onance of same frequency. Including beam stiffness, our model can fully explain the dependence of frequency and damping on length and mode index, as reflected in Fig.2. It also explains the initially surprising finding[19] that amorphous silicon nitride resonators exhibit high quality factors when stretched whereas having Q-factors in the relaxed state that reflect the typical magnitude of inter- nal friction found to be rather universal in glassy materi- als[20]. More generally we conclude that the increase in mechanical quality factors with increasing tensile stress is not bound to any specific material. Since the resonance frequency is typically easier to access in an experiment, we plot the quality factor vs. corresponding resonance frequency in Fig.3b; with both numbers being a function of stress. The resulting rela- tion of quality factor on resonance frequency is (except for verylow stress) almost linear; experimental results by another group can be seen to agree well with this find- ing[21]. In addition, we show in the Supplement that although the energy loss per oscillation increases with applied stress, the linewidth of the mechanical resonance decreases. We willnowconsidertheintrinsicphysicalmechanisms that could possibly contribute to the observed damp- ing. As explained in greater detail in the Supplement, we can safely neglect clamping losses[1, 2], thermoelastic damping[4, 6] and Akhiezer damping[5, 6] since they all predict damping constants significantly smaller than the ones observed. Instead, we would like to discuss the influence of lo- calized defect states. Similar to the Akhiezer effect, it is assumed that the energy spectra of defects are modu- lated by strain[27], which thereby drives the occupancy out of thermal equilibrium. In Ref.[27], this effect is cal- culated for a broad spectrum of two-level systems. The energy difference of two uncoupled levels as well as their separating tunnel barrier height are assumed to be uni- formly distributed, leading to a broad yet not flat dis-10-2410-2310-2210-2110-20 Energy□[J] 105106107108109 Stress□[Pa]Quality□factor□103 Resonance□frequency□[MHz]2.0 1.5 1.0 0.5 0.0Stress□□[GPa]elongation bending total□energyExperiment250 200 150 100 50 0 12 108 6 4 2Experiment(a) (b) FIG. 3.Elastic Energy and Mechanical Quality Factor of the Beam in Dependence of Stress a The elastic en- ergies of the fundamental mode of the beam with l= 35µm are displayed vs. applied overall stress separated into the con- tributions resulting from the overall elongation and the lo cal bending. The dashed line marks the strain of the experimen- tally studied resonator σ0≈830MPa, there the elongation term dominates noticeably. bQuality factor and frequency are calculated for varying stress σ0. In order to compare the calculation with other published results quality factor an d stress are displayed vs. resulting frequency. tribution of relaxation rates. In the high temperature limit the thus derived energy loss per oscillation becomes frequency-independent as assumed in our model. In ad- dition, published quality factors of relaxed silicon nitride nano resonators[19] cooled down to liquid helium tem- perature display quality factors that are well within the typical range of amorphous bulk materials[20]. More- over, on a different sample chip we measured a set of resonators showing quality factors that are uniformly de- creased by a factor of approximately 1.4 compared to the datapresentedinFig.2. Theirresponsecanstillbequan- titativelymodeledusingnowanincreasedimaginarypart of Young’s modulus E2. We attribute this reduction to a non-optimized RIE-etch step, leading to an increased density of defect states. The corresponding data is pre- sented in the Supplementary Information. These three arguments clearly favor the concept of damping via de- fect states as the dominant mechanism. We wish to point out that such a model calculation based on two-level systems cannot be rigourouslyapplied at elevated temperatures, as the concept of two-level systems should be replaced by local excitable systems. However, it seems plausible that such a system still ex- hibits abroadrangeofrelaxationrates, crucialto explain frequency-independent damping. In contrast, a mecha- nism with discrete relaxation rates will exhibit damp- ing maxima whenever the oscillation frequency matches the relaxation rate[4, 6, 17]. We further notice that in ourexperimentsbeamswithlargerwidthsexhibitslightly higher quality factors. This indicates an increased defect density near the surface being either intrinsic or caused by the micro-fabrication[28] (RIE etch). For a fixed cross-section however, the applicability of our model is not affected. In conclusion, we systematically studied the transverse4 mode frequencies and quality factors of prestressed SiN nanoscale beams. Implementing continuum theory, we reproduce the measured frequencies varying with beam length and mode index over an order of magnitude. As- suming that damping is caused by local strain variations induced by the oscillation, independent of frequency, en- ables us to calculate the observed quality factors with a single interaction strength as free parameter. We thus identify the unusually high quality factors of pre-stressed beams as being primarily caused by the increased elastic energy rather than a decrease in damping rate. Several possible damping mechanisms are discussed; because of theobservednearlyfrequency-independenceofthedamp- ing parameter E2, we attribute the mechanism to inter- action of the strain with local defects of not yet identified origin. One therefore expects that defect-free resonators exhibit even larger quality factors, as recently demon- strated for ultra-clean carbon nanotubes[11]. Financial support by the Deutsche Forschungsgemein- schaft via project Ko 416/18,the German Excellence Ini- tiative viathe NanosystemsInitiative Munich (NIM) and LMUexcellent as well as the Future and Emerging Tech- nologies programme of the European Commission, under the FET-Openproject QNEMS (233992)is gratefullyac- knowledged. We would like to thank FlorianMarquardtand Ignacio Wilson-Rae for stimulating discussions. ∗quirin.unterreithmeier@physik.uni-muenchen.de [1] K. Jensen, K. Kim, and A. Zettl, Nat Nano 3, 533 (2008). [2] B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, Nano Letters 8, 3735–3738 (2008). [3] M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab, and M. L. Roukes, Nature 459, 960–964 (2009). [4] J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. 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[25] A. A. Kiselev and G. J. Iafrate, Phys. Rev. B 77, 205436 (2008). [26] A. Akhieser, Journal of Physics-ussr 1, 277–287 (1939). [27] J. JACKLE, Zeitschrift Fur Physik 257, 212–223 (1972). [28] J. L. Yang, T. Ono, and M. Esashi, Journal of Microelec- tromechanical Systems 11, 775–783 (2002).5 SUPPLEMENTARY INFORMATION DAMPING MODEL In a Zener model, an oscillating strain ǫ(t) =ℜ[ǫ[ω]exp[iωt]] and its accompanying stress σ[t] =ℜ[σ[ω]exp[iωt]] are out-of phase, described by a frequency-dependent, complex ela stic modulus σ(ω) =E[ω]ǫ[ω] = (E1[ω]+iE2[ω])ǫ[ω]. This leads to an energy loss per oscillation in a test volume δV=δA·δsof cross-section δAand length δs. ∆UδV=/integraldisplay TdtEAǫ[t]/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright force·∂ ∂t(sǫ[t]) /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright velocity=πδAδsE 2ǫ2(3) We now employ this model for our case, namely a pre-stressed, rec tangular beam of length l, widthwand height h, corresponding here to the x,y,z-direction, respectively. The orig in of the coordinate system is centered in the beam. The resonator performs oscillations in the z-direction and, a s we consider a continuum elastic model, there will be no dependence on the y-direction. For a beam of high aspect r atiol≫hand small oscillation amplitude, the displacement of the m-th mode can be approximately written um[x,y,z] =um[x]. During oscillation, a small test volume within the beam undergoes oscillating strain ǫm[x,z,t]. This strain arises because of the bending of the beam as well as its elo ngation as it is displaced. The stress caused by the overall elongation is quadratic in displacement, therefore it occ urs at twice the oscillating frequency. ǫm[x,z,t] =1 2/parenleftbigg∂ ∂xum[x]ℜ[exp[iωt]]/parenrightbigg2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright elongation+z∂2 ∂x2um[x]ℜ[exp[iωt]] /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright bending =1 2/parenleftbigg∂ ∂xum[x]/parenrightbigg21 2(1+ℜ[exp[2iωt]])+z∂2 ∂x2um[x]ℜ[exp[iωt]] (4) Inserting this into eq.3 and integrating over the cross-section w·h, the accompanying energy losses can be seen to separate into elongation and displacement caused terms. ∆Uw·h=πsE2[2ω]wh 8/parenleftbigg∂ ∂xum[x]/parenrightbigg4 +πsE2[ω]wh3 12/parenleftbigg∂2 ∂x2um[x]/parenrightbigg2 (5) Integrating over the length yields the total energy loss of a partic ular mode ∆ U=/integraltextl/2 −l/2dx∆Uw·h. In the case that E2is only weakly frequency-dependent, it turns out that for our geo metries the elongation term is several orders of magnitude (105−107) smaller than the term arising from the bending. The energy loss the refore may be simplified and writes ∆U≈∆Ubending=πE2wh3 12/integraldisplayl/2 −l/2dx/parenleftbigg∂2 ∂x2um[x]/parenrightbigg2 (6) ELASTIC ENERGY OF A PRE-STRESSED BEAM A volume δVsubject to a longitudinal pre-stress σ0stores the energy UδVwhen strained; E1is assumed to be frequency independent in the experimental range (5-100MHz) UδV=sA/parenleftbigg σ0ǫ+1 2E1ǫ2/parenrightbigg (7) To apply this formula to the case of an oscillating pre-stressed beam , we insert eq.4 |t=0(maximum displacement) and integrate over the cross-section to obtain Uw·h=1 2E1/parenleftBigg 1 4wh/parenleftbigg∂ ∂xum[x]/parenrightbigg4 +1 12wh3/parenleftbigg∂2 ∂x2um[x]/parenrightbigg2/parenrightBigg +1 2swhσ0/parenleftbigg∂ ∂xum[x]/parenrightbigg2 (8)6 Analog to eq.5 we can omit the first term in the brackets; integratin g over the length yields U≈/integraldisplayl/2 −l/2dx/parenleftBig1 2whσ0/parenleftbigg∂ ∂xum[x]/parenrightbigg2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright elongation+1 24E1wh3/parenleftbigg∂2 ∂x2um[x]/parenrightbigg2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright bending/parenrightBig (9) Wecanthereforedividethetotalenergyintopartsarisingfromth eelongationandthebendingofthebeam. Depending on the magnitude of the pre-stress, either of the two energies ca n dominate as seen in Fig.3a of the main text. We have checked that the kinetic energy Ukin= 1/2ρ(ωm)2/integraltextl/2 l/2dx(um[x])2; (ωm/(2π),ρ: resonance frequency, material density, respectively) equals the total elastic energy, as expect ed. FREQUENCY-DEPENDENT LOSS MODULUS There is no obvious reason that the imaginary part of Young’s modulu sE2should be completely frequency- independent. We therefore assume that E2obeys a (weak) power-law and chose the ansatz: E2[f] =E2(f/f0)a(10) Fitting our data with the thus extended theory, we achieve a very p recise agreement of measured and calculated quality factors, as seen in Fig.S1. The resulting exponent is a= 0.075;E2varies therefore by 20% when fchanges by one order of magnitude. 160 140 120 100 80 60 40 20 0Quality Factor [103] 80706050403020100 Frequency [MHz] Beam lenght [µm] 35 35/5 35/2 35/5 35/3 35/6 35/4 FIG. 4. Resonance frequencies and quality factors of the resonator saMeasured quality factor and resonance frequency of several harmonics of beams with different lengths (color-co ded) are displayed as filled circles (same data as in Fig.2 of t he main text). The resonance frequencies are reproduced by a contin uum model; we calculate the quality factors using a model bas ed on the strain caused by the displacement. In contrast to Fig. 2 of the main text and Fig.S2 we here allow E2to be (weakly) frequency-dependent. LINEWIDTH OF THE MECHANICAL RESONANCE The elastic energy of a harmonic oscillator is given by U= 1/2meffω2 0x2 0withmeff,ω0,x0being effective mass, resonance frequency and displacement, respectively. If we assu me the effective mass to be energy-independent, it appliesω0∝√ U. Recalling the definition of the quality factor Q= 2πU/∆U∝U, one obtains for the for the Full Width at Half Max (FWHM) of the resonance ∆ω=ω0 Q∝√ U U/∆U=∆U√ U(11) As in the main text, the energy depends on the applied overall tensile stress. Figure6 shows a numerical calculation of the resulting linewidth vs. applied stress; one can see that increa se in energy loss per oscillation is dominated by7 the increase in energy, resulting in a decreased linewidth. The exact effective mass is included in this calculation; as it changes by less than 20%, the above assumption is justified. 160 140 120 100 80 60FWHM /(2 )Δω π 103104105106107108109 Prestress□[Pa] FIG. 5. Linewidth of the mechanical resonance The calculated linewidths (FWHM) for the fundamental mode o f the beam withl= 35µm are displayed vs. applied overall stress. MICROSCOPIC DAMPING MECHANISMS We start with clamping losses as discussed, e.g., in ref.[1, 2], i. e. the r adiation of acoustic waves into the bulk caused by inertial forces exerted by the oscillating beam. With a sou nd velocity in silicon of vSi≈8km/s, the wavelength of the acoustic waves radiated at a frequency of 10MH z from the clamps into the bulk will be greater than 500 µm, and thus substantially larger than the length of our resonators . Considering each clamping point as a source of an identical wave propagating into the substrate, one would expect that mostly constructive/destructive interference would occur for in-/out-of-phaseshear forces ex erted by the clamping points, respectively. With clamping losses being important, one would therefore expect that spatially a symmetric modes with no moving center of mass exhibit significant higher quality factors than symmetric ones[3]. Ano ther way to illuminate this difference is that symmetric modes give rise to a net force on the substrate, wherea s asymmetric modes yield a torque. Since the measurement (Fig. 2) does not display such an alternating behavior of the quality factors with mode index m(best seen for the longest beam), clamping losses are likely to be of minor imp ortance. The next damping mechanism we consider are phonon-assisted losse s within the beam. At elevated temperatures, at least two effects arise, the first being thermoelastic damping: be cause of the oscillatory bending, the beam is com- pressed and stretched at opposite sides. Since such volume chang es are accompanied by work, the local temperature in the beam will deviate from the mean. For large aspect ratios as in ou r case, the most prominent gradient is in the z direction. The resulting thermal flow leads to mechanical dissipatio n. We extend existing model calculations[4] to include the tensile stress of our beams. Using relevant macroscopic material parameters such as thermal conductivity, expansion coefficient and heat capacity we derive Q-values that are typically three to four orders of magnitudes larger than found in the experiment. Therefore, heat flow can be safely n eglected as the dominant damping mechanism. In addition, the calculated thermal relaxation rate corresponds t o approximately 2GHz, so the experiment is in the so-called adiabatic regime. Consequently, one would expect the ene rgy loss to be proportional to the oscillation frequency, in contrast to the assumption of a frequency indepen dentE2and our experimental findings. Another local phonon-based damping effect is the Akhiezer-effect [5]; it is a consequence of the fact that phonon frequencies are modulated by strain, parameterized by the Gr¨ un eisen tensor. If different phonon modes (characterized bytheirwavevectorandphononbranch)areaffecteddifferently, the occupancyofeachmodecorrespondstoadifferent temperature. This imbalance relaxes towards a local equilibrium temp erature, giving rise to mechanical damping. In a model calculation applying this concept to the oscillatory motion o f nanobeams[6], the authors find in the case of large aspect ratios length/height that the thermal heat flow re sponsible for thermoelastic damping dominates the energy loss by the Akhiezer effect. We thus can safely assume this m echanism to be also negligible in our experiment.8 REDUCED QUALITY FACTOR We fabricated a set of resonators, shown in Fig.S1a, that showed lower quality factors than the ones presented in the main text (Fig.2); we attribute this reduction to a non-optimize d RIE-etch step. As in the main article, it is possible to reproduce the quality factors using a single fit paramete r, namely the imaginary part of Young’s modulus E2. The ratio of the two sets of quality factors is displayed in Fig.S1 b an d can be seen to be around 1.4 with no obvious dependence on resonance frequency, mode index or len gth. A non-optimized etch step causes additional surface roughnessand the addition of impurities, thereby increas ingthe density of defect states. As there is no obvious reason why another damping mechanism should be thereby influence d, we interpret this as another strong indication that the dominant microscopic damping mechanism is caused by localize d defect states. 100 80 60 40 20 0Quality□Factor□[103] 70 60 50 40 30 20 10 0 Frequency□[MHz]Beam□length□[µm] 35 35/5 35/2 35/6 35/3 35/7 35/4 Quality□Factor□Ratio Frequency□[MHz]1.5 1.0 0.5 0.0 70 60 50 40 30 20 10 0Beam□length□[µm] 35 35/5 35/2 35/6 35/3 35/7 35/4 (b) (a) FIG. 6. Comparison of the resonance frequencies and quality factor s of the sets of resonators aMeasured quality factor and resonance frequency of several harmonics of beams with diffe rent lengths (color-coded) are displayed as filled circles. The resonance frequencies are reproduced by a continuum model; a model based on the strain caused by the displacement allows us to calculate the quality factors, shown as hollow squares . The uniform reduction of the Q-factors is attributed to an n on- optimized RIE-etch. bThe ratio of the quality factors of the two sets resonators (F ig.2 main article and Fig.S2a) are displayed vs. frequency, being approximately constant. ∗quirin.unterreithmeier@physik.uni-muenchen.de [1] Z. Hao, A. Erbil, and F. Ayazi, Sensors and Actuators A: Ph ysical109, 156–164 (2003). [2] I. Wilson-Rae, Phys. Rev. B 77, 245418 (2008). [3] I. Wilson-Rae, private communication [4] R. Lifshitz and M. L. Roukes, Phys. Rev. B 61, 5600–5609 (2000). [5] A. Akhieser, Journal of Physics-ussr 1, 277–287 (1939). [6] A. A. Kiselev and G. J. Iafrate, Phys. Rev. B 77, 205436 (2008).

2010-03-09

We study the transverse oscillatory modes of nanomechanical silicon nitride strings under high tensile stress as a function of geometry and mode index m <= 9. Reproducing all observed resonance frequencies with classical elastic theory we extract the relevant elastic constants. Based on the oscillatory local strain we successfully predict the observed mode-dependent damping with a single frequency independent fit parameter. Our model clarifies the role of tensile stress on damping and hints at the underlying microscopic mechanisms.

Damping of Nanomechanical Resonators

1003.1868v1

arXiv:1003.4681v1 [gr-qc] 24 Mar 2010Dynamical shift condition for unequal mass black hole binar ies Doreen M¨ uller, Jason Grigsby, Bernd Br¨ ugmann Theoretical Physics Institute, University of Jena, 07743 J ena, Germany (Dated: August 19, 2021) Certain numerical frameworks used for the evolution of bina ry black holes make use of a gamma driver, whichincludesadampingfactor. Suchsimulations t ypicallyuseaconstantvaluefor damping. However, it has been found that very specific values of the dam ping factor are needed for the calculation of unequal mass binaries. We examine carefully the role this damping plays, and provide two explicit, non-constant forms for the damping to be used w ith mass-ratios further from one. Our analysis of the resultant waveforms compares well against t he constant damping case. PACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx I. INTRODUCTION The ability to simulate the final inspiral, merger, and ring-down of black hole binaries with numerical relativ- ity [1–3] plays a key role in understanding a source of gravitational waves that may one day be observed with gravitational wave detectors. While initial simulations focused on binaries of equal-mass, zero spin, and quasi- circular inspirals, there currently is a large effort to ex- plore the parameter space of binaries, e.g. [4–7]. A key part of studying the parameter space is to simulate bina- ries with intermediate mass-ratios. Todate, themassratiofurthestfromequalmassesthat has been numerically simulated is 10:1 [8, 9]. These sim- ulations use the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation [10–12] with 1+log slicing, and the ˜Γ driver condition for the shift [13, 14]. In [8], it was noted that the stability of the simulation is sensitive to the damping factor, η, used in the ˜Γ driver condition, ∂2 0βi=3 4∂0˜Γi−η∂0βi. (1) Here,βiistheshiftvectordescribinghowthecoordinates move inside the spatial slices, ∂0≡∂t−βi∂i, and˜Γiis the contraction of the Christoffel symbol, ˜Γi jk, with the conformal metric, ˜ γjk. The standard choice for ηis to set it to a constant value, which works well even for the most demanding simulations as long as the mass ratio is sufficiently close to unity. In binary simulations, a typical choice is a con- stant value of about 2 /M, withMthe total mass of the system. This choice, however, leads to instabilities for the mass ratio 10:1 simulation [8], although stability was obtained for η= 1.375/M. The value of ηis chosen to damp an outgoing change in the shift while still yield- ing stable evolutions. As we will show, if ηis too small, there are unwanted oscillations, and values that are too large lead to instabilities. By itself, this observation is not new, see e.g. [15–18]. The key issue for unequal masses is that, as evident from (1), the damping factor η has units of inverse mass, 1 /M. Therefore, the interval of suitable values for ηdepends on the mass of the blackholes. For unequal masses, a constant ηcannot equally well accommodateboth black holes. Aconstantdamping parameter implies that the effective damping near each black hole is asymmetric since the damping parameter has dimensions 1 /M. For large mass ratios, this asym- metry in the grid can be large enough to lead to a failure of the simulations because the damping may become too large or too small for one of the black holes. To cure this problem, we need a position-dependent damping param- eter that adapts to the local mass. In particular, we want it to vary such that, in the vicinity of the ithpuncture with massMi, its value approaches 1 /Mi. A position-dependent ηwas already considered when the˜Γdriverconditionwasintroduced[8,19–22], but such constructions were not pursued further because for mod- erate mass ratios a constant ηworks well. Recently, we revived the idea of a non-constant ηfor moving punc- ture evolutions in order to remove the limitations of a constantηfor large mass ratios. In [23], we constructed a position-dependent ηusing the the conformal factor, ψ, which carries information both about the location of the black holes, and about the local puncture mass. The form ofηwas chosen to have proper fall-off rates both at the punctures and at large distance from the binary. In [9], this approach was used successfully for mass ratio 10:1. (We note in passing that damping is useful in other gauges as well, e.g. in [24] the modified harmonic gauge condition includes position-dependent damping by use of the lapse function.) In the present work, we examine one potential short- coming of the choice of [23], which leads us to suggest an alternative type of position-dependent η. Using [23], we find large fluctuations in the values of that η, and this might lead to instabilities in the simulation of larger mass-ratio binary black holes. To address this, we have tested two new explicit formulas for the damping factor designedto havepredictablebehaviorthroughoutthe do- main of computation. We find the new formulas to pro- duce only small changes in the waveforms that diminish withresolution,andthereisagreatdealoffreedominthe implementation. Independently of our discussion here, in [16] the stability issues for large ηare explained, and a non-constant ηis suggested (although not yet explored2 in actual simulations), that, in its explicit coordinate de- pendence, is similar to one of our suggestions. The paperisorganizedasfollows. We firstdescribethe reasons for the damping factor and some of the reasons forlimitingitsvalueinSecII. InSec.III, wediscusssome previous forms of ηthat have been used. We also present two new definitions and why we investigated them. In Sec. IV, we find that these new definitions agree well with the use of constant ηin the extracted gravitational waves for mass ratios up to 4:1. Finally, in Sec. V, we discuss further implications of this work. II. MOTIVATION In order to define a position-dependent form for η, it is important to determine what this damping parameter accomplishes in numerical simulations. For this reason, we examine the effects of running different simulations while varying ηbetween runs. First we use evolutions of single non-spinning black-holes to identify the key phys- ical changes. Then we examine equal-mass binaries to determine specific values desired in ηat both large and small radial coordinates. A. Numerics For all the work in this paper, we have used the BAM computer codedescribed in [17, 25, 26]. It uses the BSSN formalism with 1+log slicing and ˜Γ driver condition in the moving puncture framework [2, 27]. Puncture ini- tial data [28] with Bowen-York extrinsic curvature [29] have been used throughout this work, solving the Hamil- tonian constraint with the spectral solver described in [30]. For binaries, parameters were chosen using [31] to obtain quasi-circularorbits, while the parametersfor sin- gle black holes were chosen directly. We extract waves via the Newman-Penrose scalar Ψ 4. The wave extraction procedure is described in detail in [17]. We perform a mode decomposition using spin-weighted spherical har- monics with spin weight −2,Y−2 lm, as basis functions and calculate the scalar product Ψlm 4=/parenleftbig Y−2 lm,Ψ4/parenrightbig =/integraldisplay2π 0/integraldisplayπ 0sinθdθdϕY−2 lmΨ4.(2) We furthersplit Ψlm 4intomode amplitude Almandphase φlmin order to cleanly separate effects in these compo- nents,rex·Ψlm 4=Almeiφlm. In this paper, we focus on one of the most dominant modes, the l=m= 2 mode, and report results for this mode unless stated otherwise. The extraction radius used here is rex= 90M.B. Single, non-spinning puncture with constant damping Thedamping factor, η, in Eq.(1), isincluded to reduce dynamics in the gauge during the evolution. To examine the problem brought up in the introduction, we compare results of a single, non-spinning puncture with mass M. We use a Courant factor of 0 .5 and 9 refinement levels centered around the puncture. The resolution on the finest grid is 0 .025M, and the outer boundary is situated at 256M. Varying the damping constant between 0 .0/M and4.5/M,twomainobservationscanbemade. First, as designed, a non-zero ηattenuates emerging gauge waves efficiently. Second, an instability develops for values of η that are too large. Figs. 1, and 2 illustrate the first observation. Both figures show the x-component of the shift along the x- coordinate using η∈ {0.0/M,1.5/M,3.5/M}. Apart 0 10 20 30 40 x/M00.050.10.150.2βxη = 0.0/M η = 1.5/M η = 3.5/M FIG. 1: The x-component of the shift, βx, for a single non- spinning puncture of mass Mat timet= 15.2M. The three lines were taken for different values of the damping factor η. The solid line (black) is for η= 0.0/M. The dashed line (red) is forη= 1.5/Mand the dotted-dashed line (green) is for η= 3.5/M. This shows the beginning of a pulse in βxfor smaller values of η. from the usual shift profile, Fig. 1 shows the beginnings of a pulse in the η= 0.0/Mcase (solid line) at x≈10M after 15.2Mof evolution. Examining Fig. 2, where we zoom in at a later time, t= 30.4M, one can see that the pulse has started to travel further out (solid line). Looking carefully, one can also see a much smaller pulse in theη= 1.5/Mline (dashed). Lastly, by examination, one can find almost no traveling pulse in the η= 3.5/M curve (dotted-dashed line). The observed pulse in the shift travels to regions far away from the black hole and effects the gauge of distant observers. This might have3 0 10 20 30 40 50 x/M-0.0100.010.020.030.04βxη = 0.0/M η = 1.5/M η = 3.5/M FIG. 2: The x-component of the shift, βx, for a single non- spinning puncture of mass Mat timet= 30.4M. The three lines were taken for different values of the damping factor η with the same line type and color scheme as in Fig. 1. Here it is clear a pulse radiates outward in the shift with smaller values of η. undesirable implications for the value of such numerical data when trying to understand astrophysical sources. For values of ηlarger than 3 .5/M, an instability arises in the shift at larger radius. Fig. 3 shows the x- component of the shift vector using damping constants η= 3.5/M(solid line), η= 4.0/M(dashed line) and η= 4.5/M(dotted-dashed line). The plots show an in- stability in simulations with η >3.5/Mdeveloping in βi, which eventually leads to a failure of the simulations. Contrary to this, the simulation using η= 3.5/Mdoes not show this shift related instability. In test runs we found that by decreasing the Courant factor used, we could increase the value of the damping factor and still get stable evolutions. This agrees with [16] where it was shown that the gamma driver possesses the stiff prop- erty, which limits the size of the time-step in numerical integration based on the value of the damping. Figures 1, 2, and 3 make clear how the choice of the damping factor affects the behavior of the simulations. The value we choose for ηshould be non-zero and not largerthan 3 .5/Mto allow for effective damping and sta- ble simulations. The exact cutoff value between stable and unstable simulations is not relevant here since the position dependent form we develop in Sec. III gives us the flexibility we need to obtain stable simulations.0 50 100 15000.511.520 50 100 15000.511.52 PSfrag replacementsx/M x/Mη= 3.5/M η= 3.5/Mη= 4.0/M η= 4.0/Mη= 4.5/Mβxβxt= 16M t= 30.4M FIG. 3: The x-componentof theshift vectorin the x-direction for a single non-spinning puncture of mass Mat times t= 16.0Mandt= 30.4M. The three different lines mark three values of the damping constant η. The solid line (black) is forη= 3.5/M, the dashed line (red) for η= 4.0/Mand the dotted-dashed line (green) for η= 4.5/M. Att= 16M, the simulation using η= 4.5/Mdevelops an instability in the shift vector and fails soon afterward, the same happens for η= 4.0/Matt= 30.4M. In the simulation using η= 3.5, no such instability develops (not shown). C. Equal mass binary with constant damping To examine the effect of ηon the extraction of gravi- tational waves, we compare the results from simulations of an equal mass binary with total mass Min quasi- circular orbits with initial separation D= 10M, using η∈ {0.0/M,0.5/M,2.0/M}. Again, the Courant factor is chosen to be 0 .5 and we use, in the terminology of [17], the grid configuration χ[6×56 : 5×112 : 6] with a finest resolution of 0 .013M. Here, the extraction radius rexis chosen to be 90 M. For vanishing η, we find a lot of noise in the the real part of the 22-mode of rexΨ4, shown in the solid curve of Fig. 4. A small, but non-vanishing ηsuffices to suppress this noise, as seen in the dashed curve of this figure. The dotted-dashedcurvein this plot is the result forusing the valueη= 2.0/M. We see a difference in time between peak amplitudes of the three curves due to the change of coordinates that the alternation of ηintroduces. We did, however, find that by decreasing the Courant factor thosedifferencesbetween peakamplitudes summarilyde- creased. To understand the noise in the waves for η= 0.0/M, we look at the shift vector at different times. The first panel of Fig. 5 shows the x-component of the shift over4 0 200 400 600 800-0.008-0.006-0.004-0.00200.0020.004 PSfrag replacementsη= 0.0/M η= 0.5/M η= 2.0/MRe{Ψ224}rex·M t/M0 200 400 600 800 1000 1200-0.06-0.04-0.0200.020.040.06 PSfrag replacements ηs= 0.0M ηs= 0.5M ηs= 2.0M Re{Ψ224}rex·M t/M FIG. 4: Real part of the 22-mode of Ψ 4over time for equal mass simulations using different values for η. The inset shows the full waveform until ringdown. The solid curves (black) are forη= 0, the dashed curve (red) mark η= 0.5/Mand the dotted-dashed curves (green) are for η= 2.0/M. Without damping in the shift, the extracted waves are noisy at times when the amplitude is still small (black, solid curve). x, againforη∈ {0.0/M,0.5/M,2.0/M},shortlyafter the beginning of the simulation. The fourth panel shows the same at a time shortly before the merger, and the two panels in the middle represent intermediate times. We see clear gauge pulses in the earliest time panel for all three curves. We also observe the amplitude of this pulse decreasing with increasing η. As time goes on, the gauge pulse travelsoutwardsas in the case for a single puncture in section IIB. For vanishing η(solid line), the shift becomes more and more distorted, and the distortions do not leave the grid. For non-zero η, the amplitude of the gauge pulse decreases when traveling outwards, and the shape of βxis not distorted. There is, compared to η= 2.0/M(dotted-dashed line), only a small bump left in theη= 0.5/Mcase (dashed line), that changes its shape slightly during the simulation, but does not travel to large distances from the punctures. The coordinates are disturbed in the case where no damping is used, and thus the noise in rexRe{Ψ22 4}is not surprising. In this series, using a Courant factor of 0.5, we only obtained stable evolutions for η <3.5/Mwhich agrees with thelimits foundin sectionIIB. Ifwechosethe value ofηtoo large, the same kind of instability in the shift vectorwefoundtheredevelopsintheequalmasscaseand the simulations fail. The failure occurs relatively early, before 50Mof evolution time, whereas the stable runs lasted about 1200 M,including mergerand ringdown(we stopped the runs after ringdown). III. POSITION-DEPENDENT FORMS OF η InsectionIIB, wesawthatasufficientlevelofdamping is needed to limit gauge dynamics, and too much damp--200 -100 0 100 200-0.004-0.00200.0020.004 -200 -100 0 100 200-0.004-0.00200.0020.004 -200 -100 0 100 200-0.004-0.00200.0020.004 -200 -100 0 100 200-0.004-0.00200.0020.004PSfrag replacementsη= 0.0/M η= 0.5/M η= 2.0/Mt= 64M t= 170 M t= 456 M t= 751 Mβxβxβxβx x/Mx/Mx/Mx/M FIG. 5: x-component of the shift vector, βx, for three differ- ent choices of ηat four different times during the simulation. The physical system is the same as in Fig. 4. The merger takes place at approximately t= 1000M. In the η= 0/M case (black, solid curve), the shift vector is not damped and therefore, a pulse travels outwards and distorts the shift o ver the whole grid. The amplitude of this pulse is considerably damped when using a non-vanishing ηand therefore the dis- tortions are reduced. For η= 0.5/M(red, dashed curve), there are still small bumps traveling out which are reduced by using η= 2/M(green, dotted-dashed curve). ing can lead to numerical instabilities. In section IIC, we saw the positive effect that sufficient damping has on the resultantwaveformforequalmassbinaries. While we still need damping in the gamma driver in the unequal mass case, a constant value may not fulfill the require- ments of limiting gauge dynamics and permitting stable evolutions. Rather, we need a definition for the damping that adjusts the value to the local mass-scale. We will examine definitions, that naturally track the position, and mass of the individual black holes. The choice ofηshould provide a reasonable value both near the individual black holes, and at large distance from the binary. We will start by examining some previous5 work, that has used non-constant forms of the damping parameter, and why it may be necessary to use other formulas. We will then present the two new formulas for η, which we designed for this work. A. Previous dynamic damping parameters A position dependent damping was introduced some years ago by the authors of [20], and was later used in [21]. That formula reads η=ηpunc−ηpunc−η∞ 1+(ψ−1)2(3) withηpunc,η∞being constants, and assuming ψ= 1 +M1/(2r1) +M2/(2r2) (riis the distance to the ith puncture). Thisformulawasusedtodampgaugedynam- ics while using excision for equal-mass head-on collisions. It has since been found that using the moving puncture framework allows for constant damping in the approxi- mately equal mass case. We are looking for a formula which is suitable for the quasicircular inspiral of inter- mediate mass-ratio binaries. Previously [23], we used the formula η(/vector r) =ˆR0/radicalbig ˜γij∂iψ−2∂jψ−2 (1−ψ−2)2, (4) for determining a position dependent damping coeffi- cient instead of using a constant η. WithˆR0taken to be a unitless constant, it can be seen that Eq. (4) has units of inverse mass. The dependence on the BSSN variable, ψ, naturally tracks the position, and mass of the black holes. The application of Eq. (4) gave good values for the damping both at the punctures, and at the outer boundary, and was even found to somewhat decrease the grid-size of the larger black hole. The latter point could have positive effects on how the individual black holes are resolved on the numerical grid. It even had the addi- tional effect of keeping the horizon shapes roughly circu- lar, even close to merger - something that doesn’t hold in the constant ηcase. Most importantly, the simulations remained stable, without significantly changing the grav- itational waves. The formula was later used successfully for the 10:1 mass-ratio in [9]. Despite all this, Eq. (4) provides reason for concern. Fig.6showstheformof ηusingEq.(4)foranon-spinning binary of equal mass in quasicircular orbits starting at a separation of D= 10Mat four different times in the simulation. As can be seen, noise travels out from the origin as time progresses. This leaves steady features on the form of ηwhich could spike to higher and lower val- ues than the range determined in Sec. IIB. Additionally, these sharp features may lead to unpredictable coordi- nate drifts, and could, in some cases, affect the long-term stability of the simulation. To illuminate the origin of the disturbances in η(/vector r), we looked at the development of η(/vector r) in simulations of0 200 400 600 80000.511.522.53 0 200 400 600 80000.511.522.53 0 200 400 600 80000.511.522.53 0 200 400 600 80000.511.522.53 PSfrag replacements x/Mη(x)·M η (x)·M η (x)·M η (x)·Mt= 0M t= 79M t= 165 M t= 300 M FIG. 6: Damping factor, η, along the x-axis using Eq. (4). The simulated configuration is an equal mass binary with ini- tial separation D= 10Mand orbits lying in the ( x,y)-plane. Shown are four different times during the simulation. a single, non-spinning puncture, and a single, spinning puncture (Sz/M2= 0.25). The result for the spinning case is plotted in Fig. 7 at two different times over the x-axis. Again, we see a pulse traveling outwards. Only this time, it does not leave much noise on the grid. The fact that this pulse travels at a speed which is roughly 1.39 (in our geometric units where c=G= 1) in both the spinning and non-spinning scenario indicates that it is related to the gauge modes traveling at speed√ 2 in the asymptotic regionwhere α≃1 (see [32] and [19] for a discussion of gauge speeds). In contrast to gauge pulses in the lapse, α, or shift vector, βi, the pulse in η(x) is amplified as it walks out. We found the same result in the single puncture simulation without spin. We believe the reason for this behavior is that as the distance to the puncture increases,the conformalfactor, ψ, getscloserto unity. Therefore, the denominator in Eq. (4) approaches zero, and the gauge disturbances in the derivatives of ψ are magnified. We further observed reflections at the re-6 finement boundaries as this pulse passes through them. Thismayexplainthe fluctuationsin η(x) shownin Fig.6. While one could continue to fine-tune a formula depen- dent on the conformalfactorto deal with these problems, we looked in a different direction to determine the form of the damping parameter. -200 -100 0 100 20000.511.522.53 PSfrag replacements x/Mη(x)·Mt= 50 .56M t= 101 .25M FIG. 7: Form of η(/vector r) for a single spinning puncture sitting atx= 0 using Eq. (4) after simulation time t= 50.56M (solid black line) and t= 101.25M(dashed red line) over x−direction. B. Formulas for ηwith explicit dependence on the position and mass of the punctures Since we always know the location of a puncture, and we know what its associated mass, we chose a form of damping that uses this local information throughout the domain. To address the demands and concerns dis- cussed in Section II and IIIA, we designed two position- dependent forms of η. The two forms we tested are η(/vector r) =A+C1 1+w1(ˆr2 1)n+C2 1+w2(ˆr2 2)n,(5) and η(/vector r) =A+C1e−w1(ˆr2 1)n+C2e−w2(ˆr2 2)n.(6) InEqs.(5) and(6), w1andw2arerequiredtobe positive, unitless parameters which can be chosen to change the width of the functions. The power nis taken to be a positive integer which determines the fall-off rate. The constantsA,C1, andC2are then chosen to provide the desired values of ηat the punctures, and at at infinity. Lastly, ˆr1and ˆr2are defined as ˆ ri=|/vector ri−/vector r| |/vector r1−/vector r2|, whereiis either one or two, and /vector riis the position of the i’th black hole. The definition of ˆ riis chosen to naturally scale the fall- offtothe separationofthe blackholes. w1,w2, andncan be chosen to change the overallfall-off. Our work focuseson the choice w1=w2=wandn= 1. Following [23], we construct the damping factor to have units of inverse mass. We choose A= 2/Mtot, whereMtot≡M1+M2 is defined as the sum of the irreducible masses. We then takeCi= 1/Mi−A. It is then evident that both Eqs. (5) and (6) will give a constant value of η= 2/Mtotin the equal mass case. We designed the two formulas for ηin order to test the value of using fundamentally different functions. In our simulations, we found little noticeable difference in the application of one compared to the other. In the absence of such a difference, it becomes more beneficial to use Eq. (5), as Gaussians are computationally more expensive. It should be pointed out that Eq. (5) is very similar to Eq. (13) suggested in [16], and we believe the following results are very similar to what would be found using that form for the damping. Going into the present work, we haveno ansatz which might suggestthese forms of damping yield wave forms which are any better than the use of any previous form of η. However, as will be seen in the results sections, the waveforms we get from unequal mass binaries show noticeable improvement over the constant ηcase. IV. RESULTS For data analysis purposes, we are mainly interested in the properties of the emitted gravitational waves of the black hole binary systems under study. Hence, it is im- portant to check how the changes in the gauge alter the extracted waves. In the context of gravitational wave ex- traction, Ψ 4is only first orderinvariant under coordinate transformations. In addition, we haveto chosean extrac- tion radius rexfor the computation of modes, which is also coordinate dependent. Although the last point can be partly addressed by extrapolation of rex→ ∞, it is a priori not clear how much a change of coordinates af- fects the gravitational waves. Furthermore, a change of coordinates implies an effective change of the numerical resolution, and for practical purposes we have to ask how much waveforms differ at a given finite resolution. A. Waveform comparison using formula (5) The results in the following section refers to the use of Eq. (5). We compare numerical simulations using three different grid configurations, which correspond to three different resolutions. In the terminology of [17], the grid set-upsareφ[5×64 : 7×128 : 5],φ[5×72 : 7×144: 5], and φ[5×80 : 7×160: 5], which correspondsto resolutionson the finest grids of 3 M/320 (N= 64),M/120 (N= 72) and 3M/400 (N= 80), respectively. When referring to results from different resolutions, we will from here on use the number of grid points on the finest grid, N, to distinguish between them. In this subsection, we use w1=w2= 12 andn= 1 in Eq. (5). As test system we7 use an unequal mass black hole binary with mass ratio m2/m1= 4 andan initial separationof D= 5Mwithout spins in quasi-circular orbits. For orientation, Fig. 8 shows the amplitude of the 22- mode,A22, computed with the standard gauge η= 2/M (displayed as solid lines) and with the new η(/vector r) using Eq. (5) (displayed as non-solid lines). The three differ- ent colors correspond to the three resolutions. The inset shows a larger time range of the simulation, while the main plot concentrateson the time frame around merger. The plotgivesacourseviewofthe closenessofthe results we obtain with standard and new gauges. In Fig. 9, we plot the relative differences between the amplitudes at low and medium (solid lines), and medium and high resolution (non-solid lines) obtained withη= 2/M(light gray lines) as well as η(/vector r) (Eq. (5)) (black lines). Here, we find the maximum error be- tween the low and medium resolution of the series using η= 2/Mamounts to about 12% (solid gray curve). Be- tween medium and high resolution (dashed gray curve), we find a smaller relative error, but it still goes up to 7% at the end of the simulation. Employing Eq. (5), the maximum amplitude error between low and medium resolution (solid black line) is only about 4%, and there- fore even smaller than the error between medium and high resolution for the constant damping case. Between medium and high resolution, the relative amplitude dif- ferences for Eq. (5) are in general smaller than the ones between low and medium resolution, although the maxi- mum error is comparable to it (dot-dashed black line). We repeat the previous analysis for the phase of the 22-mode,φ22. Again, we comparethe errorsbetween res- olutions in a fixed gauge. Figure 10 shows that the error between lowest and medium resolution using η= 2/M (solid gray line) grows up to about 0.31 radians. For the differences between medium and high resolution (dashed line) we find a maximal error of 0.2 radians for η= 2/M. Forη(/vector r) following Eq. (5), the phase error between low and medium resolution is only about 0.19 radians (solid black line) and decreases to 0.1 radians between medium and high resolution (dot-dashed line). Again, employ- ing the position dependent form of η, Eq. (5), the error between lowest and medium resolution is lower than the one we obtain for constant ηbetween medium and high resolution. The results for amplitude and phase error suggest that we can achieve the same accuracy with less computationalresourcesusingaposition-dependent η(/vector r). B. Waveform comparison using formula (6) We repeated the analysis of Sec. IVA with the wave- forms we obtain using Eq. (6) (with w1=w2= 12 and n= 1). We use the same initial conditions (mass ratio 4 : 1,D= 5M, no spins), and compare the amplitudes and phases of the 22-mode of Ψ 4with the results of the η= 2/M-runs. The grid configurations remain the same. The results are very similar to the ones we obtained in160 165 170 175 180 185 1900.0150.0200.0250.0300.035 t/Slash1MA22/DotMatΗM 100 120 140 160 180 2000.0000.0050.0100.0150.0200.0250.0300.035 t/Slash1MA22/DotMatΗMN/Equal64 N/Equal72 N/Equal80 FIG. 8: Amplitude of the 22-mode of Ψ 4of a binary with mass ratio 4:1 and initial separation D= 5M. The different colors correspond to three different resolutions according to the grid setup described in the text. The solid lines are resu lts forη= 2/M, the dashed, dotted and dot-dashed ones are for η(/vector r) (Eq. (5)). The inset shows the simulation from shortly after the junk radiation passed, in the main plot we zoom into the region of highest amplitude (near the merger). 100 120 140 160 180 200/Minus0.050.000.050.100.15 t/Slash1M/CapDelta/CapAlpΗa22/Slash1/CapAlpΗa22/LParen164/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1 /LParen180/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1 /LParen164/Minus72/RParen1,Η/Equal2/Slash1M /LParen180/Minus72/RParen1,Η/Equal2/Slash1M FIG. 9: Relative differences of the amplitude of the 22-mode of Ψ4between resolutions N= 64 and N= 72 (gray solid curve) as well as N= 72 and N= 80 (gray dashed curve) when using η= 2/M. The same for η(/vector r) (Eq. (5)) between N= 64 and N= 72 (black solid curve) and N= 72 and N= 80 (black dot-dashed curve). The physical situation is the same as in Fig. 8. The maximum differences are above 10%, comparing low and medium resolution of the constant ηsimulations (gray solid line). Figs. 9 and 10, and we therefore do not show them here. Although Eqs. (5) and (6) result in different shapes for η(/vector r), Ψ22 4is very similar. Therefore, the comparison to η= 2/Mnaturally gives very similar results, too. The phase differences between results from Eqs. (5) and (6) at a given resolution are shown in Fig. 11. These are, with a maximum phase error of 0.004 radians, very small comparedto the phase errorsbetween resolutions, which, at minimum, are about 0.1 radian (see Fig. 10). Fig. 12 comparesthe phase errorbetween low and medium (solid8 100 120 140 160 180 2000.00.10.20.3 t/Slash1M/CapDeltaΦ22/LParen164/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1 /LParen180/Minus72/RParen1,Η/LParen1r/OverRVector/RParen1 /LParen164/Minus72/RParen1,Η/Equal2/Slash1M /LParen180/Minus72/RParen1,Η/Equal2/Slash1M FIG. 10: Phase differences between lowest and medium reso- lution for the series using η= 2/M(solid gray line) and η(/vector r) (Eq. (5)) (solid black line) as well as between medium and high resolution for η= 2/M(dashed gray line) and for η(/vector r) (Eq. (5)) (dot-dashed black line). The physical situation i s the one of Fig. 8. 100 120 140 160 180 200/Minus0.004/Minus0.0020.0000.0020.004 t/Slash1M/CapDeltaΦ22N/Equal64,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1 N/Equal72,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1 N/Equal80,ΗEq./LParen15/RParen1/MinusΗEq./LParen16/RParen1 FIG. 11: Phase differences between waveforms obtained with Eq. (5) and Eq. (6) in three different resolutions (solid , dashed, dotted-dashed lines) for mass ratio 4:1, D= 5M. lines), and medium and high resolution (dotted-dashed and dashed line) of Eq. (6) (gray) to the ones of Eq. (5) (black). For comparison, the error between medium and high resolution is also plotted for Eq. (4) in this figure (dotted line). The plot indicates that the errors between resolutions are in good agreement for the different posi- tion dependent formulas of η. C. Behavior of the shift vector In [23], we found an unusual behavior of the shift vec- tor. This is illustrated in Fig. 13, where we plot the x-component of the shift, βx, in thex-direction after 160Mof evolution (this means approximately 80 Maf- ter merger) for all four versions of the damping constant we used for comparison in this paper before, and for the same binary configuration as the one used in Secs. IVA100 120 140 160 180 2000.000.050.100.150.20 t/Slash1M/CapDeltaΦ22/LParen180/Minus72/RParen1,ΗEq./LParen14/RParen1 /LParen164/Minus72/RParen1,ΗEq./LParen15/RParen1 /LParen180/Minus72/RParen1,ΗEq./LParen15/RParen1 /LParen164/Minus72/RParen1,ΗEq./LParen16/RParen1 /LParen180/Minus72/RParen1,ΗEq./LParen16/RParen1 FIG. 12: Phase difference between waveforms at low and medium resolution (solid lines) and medium and high reso- lution (dotted-dashed and dashed line) using either Eq. (5) (black lines) or Eq. (6) (gray lines) for mass ratio 4:1, D= 5M. For comparison, we also show the phase difference ob- tainedwith Eq. (4) between mediumandhigh resolution (dot- ted line). 0 100 200 300 400 5000.0000.0010.0020.0030.004 x/Slash1MΒxΗ/Equal2/Slash1M Eq./LParen14/RParen1 Eq./LParen15/RParen1 Eq./LParen16/RParen1 FIG. 13: x-component of the shift vector in x-direction after 160Mof evolution of the system with mass ratio 4 : 1 and D= 10M. The black, dot-dashed line refers to the use of a constant damping η, while the black, solid line uses Eq. (4). The gray, dashed line is for the use of Eq. (6) and the gray, dotted one for Eq. (5). Except for the constant η(black, dot- dashed line), the results in this plot are indistinguishabl e. and IVB. Like in [23], we find that using Eq. (4) results in a shift which falls off to zero too slowly towards the outer boundary, and which develops a “bump” (black, solid line), while the constant damping case (black, dot- dashed line) falls off to zero quickly. Employing Eqs. (5) or (6) avoids this undesirable feature. After merger, the shift falls off to zero when going awayfrom the punctures as it does in the constant damping case (gray dashed and dotted lines). Using Eq. (5) or (6) prevents unwanted co- ordinate drifts at the end of the simulations.9 0 10 20 30 40 500102030405060 PSfrag replacements t/MAAH,1/AAH,2 n= 1, w= 0.1 n= 1, w= 0.5 n= 1, w= 6.0 n= 1, w= 12.0 n= 1, w= 16.0n= 1, w= 200 .0 n= 3, w1= 0.01, w2= 0.001 η= 2/M ηfrom Eq. (4) FIG. 14: Shown is the time dependence of the ratio between the coordinate areas of the apparent horizons of both black holes in a simulation with mass ratio 4 : 1 with initial separa - tionD= 5M. The black, blue and red lines use η(/vector r), Eq. (5) withvaryingvaluesofthewidthparameter w. Theorange line (dash-dot-dot) uses the constant damping η= 2/Mand the green (dash-dot) one refers to the result of [23] with Eq. (4) . Using Eq. (5), the coordinate areas can be varied with re- spect to each other depending on the choice of w. A ratio of 1 means the black holes have the same size on the numerical grid. V. DISCUSSION In this work, we examined the role that the damping factor,η, plays in the evolution of the shift when using the gamma driver. In particular, we examined the range of values allowed in various evolutions, and what effects showed up because of the value chosen. We then de- signed a form of ηfor the evolution of binary black holes which provides appropriate values both near the individ- ual punctures and far away from them with a smooth transition in between. In Sec. IV, we directly examined the waveformsfor the caseusingEq.(5), where w1=w2= 12andn= 1. While the form of ηis predictable, and can be easily adjusted for stability, we also saw that the waveforms produced using this definition showed less deviation with increas- ing resolution than using a constant η. When examining the waveforms produced using Eq. (6), we found simi- lar results. In the absence of a noticeable difference in the quality of the waveforms, Eq. (5) is computationally cheaper, and, as such, is our preferred definition for the damping. We have already pointed out a certain freedom to pick parameters in Eqs. (5) and (6). We did perform some experimentation along this line where we varied w=w1=w2to see if we could get a useful effect of the coordinate size of the apparent horizons on the numer- ical grid. In [17, 33], it was noticed that the damping coefficient affects the coordinate location of the apparent horizon, and therefore the resolution of the black hole onthe numerical grid. Fig. 14 plots the ratio of the grid- area of larger apparent horizon to the smaller apparent horizon as a function of time for w-values of 0 .1, 0.5 and for 200, all with n= 1. Also plotted is the relative co- ordinate size for the same binaries using a constant η in dashed, double-dotted line, and for using Eq. (4) in a blue dashed-dotted line. All the evolutions show an immediate dip, and then increase in the grid-area ratio during the course of the evolution. While a very low ra- tio was found using Eq. (4), the orange dotted line was later found for the choices of n= 3 withw1= 0.01 and w2= 0.0001 with Eq. (5). Due to this freedom in the implementation of our explicit formula for the damping, it may be possible to further reduce the relative grid size of the black holes. This effect could be important in eas- ing the computational difficulty of running a numerical simulation for unequal mass binaries. Having a form of ηthat leads to stable evolutions for any mass-ratio is an important step towards the numer- ical evolution of binary black holes in the intermediate mass-ratio. We believe the form given in Eq. (5) pro- vides such a damping factor at a low computational cost, although the test results presented are limited to mass ratio 4 : 1. We plan to examine larger mass ratios in future work. The new method should allow binary sim- ulations for mass ratio 10 : 1, or even 100 : 1. It remains to be seen whether other issues than the gauge are now the limiting factor for simulations at large mass ratios. Acknowledgments ItisapleasuretothankZhoujianCaoandErikSchnet- ter for discussions. This work was supported in part by DFG grant SFB/Transregio 7 “Gravitational Wave Astronomy” and the DLR (Deutsches Zentrum f¨ ur Luft und Raumfahrt). D. M. was additionally supported by the DFG Research Training Group 1523 “Quantum and Gravitational Fields”. Computations were performed on the HLRB2 at LRZ Munich.10 [1] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr- qc/0507014. [2] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006), gr- qc/0511048. [3] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006), gr- qc/0511103. [4] F. Pretorius (2007), arXiv:0710.1338v1 [gr-qc]. [5] M. 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Laguna (2006), gr- qc/0601026.

2010-03-24

Certain numerical frameworks used for the evolution of binary black holes make use of a gamma driver, which includes a damping factor. Such simulations typically use a constant value for damping. However, it has been found that very specific values of the damping factor are needed for the calculation of unequal mass binaries. We examine carefully the role this damping plays, and provide two explicit, non-constant forms for the damping to be used with mass-ratios further from one. Our analysis of the resultant waveforms compares well against the constant damping case.

Dynamical shift condition for unequal mass black hole binaries

1003.4681v1

arXiv:1003.5375v1 [math.AP] 28 Mar 2010DAMPED WAVE DYNAMICS FOR A COMPLEX GINZBURG-LANDAU EQUATION WITH LOW DISSIPATION EVELYNE MIOT Abstract. We consider a complex Ginzburg-Landau equation on RN, corresponding to a Gross-Pitaevskii equation with a small dissipation term. W e study an asymptotic regime for long-wave perturbations of constant maps of modulus one. We show that such solutions never vanish on RNand we derive a damped wave dynamics for the perturbation. Ou r results are obtained in the same spirit as those by Bethuel, Danchin and S mets for the Gross-Pitaevskii equation [2]. 1.Introduction We consider a complex Ginzburg-Landau equation ∂tΨ = (κ+i)[∆Ψ+Ψ(1 −|Ψ|2)], (C) whereΨ = Ψ( t,x) :R+×RN→C, withN≥1, isacomplex-valued mapandwhere0 < κ <1. Equation (C) admits elementary non-vanishing solutions, w hich are given by all constant maps of modulus equal to one. The aim of this paper is to study t he dynamics for (C) near such states. We focus on a regime in which the solutions Ψ do no t vanish on RN, so that we may write them into the form Ψ =rexp(iφ). Secondly, we assume that ( r2,∇φ) is a long-wave perturbation of (1 ,0). More precisely, we introduce a small parameter ε >0 and we define ( r2,∇φ) through the change of variables r2(t,x) = 1+ε√ 2aε(εt,εx) 2∇φ(t,x) =εuε(εt,εx),(1.1) where (aε,uε) belongs to C(R+,Hs+1×Hs), withs≥2, and satisfies suitable bounds. Our objective is two-fold. First, to define ( aε,uε) we wish to determine how long a solution initially given by (1.1) does not vanish on RN. Our second purpose is to investigate the dynamics of ( aε,uε) whenεvanishes and κis small. This asymptotic dynamics depends on the balance between the amount κof dissipation in Eq. (C) and the size εof the perturbation; to characterize this balance we introduce the ratio νε=κ ε. According to (C) we obtain the equations for the perturbatio n (aε,uε) /braceleftigg ∂taε+√ 2divuε+2νε−κε∆aε= fε(aε,uε) ∂tuε+√ 2∇aε−κε∆uε= gε(aε,uε),(1.2) Date: November 23, 2018. 12 EVELYNE MIOT where f εand gεare given by fε(aε,uε) =√ 2κ/parenleftig −2|∇ρa|2−ρ2 a|uε|2 2−a2 ε/parenrightig −εdiv(aεuε) gε(aε,uε) =κε∇/parenleftbigg∇ρ2 a ρ2a·uε/parenrightbigg +2ε∇∆ρa ρa−εuε·∇uε,(1.3) with ρ2 a(t,x) = 1+ε√ 2aε(t,x). Our first result establishes that if the initial perturbatio n is not too large, the solution Ψ never exhibits a zero so that (1.1) does hold for all time. Theorem 1.1. Letsbe an integer such that s >1 +N/2. There exist positive numbers K1(s,N),K2(s,N)and0< κ0(s,N)<1, depending only on sandN, satisfying the following property. Let0< κ≤κ0(s,N). For0< ε≤1, let(a0 ε,ϕ0 ε)∈Hs+1(RN)2such that M0:=/ba∇dbl(a0 ε,u0 ε)/ba∇dblHs+ε/ba∇dbla0 ε/ba∇dblHs+1+/ba∇dblϕ0 ε/ba∇dblL2≤min(νε,κ−1,ε−1) K1(s,N), whereu0 ε= 2∇ϕ0 ε. Then Eq. (1.2)-(1.3)has a unique global solution (aε,uε)inC(R+,Hs+1×Hs)such that (aε,uε)(0) = (a0 ε,u0 ε). Moreover /ba∇dbl(aε,uε)/ba∇dblL∞(Hs)+ε/ba∇dblaε/ba∇dblL∞(Hs+1)≤K2(s,N)M0. Finally, if Ψdenotes the corresponding solution to Eq. (C), we have for all t≥0 /vextenddouble/vextenddouble|Ψ(t)|2−1/vextenddouble/vextenddouble ∞<1 2. Remark 1.1. Fixingκ=κ0andε=ε0, Theorem 1.1 entails that for initial data Ψ0(x) =/parenleftbig 1+˜a0(x)/parenrightbig1/2exp(i˜ϕ0(x)), with/ba∇dbl(˜a0,˜ϕ0)/ba∇dblHs+1≤C, whereConly depends on sandN, the corresponding solution1Ψ to Eq.(C)remains bounded and bounded away from zero for all time. Remark 1.2. For all0< ε≤ε0and0< κ≤κ0satisfying ε≤κ, so that νε≥1, Theorem 1.1 allows to handle initial data Ψ0 ε(x) =/parenleftbigg 1+ε√ 2a0(εx)/parenrightbigg1/2 exp(iϕ0(εx)), (1.4) where(a0,ϕ0)∈Hs+1(RN)2does not depend on ε, so that M0is constant, and where M0is smaller than a number depending only on sandN. Once the question of existence for ( aε,uε) has been settled, our next task is to determine a simplified system of equations to describe its asymptotic d ynamics. From now on we focus on a regime with low dissipation, namely we further assume th at κ=κ(ε) and lim ε→0κ(ε) = 0. 1Given by Theorem 3.1 below.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 3 In view of (1.3), this is a natural ansatz in order to treat the second members f εand gεas perturbations in the limit ε→0. Eq. (1.2) then formally reduces to a damped wave equation /braceleftigg ∂ta+√ 2divu+2νεa= 0 ∂tu+√ 2∇a= 0,(1.5) with propagation speed equal to√ 2 and damping coefficient equal to 2 νε. As a consequence of Theorem 1.1 we can compare the solution ( aε,uε) to the one of the linear damped wave equation (1.5) with loss of three derivat ives. Theorem 1.2. Letsbe an integer such that s >1+N/2. Let(a0 ε,ϕ0 ε)∈Hs+1(RN)2satisfy the assumptions of Theorem 1.1. Let u0 ε= 2∇ϕ0 ε. We denote by (aℓ,uℓ)∈C(R+,Hs+1×Hs)the solution of Eq. (1.5)with initial datum (a0 ε,u0 ε). There exists a constant K3(s,N)depending only on sandNsuch that for all t≥0 /ba∇dbl(aε−aℓ,uε−uℓ)(t)/ba∇dblHs−2≤K3(s,N)(εκt)1/2max(1,ν−1 ε)(M2 0+M0), whereM0is defined in Theorem 1.1. In particular, for initial data given by (1.4), the approxim ation by the damped wave equa- tion is optimal when κandεare comparable. Moreover, Theorem 1.2 yields a correct appr ox- imation up to times of order C(κε)−1. In order to handle larger times, it is helpful to take into account the linear parabolic terms in (1.2): /braceleftigg ∂ta+√ 2divu+2νεa−κε∆a= 0 ∂tu+√ 2∇a−κε∆u= 0.(1.6) Our next result presents uniform in time comparison estimat es with the solution of Eq. (1.6) for high order derivatives. Theorem 1.3. Letsbe an integer such that s >1+N/2. Let(a0 ε,ϕ0 ε)∈Hs+1(RN)2satisfy the assumptions of Theorem 1.1. We denote by (aℓ,uℓ)∈C(R+,Hs+1×Hs)the solution of Eq. (1.6)with initial datum (a0 ε,u0 ε). There exists a constant K4(s,N)depending only on sandNsuch that •/ba∇dbl(aε−aℓ,uε−uℓ)/ba∇dblL∞(Hs−2)≤K4(s,N)/parenleftbig κmax(1,ν−1 ε)2M2 0+εmax(1,ν−1 ε)M0/parenrightbig , •/ba∇dbl(aε−aℓ,uε−uℓ)/ba∇dblL∞(Hs−1)≤K4(s,N)/parenleftig max(1,ν−1 ε)/parenleftbig max(κ,ε)+ν−1 ε/parenrightbig M2 0+ν−1 εM0/parenrightig , •/ba∇dbl(aε−aℓ,uε−uℓ)/ba∇dblL∞(Hs)≤K4(s,N)/parenleftig (ν−1 εmax(1,ν−1 ε)+κ−1)M2 0+κ−1M0/parenrightig . Finally, for all t≥0 •/ba∇dbl(aε−aℓ,uε−uℓ)(t)/ba∇dblHs−2≤K4(s,N)(εκt)1/2/parenleftbig max(1,ν−1 ε)M2 0+ν−1 εM0/parenrightbig , •/ba∇dbl(aε−aℓ,uε−uℓ)(t)/ba∇dblHs−1≤K4(s,N)(εκ−1t)1/2M0. We come back to initial data given by (1.4). Since κ−1diverges when ε→0, Theorem 1.3 does not provide a correct approximation for s-order derivatives. However, Eq. (1.6) yields a satisfactory large in time approximation for the de rivatives of order s−1 ifν−1 ε vanishes with ε. In fact, the corresponding comparison estimate is optimal whenever κand√εare proportional. This is due to the fact that the regularizi ng properties of the parabolic contributions in (1.6) become less efficient when κis small. On the other hand, as in Theorem4 EVELYNE MIOT 1.2, the global in time comparison estimates involving the l ower (s−2)-order derivatives are more efficient when κandεare proportional. The complex Ginzburg-Landau equations are widely used in th e physical literature as a model for various phenomena such as superfluidity, Bose-Ein stein condensation or supercon- ductivity, see [1]. In the specific form considered here, Eq. (C) corresponds to a dissipative extension of the purely dispersive Gross-Pitaevskii equat ion ∂tΨ =i[∆Ψ+Ψ(1 −|Ψ|2)]. (GP) A similar asymptotic regime for (GP) has been recently inves tigated by Bethuel, Danchin and Smets [2]. The analysis of [2] exhibits a lower bound for the fi rst time Tεwhere the solution vanishes and shows that ( aε,uε) essentially behaves according to the free wave equation ( νε≡ 0), or to a similar version, until then. In the two-dimensional case N= 2, there exists a formal analogy between Eq. (C) and the Landau-Lifschitz-Gilbert equation for sphere-valued magnetizations in three-dimensional ferromagnetics, see [3, 7]. We mention that a thin-film regim e leading to a damped wave dynamics for the in-plane components of the magnetization h as been studied by Capella, Melcher and Otto [3]. Finally, still in the two-dimensional case N= 2, Eq. (C) presents another remarkable regime in which the solutions exhibit zeros (vortices). Thi s regime has been investigated by Kurtzke, Melcher, Moser and Spirn [6] and the author [9] when κis proportional to |lnε|−1. In this setting, Eq. (C) is considered under the form ∂tΨε= (κ+i)[∆Ψε+1 ε2Ψε(1−|Ψε|2)], (Cε) which is obtained from the original equation via the parabol ic scaling Ψε(t,x) = Ψ/parenleftbiggt ε2,x ε/parenrightbigg . (1.7) A natural extension of the results in [6, 9] would consist in a llowing for superpositions of vortices and oscillating phases in the initial data. This di fficult issue was a strong motivation to analyze the behavior of the phase in the regime (1.1), excl uding vortices, as a first attempt to tackle the general situation where it is coupled with vort ices. 2.General strategy We now present our approach for proving Theorems 1.1, 1.2 and 1.3, which will be partly borrowed from the analysis in [2] for the Gross-Pitaevskii e quation. First, we handle Eq. (C) in its parabolic scaling (1.7) yield ing Eq. (C ε). We define the variables bε(t,x) =aε/parenleftbiggt ε,x/parenrightbigg vε(t,x) =uε/parenleftbiggt ε,x/parenrightbigg , so that in the regime (1.1) we have Ψε(t,x) =ρε(t,x)exp(iϕε(t,x)) on R+×RN, (2.1)DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 5 where ρ2 ε(t,x) = 1+ε√ 2bε(t,x) 2∇ϕε(t,x) =εvε(t,x).(2.2) The system for ( bε,vε) translates into ∂tbε+√ 2 εdivvε+2νε εbε−κ∆bε=˜fε(bε,vε) ∂tvε+√ 2 ε∇bε−κ∆vε= ˜gε(bε,vε),(2.3) where ˜fε(bε,vε) =√ 2νε/parenleftbigg −2|∇ρε|2−ρ2 ε|vε|2 2−b2 ε/parenrightbigg −div(bεvε) ˜gε(bε,vε) =κ∇/parenleftbigg∇ρ2 ε ρ2ε·vε/parenrightbigg +2∇/parenleftbigg∆ρε ρε/parenrightbigg −vε·∇vε.(2.4) For a map Ψ ∈H1 loc, the Ginzburg-Landau energy of Ψ is defined by Eε(Ψ) =/integraldisplay RN/parenleftig|∇Ψ|2 2+(1−|Ψ|2)2 4ε2/parenrightig dx, andEdenotesthecorrespondingspaceoffiniteenergyfields. Fort heGross-Pitaevskiiequation the Ginzburg-Landau energy is an Hamiltonian, whereas for s olutions to Eq. (C ε) it decreases in time. Note that, in the regime (2.1)-(2.2), the solution Ψ εbelongs to Esince (bε,vε)∈ H1×L2. In fact, one has Eε(Ψε)≃C(/ba∇dbl(bε,vε)/ba∇dbl2 L2+ε2/ba∇dbl∇bε/ba∇dbl2 L2) provided that /ba∇dbl|Ψε|−1/ba∇dbl∞<1. Our first issue is to solve the Cauchy problem for (C ε) so that ( bε,vε) being defined by (2.2), as long as Ψ εdoes not vanish, does belong to C(Hs+1×Hs). As mentioned, the initial field Ψ0 εhas finite Ginzburg-Landau energy. In [4] (see also [5]) it ha s been shown that E ⊂ W+H1(RN). Here the space W, which will be defined in Section 3 below, contains in particu lar all constant maps of modulus one. It is therefore natural to determine the solution Ψ εinC(W+Hs+1). This is done in Section 3. In Theorems 1.1, 1.2 and 1.3 one assumes that /ba∇dblb0 ε/ba∇dbl∞is bounded in such a way that |Ψ0 ε|is bounded and bounded away from zero. More precisely, the cons tantK1(s,N) can be adjusted so that c(s,N)ε√ 2/ba∇dblb0 ε/ba∇dblHs<1 2. (2.5) Here the constant c(s,N) corresponds to the Sobolev embedding Hs(RN)⊂L∞(RN) for s > N/2. Hence (2.5) guarantees that /ba∇dbl|Ψ0 ε|2−1/ba∇dbl∞<1/2. As long as infRN|Ψε(t)|>0, one may define ( bε,vε)(t) explicitely as a function of Ψ ε(t). In fact, to prove that Ψ εand (bε,vε) are globally defined, and to establish Theorems 1.2 and 1.3 it suffices to show that /ba∇dbl(bε,vε)/ba∇dblHs+1×Hsremains bounded. Moreover, to obtain the bound /ba∇dbl|Ψε(t)|2−1/ba∇dbl∞<1/2, it suffices to show that (2.5) holds as long as bεis defined. Due to the presence of higher order derivatives in the right- hand sides in (2.3), controlling /ba∇dbl(bε,vε)/ba∇dblHs+1×Hsis however a difficult issue. As in [2], this control will be car ried out by6 EVELYNE MIOT incorporating the equation satisfied by ∇ln(ρ2 ε). More precisely, we focus on the new variable (bε,zε), where zε=vε−i∇ln(ρ2 ε) =∇/parenleftbig 2ϕε−iln(ρ2 ε)/parenrightbig ∈CN. We remark that ( bε,zε) is well-suited to our analysis since Eε(Ψε) =1 8/parenleftig /ba∇dblbε/ba∇dbl2 L2+/ba∇dblzε/ba∇dbl2 L2((1+εb/√ 2)dx)/parenrightig . Moreover, there exists a constant C=C(s,N) such that2 C−1/ba∇dbl(bε,zε)/ba∇dblHs≤ /ba∇dbl(bε,vε)/ba∇dblHs+ε/ba∇dblbε/ba∇dblHs+1≤C/ba∇dbl(bε,zε)/ba∇dblHs. From now on we will sometimes omit the subscript εfor more clarity in the notations. The equations for ( b,z) are given in the following Proposition 2.1. Lets≥2,T0>0andΨbe a solution to (Cε)on[0,T0]satisfying inf (t,x)∈[0,T0]×RN|Ψ(t,x)| ≥m >0 and such that (b,v)∈C1([0,T0],Hs+1×Hs). Then3 ∂tb+√ 2 εdivRez=κ/parenleftig −(√ 2 ε+b)div(Imz)−1 2(√ 2 ε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht −√ 2 ε(√ 2 ε+b)b/parenrightig −div(bRez) ∂tz+√ 2 ε∇b= (κ+i)∆z+−1+κi 2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht+κ√ 2 εi∇b. Dealing with ( b,z) instead of ( b,v) presents many advantages when computing energy estimates. Indeed, in contrast with System (2.3) for ( b,v), the equations for ( b,z) involve only non linear first-order quadratic terms and a linear seco nd-order operator ( κ+i)∆z. This is due to the identityε√ 2∇b=−(1+ε√ 2b)Imz, which enables to save one derivative. For the Gross-Pitaevskii equation (GP), the energy estimat es performed in [2] for ( b,z) involve a family of semi-norms with a suitable weight Γk(b,z) :=/integraldisplay RN|Dkb|2+/integraldisplay RN(1+ε√ 2b)|Dkz|2, k= 0,...,s. In particular, we have the remarkable identity Γ0(b,z) = 8Eε(Ψ), which in fact was the principal motivation to add the imagina ry part of z. Moreover we remark that Γk(b,z) and/ba∇dbl(Dkb,Dkz)/ba∇dbl2 L2are comparable as long as |Ψ|is close to one. ForthecomplexGinzburg-Landauequation(C ε)wewillpartlyrelyontheestimates already stated in [2] to establish the following 2See (5.4) below. 3Here/angbracketleftz,z/angbracketright=N/summationdisplay i=1z2 i, wherez= (z1,...,z N)∈CN.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 7 Proposition 2.2. Lets > N/2andT0>0. LetΨbe a solution to (Cε)on[0,T0]such that /ba∇dbl|Ψ|2−1/ba∇dblL∞([0,T0]×RN)<1 2 and such that (b,z)∈C1([0,T0],Hs+1). There exists a constant K=K(s,N)depending only onsandNsuch that for 1≤k≤sandt∈[0,T0] d dt/parenleftbig Γk(b,z)+Eε(Ψ)/parenrightbig +κ 2/parenleftbig Γk+1(b,z)+1 ε2Γk(b,0)/parenrightbig ≤K/parenleftbig νε/ba∇dblb/ba∇dbl∞+κ/ba∇dbl(b,z)/ba∇dbl2 ∞+/ba∇dbl(Db,Dz)/ba∇dbl∞/parenrightbig/parenleftbig Γk(b,z)+Eε(Ψ)/parenrightbig . We further assume that s >1+N/2. Combining Proposition 2.2 and Sobolev embedding we readily find /ba∇dbl(b,z)(t)/ba∇dblHs≤C/ba∇dbl(b,z)(0)/ba∇dblHs+C(ε)/integraldisplayt 0/ba∇dbl(b,z)(τ)/ba∇dbl3 Hsdτ. Thisprovidesafirstcontrol of thenorm /ba∇dbl(b,z)(t)/ba∇dblHsuptotimes of order C(ε)−1/ba∇dbl(b,z)(0)/ba∇dbl−2 Hs. However, we need to refine this control since C(ε) diverges as εtends to zero. In fact, one may also apply Cauchy-Schwarz inequality and Sobolev imbed ding together with Proposition 2.2 to infer an estimate for /ba∇dbl(b,z)/ba∇dblL∞ t(Hs)in terms of the norms /ba∇dbl(b,z)/ba∇dblL2 t(Hs)and/ba∇dblb/ba∇dblL2 t(L∞). Proposition 2.3. Under the assumptions of Proposition 2.2, we assume moreove r thats > 1 +N/2. There exists a constant K=K(s,N)depending only on sandNsuch that for [0,T0] K−1/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)≤ /ba∇dbl(b,z)(0)/ba∇dblHs +νε/ba∇dbl(b,z)/ba∇dblL2 t(Hs)/ba∇dblb/ba∇dblL2 t(L∞)+/parenleftbig κ/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)+1/parenrightbig /ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs) and K−1κ/ba∇dbl(Db,Dz)/ba∇dbl2 L2 t(Hs)≤ /ba∇dbl(b,z)(0)/ba∇dbl2 Hs +/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)/parenleftbig νε/ba∇dbl(b,z)/ba∇dblL2 t(Hs)/ba∇dblb/ba∇dblL2 t(L∞)+/parenleftbig κ/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)+1/parenrightbig /ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs)/parenrightbig . In the second step of the proofs, we will exploit the decreasi ng properties of the semi- group operator associated to System (2.3) to derive estimat es for the norms /ba∇dbl(b,z)/ba∇dblL2 t(Hs) and/ba∇dblb/ba∇dblL2 t(L∞)in terms of /ba∇dbl(b,z)/ba∇dblL∞ t(Hs). These estimates are summarized in the following Proposition 2.4. Under the assumptions of Proposition 2.3, there exists a con stantK= K(s,N)depending only on sandNsuch that for t∈[0,T0] K−1/ba∇dbl(b,z)/ba∇dblL2 t(Hs)≤κ1/2max(1,ν−1 ε)M0 +/parenleftbig 1+ε/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)/parenrightbig /ba∇dbl(b,z)/ba∇dblL2 t(Hs)/parenleftbig κ1/2/ba∇dbl(b,z)/ba∇dblL2 t(Hs)+(ε+ν−1 ε)/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)/parenrightbig and K−1/ba∇dblb/ba∇dblL2 t(L∞)≤(εν−1 ε)1/2M0 +/parenleftbig 1+ε/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)/parenrightbig /ba∇dbl(b,z)/ba∇dblL2 t(Hs)εmax(1,ν−1 ε)/ba∇dbl(b,z)/ba∇dblL∞ t(Hs), whereM0is defined in Theorem 1.1. Combining Propositions 2.3 and 2.4 yields an improved estim ate for/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)which, in turn, leads to Theorems 1.1, 1.2 and 1.3. The remainder of this work is organized in the following way. In Section 3 we study the Cauchy problem for (C ε) and prove local well-posedness for ( b,z). Propositions 2.1, 2.2 and8 EVELYNE MIOT 2.3 are established in Section 4. Section 5 is devoted to the p roof of Proposition 2.4 by means of a Fourier analysis. We finally turn to the proof of Theorems 1.1 and 1.3 in Section 6. We omit the proof of Theorem 1.2, which can be obtained with some minor modifications. At some places, we will rely on helpful estimates that are recal led or established in the appendix. 3.The Cauchy problem for the complex Ginzburg-Landau equatio n In this section, we address the Cauchy problem for (C ε) in a space including the fields Ψ = (1+ a)1/2exp(iϕ), where ( a,ϕ)∈Hs+1(RN)2ands+1≥N/2. We consider the set W=/braceleftbig U∈L∞(RN),∇U∈H∞(RN) and 1 −|U|2∈L2(RN)/bracerightbig . Applying a standard fixed point argument (see, e.g., the proo f of Theorem 1 in [9]) and using the Sobolev embedding Hs+1⊂L∞ifs+1> N/2, it can be shown the following Theorem 3.1. Lets+ 1> N/2andU0∈ W. For any ω0∈Hs+1(RN)there exists T∗= T(U0,ω0)>0and a unique maximal solution Ψ∈ {U0}+C([0,T∗),Hs+1(RN)) to Eq.(Cε)such that Ψ(0) =U0+ω0. The Ginzburg-Landau energy of Ψis finite and satisfies Eε(Ψ(t))≤Eε(Ψ(0)),∀t∈[0,T∗). Moreover, there exists a number Cdepending only on Eε(Ψ(0))such that /ba∇dblΨ(t)−Ψ(0)/ba∇dblL2(RN)≤Cexp(Ct),∀t∈[0,T∗). Finally, either T∗= +∞orlimsup t→T∗/ba∇dbl∇Ψ(t)/ba∇dblHs= +∞. We recall that Edenotes the space of finite energy fields. Thanks to the alread y mentioned inclusion (see [4]) E ⊂ W+H1(RN), a consequence of Theorem 3.1 is the Corollary 3.1. Lets+1> N/2. Let(a0,ϕ0)∈Hs+1(RN)2. We assume that ε√ 2/ba∇dbla0/ba∇dbl∞<1. There exists T0>0and a unique solution (b,v)∈C([0,T0],Hs+1×Hs)to System (2.3)with initial datum (a0,u0= 2∇ϕ0). Moreover, there exists ϕ∈C([0,T0],H1 loc)such that v= 2∇ϕ. Proof.Set Ψ0(x) =/parenleftbig 1+ε√ 2a0(x)/parenrightbig1/2exp(iϕ0(x)). By assumption on ( a0,ϕ0), Ψ0belongs to Eand /ba∇dbl|Ψ0|2−1/ba∇dbl∞<1. (3.1) SinceE ⊂ W+H1(RN), wehave Ψ0∈ {U0}+H1(RN) for some U0∈ W. Usingtheembedding Hs+1(RN)⊂L∞(RN), we check that /ba∇dbl∇Ψ0/ba∇dblHs≤C(1+/ba∇dbl(a0,u0)/ba∇dbl2 Hs+1×Hs). This shows that actually Ψ0∈ {U0}+Hs+1(RN). Hence, by virtue of Theorem 3.1 there existsT∗>0 and a unique maximal solution Ψ ∈ {U0}+C([0,T∗),Hs+1) to (C ε) such that Ψ(0) = Ψ0.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 9 Next, thanks to (3.1) and to the inclusion Hs+1(RN)⊂L∞(RN), there exists by time continuity a non trivial interval [0 ,T0]⊂[0,T∗) for which inf (t,x)∈[0,T0]×RN|Ψ(t,x)| ≥m >0. Consequently, we may find a lifting for Ψ on [0 ,T0] : Ψ(t,x) =/parenleftbig 1+ε√ 2b(t,x)/parenrightbig1/2exp(iϕ(t,x)),whereϕ∈L2 loc. Setting then v= 2∇ϕ, we determine bandvin a unique way through the identities b=√ 2 ε(|Ψ|2−1) and v=2 |Ψ|2(Ψ×∇Ψ). In view of the regularity of Ψ we have ( b,v)∈C([0,T0],Hs+1×Hs). In addition, ( b,v) is a solution to System (2.3) on [0 ,T0], and the conclusion follows. /square 4.Proofs of Propositions 2.1, 2.2 and 2.3. 4.1.Notations. We use this paragraph to fix some notations. The notation a·bdenotes the standard scalar product on RNorR2N, which we extend to complex vectors by setting z·ζ= (Rez,Imz)·(Reζ,Imζ)∈R,∀z,ζ∈CN. We define the complex product of z= (z1,...,zN) andζ= (ζ1,...,ζN)∈CNby /an}b∇acketle{tz,ζ/an}b∇acket∇i}ht=N/summationdisplay j=1zjζj∈C. Therefore when z=a+ib∈CNandζ=x+iy∈CNwitha,b,x,y∈RNwe have /an}b∇acketle{tz,ζ/an}b∇acket∇i}ht=a·x−b·y+i(a·y+b·x) and z·ζ=a·x+b·y. With the same notations as above we finally introduce ∇z=∇a+i∇b∈CN×N and ∇z:∇ζ=∇a:∇x+∇b:∇y∈R, where for A,B∈RN×Nwe have set A:B= tr(AtB). 4.2.Proof of Proposition 2.1. Since Ψ = ρexp(iϕ) is a solution to (C ε), we have, with v= 2∇ϕ, ∂tρ2 ρ2= 2κ/parenleftbigg∆ρ ρ−|v|2 4+1−ρ2 ε2/parenrightbigg −div(ρ2v) ρ2 ∂t(2ϕ) = 2/parenleftbigg∆ρ ρ−|v|2 4+1−ρ2 ε2/parenrightbigg +κdiv(ρ2v) ρ2. Taking the gradient in both equations we obtain ∇∂tρ2 ρ2= 2κ∇∆ρ ρ−κ∇|v|2 2+2κ∇1−ρ2 ε2−∇div(ρ2v) ρ2 ∂tv= 2∇∆ρ ρ−∇|v|2 2+2∇1−ρ2 ε2+κ∇div(ρ2v) ρ2.10 EVELYNE MIOT Since∂tz=∂tv−i∇∂tρ2 ρ2, we have ∂tz= (1−κi)2∇∆ρ ρ−(1−κi)∇|v|2 2+2(1−κi)∇1−ρ2 ε2+(κ+i)∇div(ρ2v) ρ2. Next, expanding ∆lnρ=∆ρ ρ−|∇ρ|2 ρ2, we obtain 2∇∆ρ ρ=∇∆lnρ2+2∇|∇lnρ|2=−∆Imz+1 2∇|Imz|2. On the other hand, since vis a gradient we have ∇div(ρ2v) ρ2=∇divv+∇/parenleftig v·∇ρ2 ρ2/parenrightig = ∆Rez−∇/parenleftig Imz·v/parenrightig . Finally, using the fact that 2∇1−ρ2 ε2=−√ 2 ε∇b, we are led to the equation for z ∂tz= (κ+i)∆z−1−κi 2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht−√ 2 ε(1−κi)∇b. We next turn to the equation for b, recalling that ρ2verifies ∂tρ2=κ/parenleftbigg 2ρ∆ρ−ρ2|v|2 2+2ρ2(1−ρ2) ε2/parenrightbigg −div(ρ2v). Expanding the expression 2ρ∆ρ=ρ2∆lnρ2+ρ2 2|Imz|2=−ρ2divImz+ρ2 2|Imz|2, we find ∂tρ2=κ/parenleftig −(1+ε√ 2b)divImz−1 2(1+ε√ 2b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht−2(1+ε√ 2b)ε√ 2 ε2b/parenrightig −div/parenleftbig (1+ε√ 2b)Rez/parenrightbig , as we wanted. /square 4.3.Proof of Proposition 2.2. We present now the proof of Proposition 2.2. In all this paragraph, Cstands for a number depending only on sandN, which possibly changes from a line to another. We will make use of the identity ε√ 2∇b=−(1+ε√ 2b)Imz. (4.1) As we want to rely on the estimates already performed for the G ross-Pitaevskii equation in [2], it is convenient to write the equations for ( b,z) as follows /braceleftigg ∂tb=κfd(b,z)+fs(b,z) ∂tz=κgd(b,z)+gs(b,z),DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 1 where we have introduced the dissipative part fd(b,z) =−(√ 2 ε+b)div(Imz)−1 2(√ 2 ε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht−√ 2 ε(√ 2 ε+b)b, gd(b,z) = ∆z+i 2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht+i√ 2 ε∇b and the dispersive part fs(b,z) =−div/parenleftbig (√ 2 ε+b)Rez/parenrightbig , gs(b,z) =i∆z−1 2∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht−√ 2 ε∇b. Letk∈N∗. We compute d dtΓk(b,z) =d dt/integraldisplay RN(1+ε√ 2b)Dkz·Dkz+DkbDkb = 2/integraldisplay RN(1+ε√ 2b)Dkz·Dk∂tz+DkbDk∂tb+/integraldisplay RNε∂tb√ 2Dkz·Dkz =Is+Id, where Is= 2/integraldisplay RN(1+ε√ 2b)Dkz·Dkgs+DkbDkfs+/integraldisplay RNεfs√ 2Dkz·Dkz and κ−1Id= 2/integraldisplay RN(1+ε√ 2b)Dkz·Dkgd+DkbDkfd+/integraldisplay RNεfd√ 2Dkz·Dkz. To estimate the first term Iswe invoke Proposition 1 in [2] : |Is| ≤C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dbl(Db,Dz)/ba∇dblL∞/parenleftig Γk(b,z)+Eε(Ψε)/parenrightig , so we only need to estimate the term Id. Inserting the expressions of fdandgdwe find Id=κ(2I+2J+K), where I=/integraldisplay RN(1+ε√ 2b)/parenleftig Dkz·Dk∆z+1 2Dkz·iDk∇/an}b∇acketle{tz,z/an}b∇acket∇i}ht+√ 2 εDkz·iDk∇b/parenrightig =I1+I2+I3, J=/integraldisplay RN−DkbDk/parenleftig (√ 2 ε+b)div(Imz)/parenrightig −1 2DkbDk/parenleftig (√ 2 ε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht/parenrightig −DkbDk/parenleftig√ 2 ε(√ 2 ε+b)b/parenrightig =J1+J2+J3, and K=−/integraldisplay RN(1+ε√ 2b)/parenleftig div(Imz)+1 2Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht+√ 2 εb/parenrightig Dkz·Dkz.12 EVELYNE MIOT Step 1: estimate for I1. Integrating by parts in I1, then inserting (4.1) we find I1=−/integraldisplay RN(1+ε√ 2b)∇Dkz:∇Dkz−ε√ 2∇b·(Dkz·∇Dkz) =−/integraldisplay RN(1+ε√ 2b)|∇Dkz|2+/integraldisplay RN(1+ε√ 2b)Imz·(Dkz·∇Dkz) ≤ −/integraldisplay RN(1+ε√ 2b)|∇Dkz|2+/integraldisplay RN(1+ε√ 2b)1/2|Imz||Dkz|(1+ε√ 2b)1/2|∇Dkz|. Applying Young inequality to the second term in the right-ha nd side, we obtain I1≤ −1 2/integraldisplay RN(1+ε√ 2b)|∇Dkz|2+1 2/integraldisplay RN(1+ε√ 2b)|Imz|2|Dkz|2, so finally I1≤ −1 2/integraldisplay RN(1+ε√ 2b)|∇Dkz|2+C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblImz/ba∇dbl2 ∞/ba∇dblz/ba∇dbl2 Hk. Step 2: estimate for I2. Expanding I2thanks to Leibniz formula, we obtain I2=/integraldisplay RN(1+ε√ 2b)Dkz·Dk(i/an}b∇acketle{tz,∇z/an}b∇acket∇i}ht) =/integraldisplay RN(1+ε√ 2b)Dkz·i/an}b∇acketle{tz,∇Dkz/an}b∇acket∇i}ht+k−1/summationdisplay j=0Cj k/integraldisplay RN(1+ε√ 2b)Dkz·i/an}b∇acketle{tDk−jz,Dj(∇z)/an}b∇acket∇i}ht. Applying then Young inequality to the first term in the right- hand side, we infer that I2≤1 4/integraldisplay RN(1+ε√ 2b)|∇Dkz|2+C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblz/ba∇dbl2 ∞/ba∇dblz/ba∇dbl2 Hk +Ck−1/summationdisplay j=0/vextendsingle/vextendsingle/vextendsingle/integraldisplay RN(1+ε√ 2b)Dkz·i/an}b∇acketle{tDk−jz,Dj(∇z)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle. For each 0 ≤j≤k−1, we apply first Cauchy-Schwarz, then Gagliardo-Nirenberg (see Lemma 7.4 in the appendix) inequalities. This yields /vextendsingle/vextendsingle/vextendsingle/integraldisplay RN(1+ε√ 2b)Dkz·i/an}b∇acketle{tDk−jz,Dj(∇z)/an}b∇acket∇i}ht/vextendsingle/vextendsingle/vextendsingle≤C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblDkz/ba∇dblL2/ba∇dbl|Dk−jz||Dj(∇z)|/ba∇dblL2 ≤C(1+ε/ba∇dblb/ba∇dbl∞)/ba∇dblDkz/ba∇dblL2/ba∇dblDz/ba∇dbl∞/ba∇dblz/ba∇dblHk, and we are led to I2≤1 4/integraldisplay RN(1+ε√ 2b)|∇Dkz|2+C(1+ε/ba∇dblb/ba∇dbl∞)(/ba∇dblz/ba∇dbl2 ∞+/ba∇dblDz/ba∇dbl∞)/ba∇dblz/ba∇dbl2 Hk. Step 3: estimate for I3. SinceDk∇b∈RNwe have by definition of the complex product I3=/integraldisplay RN(1+ε√ 2b)√ 2 εDkz·iDk∇b=/integraldisplay RN(1+ε√ 2b)√ 2 εDkImz·Dk∇b.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 3 Inserting first (4.1) and using then Leibniz formula we get I3=−2 ε2/integraldisplay RN(1+ε√ 2b)DkImz·Dk/parenleftbig (1+ε√ 2b)Imz/parenrightbig =−2 ε2/integraldisplay RN(1+ε√ 2b)2|DkImz|2−2 ε2k/summationdisplay j=1Cj k/integraldisplay RN(1+ε√ 2b)DkImz·/parenleftbig Dj(1+ε√ 2b)Dk−jImz/parenrightbig . Now, we observe that for each j≥1, we have Dj(1+ε√ 2b) =ε√ 2Djb. Consequently, applying Young inequality to each term of the sum we find I3≤ −1 ε2/integraldisplay RN(1+ε√ 2b)2|DkImz|2+Ck/summationdisplay j=1/integraldisplay RN|DjbDk−jImz|2, and we finally infer from Gagliardo-Nirenberg inequality th at I3≤C/parenleftbig /ba∇dblb/ba∇dbl2 ∞+/ba∇dblImz/ba∇dbl2 ∞/parenrightbig /ba∇dbl(b,z)/ba∇dbl2 Hk. Step 4: estimate for J1. A short calculation using (4.1) yields J1=−/integraldisplay RNDkbDk/parenleftig (√ 2 ε+b)div(Imz)/parenrightig =−/integraldisplay RNDkbDkdiv/parenleftig (√ 2 ε+b)Imz/parenrightig +/integraldisplay RNDkbDk(∇b·Imz) =/integraldisplay RNDkbDkdiv(∇b)+/integraldisplay RNDkbDk(∇b·Imz). After integrating by parts in the first term in the right-hand side and expanding the second term by means of Leibniz formula we obtain J1=−/integraldisplay RN|∇Dkb|2+/integraldisplay RNDkb(Dk∇b)·Imz+k/summationdisplay j=1Cj k/integraldisplay RNDkb(Dk−j∇b)·DjImz. Next, combining Young, Cauchy-Schwarz and Gagliardo-Nire nberg inequalities we find J1≤ −1 2/integraldisplay RN|∇Dkb|2+C/ba∇dblImz/ba∇dbl2 ∞/ba∇dblb/ba∇dbl2 Hk+C/ba∇dblb/ba∇dblHk(/ba∇dbl∇b/ba∇dbl∞+/ba∇dblDz/ba∇dbl∞)/ba∇dbl(b,z)/ba∇dblHk, so that J1≤ −1 2/integraldisplay RN|∇Dkb|2+C/parenleftbig /ba∇dblImz/ba∇dbl2 ∞+/ba∇dbl(∇b,Dz)/ba∇dbl∞/parenrightbig /ba∇dbl(b,z)/ba∇dbl2 Hk.14 EVELYNE MIOT Step 5: estimate for J2. Similarly, we compute thanks to Leibniz formula J2=−1 2/integraldisplay RNDkbDk/parenleftig (√ 2 ε+b)Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht/parenrightig =−1 2/integraldisplay RNDkb(√ 2 ε+b)Dk(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht)+1 2k/summationdisplay j=1Cj k/integraldisplay RNDkbDjbDk−j(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht) =−1 ε√ 2/integraldisplay RNDkbDk(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht)−1 2/integraldisplay RNbDkbDk(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht) +1 2k/summationdisplay j=1Cj k/integraldisplay RNDkbDjbDk−j(Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht). Invoking Young and Cauchy-Schwarz inequalities, we obtain J2≤1 ε2/integraldisplay RN|Dkb|2+C/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dbl2 Hk +C/parenleftbig /ba∇dblb/ba∇dbl∞/ba∇dblb/ba∇dblHk/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dblHk+/ba∇dblb/ba∇dblHkk/summationdisplay j=1/ba∇dblDjbDk−j/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dblL2/parenrightbig , so that by virtue of Lemma 7.4, J2≤1 ε2/integraldisplay RN|Dkb|2+C/ba∇dbl(b,z)/ba∇dbl2 ∞/ba∇dbl(b,z)/ba∇dbl2 Hk. Step 6: estimate for J3. We have J3=−√ 2 ε/integraldisplay RNDkbDk/parenleftig b(√ 2 ε+b)/parenrightig =−2 ε2/integraldisplay RN|Dkb|2−√ 2 ε/integraldisplay RNDkbDk(b2), so, thanks to Cauchy-Schwarz inequality and Lemma 7.4, J3≤ −2 ε2/integraldisplay RN|Dkb|2+C ε/ba∇dblb/ba∇dbl∞/ba∇dblb/ba∇dbl2 Hk. Step 7: estimate for K. We readily obtain |K| ≤C(1+ε/ba∇dblb/ba∇dbl∞)/parenleftig/ba∇dblb/ba∇dbl∞ ε+/ba∇dblDz/ba∇dbl∞+/ba∇dblz/ba∇dbl2 ∞/parenrightig /ba∇dblz/ba∇dbl2 Hk. Gathering the previous steps we obtain d dtΓk(b,z)+κ 2Γk+1(b,z)+2κ ε2Γk(b,0) ≤C(1+ε/ba∇dblb/ba∇dbl∞)/parenleftig κ/parenleftbig /ba∇dbl(b,z)/ba∇dbl2 ∞+ε−1/ba∇dblb/ba∇dbl∞/parenrightbig +/ba∇dbl(∇b,Dz)/ba∇dbl∞/parenrightig /ba∇dbl(b,z)/ba∇dbl2 Hk,DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 5 holding for any 1 ≤k≤s. Following step by step the previous computations we readil y check that it also holds for k= 0. Finally, we have by assumption 1 2≤1+εb√ 2≤3 2on [0,T0]×RN, from which we infer that /ba∇dbl(b,z)/ba∇dbl2 Hk≤CΓk(b,z) for all 0 ≤k≤s. Therefore the proof of Proposition 2.2 is complete. /square 4.4.Proof of Proposition 2.3. Toshow thefirstinequality weaddtheinequalities obtained in Proposition 2.2 for kvarying from 1 to s. Since 1 /2≤1+εb/√ 2≤3/2, this yields d dt/ba∇dbl(b,z)/ba∇dbl2 Hs≤C(νε/ba∇dblb/ba∇dbl∞+κ/ba∇dbl(b,z)/ba∇dbl2 ∞+/ba∇dbl(Db,Dz)/ba∇dbl∞)/ba∇dbl(b,z)/ba∇dbl2 Hs ≤C/parenleftbig νε/ba∇dblb/ba∇dbl∞+(κ/ba∇dbl(b,z)/ba∇dblHs+1)/ba∇dbl(b,z)/ba∇dblHs/parenrightbig /ba∇dbl(b,z)/ba∇dbl2 Hs. After integrating on [0 ,T] and using Cauchy-Schwarz inequality this leads to /ba∇dbl(b,z)(T)/ba∇dbl2 Hs≤ /ba∇dbl(b,z)(0)/ba∇dbl2 Hs +C/ba∇dbl(b,z)/ba∇dblL∞ T(Hs)/parenleftig νε/ba∇dblb/ba∇dblL2 T(L∞)/ba∇dbl(b,z)/ba∇dblL2 T(Hs)+(κ/ba∇dbl(b,z)/ba∇dblL∞ T(Hs)+1)/ba∇dbl(b,z)/ba∇dbl2 L2 T(Hs)/parenrightig , for allT∈[0,T0]. Considering the supremum over T∈[0,t] and applying Young inequality in the right-hand-side we find the result. Finally the second inequality in Proposition 2.3 is obtaine d by integrating on [0 ,t] and using Sobolev and Cauchy-Schwarz inequalities. /square 5.Proof of Proposition 2.4. In this paragraph again, Crefers to a constant depending only on sandNand possibly changing from a line to another. First, we formulate System (2.3)-(2.4) with second members involving only bandz. By the same computations as those in Paragraph 4.2 we find ∂tb+√ 2 εdivv+2νε εb−κ∆b=f(b,z) ∂tv+√ 2 ε∇b−κ∆v−ε√ 2∇∆b=g(b,z),(5.1) wheref=˜fandg= ˜g−ε√ 2∇∆bare defined by f(b,z) =νε/parenleftbigg −1√ 2(1+ε√ 2b)|z|2−√ 2b2/parenrightbigg −div(bRez) g(b,z) =−κ∇(Rez·Imz)+ε√ 2∇div(bImz)−1 2∇Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht.(5.2) 5.1.Some notations and preliminary results. As in [2], we symmetrize System (5.1) by introducing the new functions c= (1−ε2 2∆)1/2b, d= (−∆)−1/2divv, and F= (1−ε2 2∆)1/2f, G= (−∆)−1/2divg.16 EVELYNE MIOT We remark that, knowing d, one can retrieve vsincevis a gradient. We have ∂tc+2νε εc−κ∆c+√ 2 ε(−∆)1/2(1−ε2 2∆)1/2d=F ∂td−κ∆d−√ 2 ε(−∆)1/2(1−ε2 2∆)1/2c=G.(5.3) In the following, we denote by ξ∈RNthe Fourier variable, by ˆfthe Fourier transform of fand byF−1the inverse Fourier transform. In view of the definition of ( c,d), it is useful to introduce the frequency threshold |ξ| ∼ε−1. More precisely, let us fix some R >0 and let χdenote the characteristic function on B(0,R). Forf∈L2(RN), we define the low and high frequencies parts of f fl=F−1/parenleftbig χ(εξ)ˆf/parenrightbig andfh=F−1/parenleftbig (1−χ(εξ))ˆf/parenrightbig , so that/hatwidefland/hatwidefhare supported in {|ξ| ≤Rε−1}and{|ξ| ≥Rε−1}respectively. Lemma 5.1. There exists C=C(s,N,R)>0such that the following holds for all 0≤m≤s andt∈[0,T0]: /ba∇dblg(t)/ba∇dblHm≈ /ba∇dblG(t)/ba∇dblHm,/ba∇dblfl(t)/ba∇dblHm≈ /ba∇dblFl(t)/ba∇dblHmand/ba∇dbl(ε∇f)h(t)/ba∇dblHm≈ /ba∇dblFh(t)/ba∇dblHm. In addition, /ba∇dblv(t)/ba∇dblHm≈ /ba∇dbld(t)/ba∇dblHm,/ba∇dblbl(t)/ba∇dblHm≈ /ba∇dblcl(t)/ba∇dblHmand/ba∇dbl(ε∇b)h(t)/ba∇dblHm≈ /ba∇dblch(t)/ba∇dblHm. Finally, /ba∇dbl(b,z)(t)/ba∇dblHm≈ /ba∇dbl(b,v)l(t)/ba∇dblHm+/ba∇dbl(ε∇b,v)h(t)/ba∇dblHm. Here we have set for f1,f2∈Hm /ba∇dblf1/ba∇dblHm≈ /ba∇dblf2/ba∇dblHmif and only if C−1/ba∇dblf1/ba∇dblHm≤ /ba∇dblf2/ba∇dblHm≤C/ba∇dblf1/ba∇dblHm. Proof.For the first two statements it suffices to consider the Fourier transforms of the func- tions and to use their support properties. The last statemen t is already established in [2], Lemma 1. /square Lemma 5.1 guarantees that for 0 ≤m≤s, /ba∇dbl(b,v)(t)/ba∇dblHm+ε/ba∇dblb(t)/ba∇dblHm+1≈ /ba∇dbl(b,z)(t)/ba∇dblHmand/ba∇dbl(b,z)(t)/ba∇dblHm≈ /ba∇dbl(c,d)(t)/ba∇dblHm,(5.4) therefore we have /ba∇dbl(c,d)(0)/ba∇dblHs≤CM0, whereM0is defined in Theorem 1.1. On the other side, when s−1> N/2, Sobolev embedding yields /ba∇dblbl(t)/ba∇dbl∞≤C/ba∇dblbl(t)/ba∇dblHs−1≤C/ba∇dblcl(t)/ba∇dblHs−1 and /ba∇dblbh(t)/ba∇dbl∞≤C/ba∇dblbh(t)/ba∇dblHs−1≤C/ba∇dbl(ε∇b)h(t)/ba∇dblHs−1≤C/ba∇dblch(t)/ba∇dblHs−1. Therefore it suffices to establish the first inequality of Prop osition 2.4 for /ba∇dbl(c,d)/ba∇dblL2 t(Hs)and the second inequality for /ba∇dblc/ba∇dblL2 t(Hs−1). Next, we have d dt/parenleftbiggˆc ˆd/parenrightbigg +M(ξ)/parenleftbiggˆc ˆd/parenrightbigg =/parenleftbiggˆF ˆG/parenrightbigg ,DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 7 where M(ξ) =νε ε 2+ε2|ξ|2|ξ| νε(2+ε2|ξ|2)1/2 −|ξ| νε(2+ε2|ξ|2)1/2ε2|ξ|2 . By Duhamel formula we have /hatwide(c,d)(t,ξ) =e−tM(ξ)/hatwide(c,d)(0,ξ)+/integraldisplayt 0e−(t−τ)M(ξ)/hatwider(F,G)(τ,ξ)dτ. Ournextresult, whichisprovedintheappendix,establishe spointwiseestimates for e−tM(ξ). Lemma 5.2. There exist positive numbers κ0,r,candCsuch that for all (a,b)∈C2, we have for 0< ε≤1,κ < κ0andt≥0 (1)If|ξ| ≤rνεthen /vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle≤Cexp(−νεε|ξ|2t)/bracketleftbigg exp/parenleftig −νε εt/parenrightig (|a|+|b|)+exp/parenleftbigg −c|ξ|2 νεεt/parenrightbigg (ν−1 ε|ξ||a|+|b|)/bracketrightbigg . (2)If|ξ| ≥rνεthen /vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle≤Cexp/parenleftbigg −νε(1+ε2|ξ|2) 2εt/parenrightbigg (|a|+|b|). Here for A= (a,b)∈C2we have set |A|=|a|+|b|. Lemma 5.2 reveals the new frequency threshold |ξ| ∼νε. We may choose R > r, so that rνε< Rε−1. We are therefore led to split the frequency space into three regions RN=R1∪R2∪R3, where • R1={|ξ| ≤rνε}denotesthelow frequenciesregion, inwhichthesemi-group iscomposed of a parabolic part (exp( −(νεε)−1|ξ|2t)), and a damping part (exp( −νεε−1t)). • R2={rνε≤ |ξ| ≤Rε−1}denotestheintermediatefrequenciesregion, inwhichthed amp- ing effect exp( −νεε−1t) is prevalent with respect to the parabolic contribution ex p(−νεε|ξ|2t). • R3={|ξ| ≥Rε−1}denotes the high frequencies region, in which the parabolic contribu- tion is strong and dominates the damping. With respect to this decomposition we introduce the small, i ntermediate and high frequen- cies parts of f∈L2(RN) as follows fs=F−1/parenleftbig χ|ξ|≤rνεˆf/parenrightbig , fm=F−1/parenleftbig χrνε≤|ξ|≤Rε−1ˆf/parenrightbig andfh=F−1/parenleftbig χ|ξ|≥Rε−1ˆf/parenrightbig , whereχEdenotes the characteristic function on the set E. Note that we have f=fs+fm+fh=fl+fh. 5.2.Proof of Proposition 2.4. We first introduce some notations. Let L(b,z)(t) =/ba∇dbl(1+εb(t))|z(t)|2/ba∇dblHs+/ba∇dblb2(t)/ba∇dblHs+/ba∇dblb(t)z(t)/ba∇dblHs+/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht(t)/ba∇dblHs. Next, we sort the terms in the definitions of f(b,z) andg(b,z) in System (5.2) as follows. We set f(b,z) =νεf0(b,z)+f1(b,z) and g(b,z) =g1(b,z)+εg2(b,z) =∇h0(b,z)+ε∇h1(b,z),18 EVELYNE MIOT where the subscript j= 0,1,2 denotes the order of the derivative, so that f0(b,z) =−1√ 2(1+ε√ 2b)|z|2−√ 2b2 f1(b,z) =−div(bRez) and g1(b,z) =−κ∇(Rez·Imz)−1/2∇Re/an}b∇acketle{tz,z/an}b∇acket∇i}ht=∇h0(b,z) g2(b,z) =1√ 2∇div(bImz) =∇h1(b,z). The proof of Proposition 2.4 relies on several lemmas which w e present now separately. Lemma 5.3. Under the assumptions of Proposition 2.4 we have for T∈[0,T0] C−1/ba∇dbl(c,d)s/ba∇dblL2 T(Hs)≤κ1/2max(1,ν−1 ε)M0+ε/ba∇dblL(b,z)/ba∇dblL2 T+κ1/2/ba∇dblL(b,z)/ba∇dblL1 T. Proof.By virtue of Lemma 5.2 we have |/hatwide(c,d)s(t,ξ)| ≤C(I(t,ξ)+J(t,ξ)), where I(t,ξ) =e−νε εt|/hatwide(c,d)s(0,ξ)|+/integraldisplayt 0e−νε ε(t−τ)|/hatwider(F,G)s(τ,ξ)|dτ and J(t,ξ) =e−c|ξ|2 νεεt/vextendsingle/vextendsingle(|ξ|ν−1 ε/hatwidecs(0),/hatwideds(0))/vextendsingle/vextendsingle+/integraldisplayt 0e−c|ξ|2 νεε(t−τ)/vextendsingle/vextendsingle(|ξ|ν−1 ε/hatwiderFs,/hatwiderGs)/vextendsingle/vextendsingledτ =JL(t,ξ)+JNL(t,ξ). We setˇI=F−1IandˇJ=F−1J, so that /ba∇dbl(c,d)s/ba∇dblL2 T(Hs)≤C(/ba∇dblˇI/ba∇dblL2 T(Hs)+/ba∇dblˇJ/ba∇dblL2 T(Hs)). First step : estimate for /ba∇dblˇI/ba∇dblL2 T(Hs). Invoking Lemma 7.3 we obtain /ba∇dblˇI/ba∇dblL2 T(Hs)≤C/parenleftbig (εν−1 ε)1/2/ba∇dbl(c,d)s(0)/ba∇dblHs+εν−1 ε/ba∇dbl(f,g)s/ba∇dblL2 T(Hs)/parenrightbig . Leth∈Hs. We observe that thanks to the support properties of /hatwidehs, we have /ba∇dblDkhs/ba∇dblHs≤Cνk ε/ba∇dblhs/ba∇dblHs, k∈N. Applying this inequality to the higher order derivatives f1,g1andg2, we see that /ba∇dbl(f,g)s(t)/ba∇dblHs≤C(νε+εν2 ε)L(b,z)(t)≤CνεL(b,z)(t), and we conclude that /ba∇dblˇI/ba∇dblL2 T(Hs)≤C/parenleftbig (εν−1 ε)1/2M0+ε/ba∇dblL(b,z)/ba∇dblL2 T/parenrightbig . (5.5) Second step : estimate for /ba∇dblˇJ/ba∇dblL2 T(Hs). We have /ba∇dblˇJ/ba∇dblL2 T(Hs)≤C(/ba∇dblˇJL/ba∇dblL2 T(Hs)+/ba∇dblˇJNL/ba∇dblL2 T(Hs)).DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 1 9 For the linear term we obtain /ba∇dblˇJL/ba∇dblL2 T(Hs)≤/vextenddouble/vextenddouble(1+|ξ|s)e−c|ξ|2 νεεt(|ξ|ν−1 ε|/hatwidecs(0)|+|/hatwideds(0)|)/vextenddouble/vextenddouble L2 T(L2) ≤C/vextenddouble/vextenddouble(1+|ξ|s)e−c|ξ|2 νεεt|ξ|(ν−1 ε|/hatwidecs(0)|+|ξ|−1|/hatwideds(0)|)/vextenddouble/vextenddouble L2 T(L2) ≤Cmax(1,ν−1 ε)/vextenddouble/vextenddouble(1+|ξ|s)e−c|ξ|2 νεεt|ξ|(|/hatwidecs(0)|+|/hatwiderϕs(0)|)/vextenddouble/vextenddouble L2 T(L2), becaused(0) =−2(−∆)1/2ϕ(0). By virtue of Lemma 7.1 in the appendix, this yields /ba∇dblˇJL/ba∇dblL2 T(Hs)≤Cmax(1,ν−1 ε)(ενε)1/2/parenleftbig /ba∇dblcs(0)/ba∇dblHs+/ba∇dblϕs(0)/ba∇dblHs/parenrightbig ≤Cmax(1,ν−1 ε)κ1/2M0. On the other side, Lemma 5.1 yields /ba∇dblˇJNL/ba∇dblL2 T(Hs)≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0(1+|ξ|s)e−c|ξ|2 νεε(t−τ)/parenleftbig |ξ|ν−1 ε|/hatwiderFs|+|/hatwiderGs|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2) ≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0(1+|ξ|)se−c|ξ|2 νεε(t−τ)/parenleftbig |ξ|ν−1 ε|/hatwidefs|+|/hatwidegs|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2). Inserting the expressions f=νεf0+f1andg=∇h0+ε∇h1we obtain /vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)(1+|ξ|s)/parenleftbig |ξ|ν−1 ε|/hatwidefs|+|/hatwidegs|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2) ≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)|ξ|2(1+|ξ|s)/parenleftbig ν−1 ε|ξ|−1|/hatwidef1|+ε|ξ|−1|/hatwiderh1|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2) +/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)|ξ|(1+|ξ|s)/parenleftbig |/hatwidef0|+|/hatwiderh0|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2). First, invoking Lemma 7.1, we find /vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)|ξ|2(1+|ξ|s)/parenleftbig ν−1 ε|ξ|−1|/hatwidef1|+ε|ξ|−1|/hatwiderh1|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2) ≤Cενε/ba∇dbl(1+|ξ|s)(ν−1 ε|ξ|−1/hatwidef1,ε|ξ|−1/hatwiderh1)/ba∇dblL2 T(L2) ≤Cενε(ν−1 ε+ε)/ba∇dbl(1+|ξ|s)(|/hatwidebRez|+|/hatwidebImz|)/ba∇dblL2 T(L2) ≤Cενε(ν−1 ε+ε)/ba∇dblb·z/ba∇dblL2 T(Hs) ≤Cε/ba∇dblL(b,z)/ba∇dblL2 T. Next, we infer from Lemma 7.2 in the appendix that /vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)|ξ|(1+|ξ|s)/parenleftbig |/hatwidef0|+|/hatwiderh0|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2) ≤C(ενε)1/2/ba∇dbl(1+|ξ|s)(|/hatwidef0|+|/hatwiderh0|)/ba∇dblL1 T(L2) ≤Cκ1/2/ba∇dblL(b,z)/ba∇dblL1 T. Gathering the previous steps and noticing that ( εν−1 ε)1/2≤κ1/2max(1,ν−1 ε), we conclude the proof of the lemma. /square Lemma 5.4. Under the assumptions of Proposition 2.4 we have for T∈[0,T0] C−1/parenleftig /ba∇dbl(c,d)m/ba∇dblL2 T(Hs)+/ba∇dbl(c,d)h/ba∇dblL2 T(Hs)/parenrightig ≤(εν−1 ε)1/2M0+(ε+ν−1 ε)/ba∇dblL(b,z)/ba∇dblL2 T.20 EVELYNE MIOT Proof.We divide the proof into several steps. First step : intermediate frequencies rνε≤ |ξ| ≤Rε−1. Another application of Lemma 5.2 yields |/hatwide(c,d)m(t,ξ)| ≤Ce−νε 2εt|/hatwide(c,d)m(0,ξ)|+C/integraldisplayt 0e−νε 2ε(t−τ)|/hatwider(F,G)m(τ,ξ)|dτ, whence, according to Lemma 7.3, /ba∇dbl(c,d)m/ba∇dblL2 T(Hs)≤C(εν−1 ε)1/2/ba∇dbl(c,d)(0)/ba∇dblHs+Cεν−1 ε/ba∇dbl(F,G)m/ba∇dblL2 T(Hs). Let us set (F,G)m=Am+Bm, whereAmandBm∈L2 T(Hs×Hs), to be determined later on, are such that /hatwidestAm(t,·) and /hatwiderBm(t,·) are compactly supported in/parenleftbig R1∪ R2={|ξ| ≤Rε−1}/parenrightbig2. Owing to these support properties we find /ba∇dbl(F,G)m/ba∇dblL2 T(Hs)≤ /ba∇dblAm/ba∇dblL2 T(Hs)+/ba∇dblBm/ba∇dblL2 T(Hs)≤C(ε−1/ba∇dblAm/ba∇dblL2 T(Hs−1)+ε−2/ba∇dblBm/ba∇dblL2 T(Hs−2)), so finally C−1/ba∇dbl(c,d)m/ba∇dblL2 T(Hs)≤(εν−1 ε)1/2M0+ν−1 ε/parenleftbig /ba∇dblAm/ba∇dblL2 T(Hs−1)+ε−1/ba∇dblBm/ba∇dblL2 T(Hs−2)/parenrightbig .(5.6) Second step : high frequencies |ξ| ≥Rε−1. For the high frequencies we neglect the contribution of the d ampinge−νε 2εtand only take the contribution of e−νεε|ξ|2tinto account. Exploiting again Lemma 5.2 we have |/hatwide(c,d)h(t,ξ)| ≤Ce−νεε|ξ|2t|/hatwide(c,d)h(0,ξ)|+C/integraldisplayt 0e−νεε|ξ|2(t−τ)|/hatwider(F,G)h(τ,ξ)|dτ ≤Cε|ξ|e−νεε|ξ|2t|/hatwide(c,d)h(0,ξ)|+C/integraldisplayt 0e−νεε|ξ|2(t−τ)|/hatwider(F,G)h(τ,ξ)|dτ, where the second inequality is due to the fact that 1 ≤Cε|ξ|on the support of /hatwide(c,d)h. By virtue of Lemma 7.1 we obtain /ba∇dbl(c,d)h/ba∇dblL2 T(Hs)≤C/parenleftbig (εν−1 ε)1/2/ba∇dbl(c,d)h(0)/ba∇dblHs+(νεε)−1/ba∇dbl(F,G)h/ba∇dblL2 T(Hs−2)/parenrightbig .(5.7) As in the first step, we set (F,G)h=Ah+Bh, whereAhandBh∈L2 T(Hs−1×Hs−1) will be set in such a way that /hatwiderAh(t,·) and/hatwiderBh(t,·) are supported in the region/parenleftbig R3={|ξ| ≥Rε−1}/parenrightbig2. Thanks to these support properties we can save one factor εto the detriment of one derivative : /ba∇dbl(F,G)h/ba∇dblL2 T(Hs−2)≤ /ba∇dblAh/ba∇dblL2 T(Hs−2)+/ba∇dblBh/ba∇dblL2 T(Hs−2)≤C(ε/ba∇dblAh/ba∇dblL2 T(Hs−1)+/ba∇dblBh/ba∇dblL2 T(Hs−2)). Therefore in view of (5.7) we are led to C−1/ba∇dbl(c,d)h/ba∇dblL2 T(Hs)≤(εν−1 ε)1/2M0+ν−1 ε/parenleftbig /ba∇dblAh/ba∇dblL2 T(Hs−1)+ε−1/ba∇dblBh/ba∇dblL2 T(Hs−2)/parenrightbig .(5.8) Third step . The last step consists in choosing suitable AandB. We recall that (F,G) = ((1−2−1ε2∆)1/2f,(−∆)1/2divg),DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 1 and f(b,z) =νεf0(b,z)+f1(b,z), g(b,z) =g1(b,z)+εg2(b,z). Now, for the intermediate frequencies we define /braceleftigg Am=/parenleftbig (1−2−1ε2∆)1/2fm,(−∆)−1/2div(g1)m/parenrightbig Bm=/parenleftbig 0,ε(−∆)−1/2div(g2)m/parenrightbig , and for the high frequencies /braceleftigg Ah=/parenleftbig νε(1−2−1ε2∆)1/2(f0)h,(−∆)−1/2div(g1)h/parenrightbig Bh=/parenleftbig (1−2−1ε2∆)1/2(f1)h,ε(−∆)−1/2div(g2)h/parenrightbig . ClearlyAm+Bm= (F,G)mandAh+Bh= (F,G)h. Moreover, we readily check that /ba∇dblAm/ba∇dblHs−1≈ /ba∇dbl(f,g1)m/ba∇dblHs−1and/ba∇dblAh/ba∇dblHs−1≈ /ba∇dbl(νεε∇f0,g1)h/ba∇dblHs−1(5.9) and /ba∇dblBm/ba∇dblHs−2≈ε/ba∇dbl(g2)m/ba∇dblHs−2and/ba∇dblBh/ba∇dblHs−2≈ /ba∇dbl(ε∇f1,εg2)h/ba∇dblHs−2.(5.10) On the one hand we have /ba∇dblg1/ba∇dblHs−1+/ba∇dblg2/ba∇dblHs−2≤C(/ba∇dblz·z/ba∇dblHs+/ba∇dblbImz/ba∇dblHs)≤CL(b,z). (5.11) On the other hand, the support properties of/hatwidest(f0)mimply that /ba∇dbl(f0)m/ba∇dblHs−1≤Cmin(1,ν−1 ε)/ba∇dbl(f0)m/ba∇dblHs, so that /ba∇dblfm/ba∇dblHs−1≤νε/ba∇dbl(f0)m/ba∇dblHs−1+/ba∇dbl(f1)m/ba∇dblHs−1≤C(/ba∇dbl(f0)m/ba∇dblHs+/ba∇dbl(f1)m/ba∇dblHs−1), and finally /ba∇dblfm/ba∇dblHs−1≤CL(b,z). (5.12) Arguing similarly we obtain νε/ba∇dbl(ε∇f0)h/ba∇dblHs−1≤Cνεε/ba∇dblf0/ba∇dblHs≤CL(b,z) (5.13) and /ba∇dbl(ε∇f1)h/ba∇dblHs−2≤ε/ba∇dblf1/ba∇dblHs−1≤CεL(b,z). (5.14) We infer from (5.9), (5.11), (5.12) and (5.13) that /ba∇dblAm/ba∇dblHs−1+/ba∇dblAh/ba∇dblHs−1≤CL(b,z). (5.15) Moreover (5.10), (5.11) and (5.14) yield /ba∇dblBm/ba∇dblHs−2+/ba∇dblBh/ba∇dblHs−2≤CεL(b,z), (5.16) so that the conclusion of Lemma 5.4 finally follows from (5.6) , (5.8), (5.15) and (5.16). /square Next, in order to establish the second part of Proposition 2. 4 involving the norm /ba∇dblb/ba∇dblL2(L∞), we show the following analogs of Lemmas 5.3 and 5.4 involving /ba∇dblc/ba∇dblL2(Hs−1). Lemma 5.5. Under the assumptions of Proposition 2.4 we have for T∈[0,T0] C−1/ba∇dblc/ba∇dblL2 T(Hs−1)≤(εν−1 ε)1/2M0+εmax(1,ν−1 ε)/ba∇dblL(b,z)/ba∇dblL2 T.22 EVELYNE MIOT Proof.We closely follow the proofs of Lemmas 5.3 and 5.4, handling a gain the regions R1, R2andR3separately. First step : low frequencies |ξ| ≤rνε. For low frequencies one may even improve the estimates given by Lemma 5.2 for the semi- group acting on c. Indeed, according to identity (7.1) stated in the proof of L emma 5.2, we get the bound |/hatwidecs(t,ξ)| ≤C(I(t,ξ)+J(t,ξ)), where I(t,ξ) =e−νε 2εt/vextendsingle/vextendsingle/hatwide(c,d)s(0,ξ)/vextendsingle/vextendsingle+/integraldisplayt 0e−νε 2ε(t−τ)|/hatwider(F,G)(τ,ξ)|dτ and J(t,ξ) =e−c|ξ|2 νεεt/vextendsingle/vextendsingle(|ξ|2ν−2 ε/hatwidecs,|ξ|ν−1 ε/hatwideds)(0)/vextendsingle/vextendsingle+/integraldisplayt 0e−c|ξ|2 νεε(t−τ)|(|ξ|2ν−2 ε/hatwiderFs,|ξ|ν−1 ε/hatwiderGs)|dτ =JL(t,ξ)+JNL(t,ξ). Here again we set ˇI=F−1IandˇJ=F−1J. In view of the first step in the proof of Lemma 5.3 (see (5.5)) we already know that /ba∇dblˇI/ba∇dblL2 T(Hs)≤C/parenleftbig (εν−1 ε)1/2M0+ε/ba∇dblL(b,z)/ba∇dblL2 T/parenrightbig . Next, since |ξ|ν−1 ε≤rwe have /ba∇dblˇJL/ba∇dblL2 T(Hs−1)≤/vextenddouble/vextenddoublee−c|ξ|2 νεεt(1+|ξ|s−1)/parenleftbig |ξ|2ν−2 ε|/hatwidecs(0)|+|ξ|ν−1 ε|/hatwideds(0)|/parenrightbig/vextenddouble/vextenddouble L2 T(L2) ≤Cν−1 ε/vextenddouble/vextenddoublee−c|ξ|2 νεεt|ξ|(1+|ξ|s−1)/parenleftbig |/hatwidecs(0)|+|/hatwideds(0)|/parenrightbig/vextenddouble/vextenddouble L2 T(L2) ≤Cν−1 ε(ενε)1/2M0, where the last inequality is a consequence of Lemma 7.1. On the other side we have /ba∇dblˇJNL/ba∇dblL2 T(Hs−1)≤C/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)(1+|ξ|s−1)/parenleftbig |ξ|2ν−2 ε|/hatwiderFs|+|ξ|ν−1 ε|/hatwiderGs|/parenrightbig dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2) ≤Cν−2 ε/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)|ξ|2(1+|ξ|s−1)|/hatwiderFs|dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2) +Cν−1 ε/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−c|ξ|2 νεε(t−τ)|ξ|2(1+|ξ|s−1)|ξ|−1|/hatwiderGs|dτ/vextenddouble/vextenddouble/vextenddouble L2 T(L2). Applying Lemma 7.1 to each term we obtain /ba∇dblˇJNL/ba∇dblL2 T(Hs−1)≤C(νεεν−2 ε/ba∇dblFs/ba∇dblL2 T(Hs−1)+νεεν−1 ε/ba∇dblD−1Gs/ba∇dblL2 T(Hs−1)) ≤C(εν−1 ε/ba∇dblfs/ba∇dblL2 T(Hs−1)+ε/ba∇dblD−1gs/ba∇dblL2 T(Hs−1)) ≤Cε/ba∇dblL(b,z)/ba∇dblL2 T. We have used the support properties of fsin the last inequality above. Finally, we gather the previous inequalities to find C−1/ba∇dblcs/ba∇dblL2 T(Hs−1)≤(εν−1 ε)1/2M0+ε/ba∇dblL(b,z)/ba∇dblL2 T. (5.17)DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 3 Second step : intermediate frequencies rνε≤ |ξ| ≤Rε−1. In contrast with the previous step, we may here imitate the fir st step of the proof of Lemma 5.4, estimating the Hs−1norm instead : /ba∇dblcm/ba∇dblL2 T(Hs−1)≤ /ba∇dbl(c,d)m/ba∇dblL2 T(Hs−1)≤C/parenleftbig (εν−1 ε)1/2/ba∇dbl(c,d)(0)/ba∇dblHs−1+εν−1 ε/ba∇dbl(F,G)m/ba∇dblL2 T(Hs−1)/parenrightbig . Recalling that ( F,G)m=Am+Bm, where/hatwidestAmand/hatwiderBmare compactly supported in the region {|ξ| ≤Rε−1}, we obtain /ba∇dbl(F,G)m/ba∇dblL2 T(Hs−1)≤ /ba∇dblAm/ba∇dblL2 T(Hs−1)+/ba∇dblBm/ba∇dblL2 T(Hs−1)≤ /ba∇dblAm/ba∇dblL2 T(Hs−1)+Cε−1/ba∇dblBm/ba∇dblL2 T(Hs−2). In view of the third step of the proof of Lemma 5.4 (see (5.15) a nd (5.16)) we get /ba∇dbl(F,G)m/ba∇dblHs−1≤CL(b,z) and we conclude that C−1/ba∇dblcm/ba∇dblL2 T(Hs−1)≤(εν−1 ε)1/2M0+εν−1 ε/ba∇dblL(b,z)/ba∇dblL2 T. (5.18) Third step : high frequencies |ξ| ≥Rε−1. With (F,G)h=Ah+Bhwe obtain, arguing exactly as in the second step of the proof o f Lemma 5.4, the analog of (5.7): /ba∇dbl(c,d)h/ba∇dblL2 T(Hs−1)≤C/parenleftbig (εν−1 ε)1/2M0+(ενε)−1/ba∇dbl(F,G)h/ba∇dblL2 T(Hs−3)/parenrightbig ≤C/parenleftbig (εν−1 ε)1/2M0+ν−1 ε/ba∇dbl(F,G)h/ba∇dblL2 T(Hs−2)/parenrightbig ≤C/parenleftbig (εν−1 ε)1/2M0+ν−1 εε(/ba∇dblAh/ba∇dblL2 T(Hs−1)+ε−1/ba∇dblBh/ba∇dblL2 T(Hs−2))/parenrightbig . Hence we infer from estimates (5.15) and (5.16) for AhandBhthat C−1/ba∇dblch/ba∇dblL2 T(Hs−1)≤(εν−1 ε)1/2M0+εν−1 ε/ba∇dblL(b,z)/ba∇dblL2 T. (5.19) The conclusion finally follows from estimates (5.17), (5.18 ) and (5.19). /square Invoking the previous results we may now complete the Proof of Proposition 2.4 . First, Cagliardo-Nirenberg inequality yields /ba∇dbl|z|2/ba∇dblHs+/ba∇dblb2/ba∇dblHs+/ba∇dblbz/ba∇dblHs+/ba∇dbl/an}b∇acketle{tz,z/an}b∇acket∇i}ht/ba∇dblHs≤C/ba∇dbl(b,z)/ba∇dbl∞/ba∇dbl(b,z)/ba∇dblHs and /ba∇dblεb|z|2/ba∇dblHs≤Cε/ba∇dbl(b,z)/ba∇dbl2 ∞/ba∇dbl(b,z)/ba∇dblHs, so that L(b,z)≤C(1+ε/ba∇dbl(b,z)/ba∇dbl∞)/ba∇dbl(b,z)/ba∇dbl∞/ba∇dbl(b,z)/ba∇dblHs. By Sobolev embedding and Cauchy-Schwarz inequality we obta in /ba∇dblL(b,z)/ba∇dblL2 T≤C/parenleftbig 1+ε/ba∇dbl(b,z)/ba∇dblL∞ T(Hs)/parenrightbig /ba∇dbl(b,z)/ba∇dblL∞ T(Hs)/ba∇dbl(b,z)/ba∇dblL2 T(Hs) and /ba∇dblL(b,z)/ba∇dblL1 T≤C/parenleftbig 1+ε/ba∇dbl(b,z)/ba∇dblL∞ T(Hs)/parenrightbig /ba∇dbl(b,z)/ba∇dbl2 L2 T(Hs). Proposition 2.4 finally follows from both estimates above to gether with Lemmas 5.3, 5.4 and 5.5. /square We conclude this section with a result that will be needed in t he course of the next section. We omit the proof, which is a straightforward adaptation of t he proof of Lemma 5.5.24 EVELYNE MIOT Proposition 5.1. Under the assumptions of Proposition 2.4 we have for all T∈[0,T0] C−1/ba∇dblc/ba∇dblL2 T(Hs)≤(εν−1 ε)1/2M0+ν−1 ε/ba∇dbl(b,z)/ba∇dblL∞ T(Hs)/ba∇dbl(b,z)/ba∇dblL2 T(Hs)(1+ε/ba∇dbl(b,z)/ba∇dblL∞ T(Hs)). 6.Proofs of Theorems 1.1 and 1.3. 6.1.Proof of Theorem 1.1. This paragraph is devoted to the proof of Theorem 1.1. Let Ψ0∈ W+Hs+1such that Ψ0=ρ0exp(iϕ0) =/parenleftbig 1+ε√ 2a0/parenrightbig1/2exp(iϕ0), where (a0,ϕ0) satisfies the assumptions of Theorem 1.1. Let Ψ ∈ W+C([0,T∗),Hs+1) denote the corresponding solution to (C ε) provided by Theorem 3.1. Withc(s,N) denoting a constant corresponding to the Sobolev embeddin gHs(RN)⊂ L∞(RN), we first assume that the constant K1(s,N) in Theorem 1.1 satisfies K1(s,N)>√ 2c(s,N). (6.1) Hence /ba∇dbl|Ψ0|2−1/ba∇dbl∞=ε√ 2/ba∇dbla0/ba∇dbl∞<1 2, so that the assumptions of Corollary 3.1 are satisfied. Let ( b,v) be the solution given by Corollary 3.1 on [0 ,T0), withT0≤T∗maximal. We introduce the following control function H(t) =/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)+/ba∇dbl(b,z)/ba∇dblL2 t(Hs) κ1/2max(1,ν−1ε)+/ba∇dblb/ba∇dblL2 t(L∞) (εν−1ε)1/2, H0=H(0).(6.2) Note that, according to (5.4) we have H0≤C1(s,N)M0and/ba∇dbl(b,v)(t)/ba∇dblHs+ε/ba∇dblb(t)/ba∇dblHs+1≤C1(s,N)H(t), where the constant C1(s,N) depends only on sandN. We recall that M0is defined in Theorem 1.1. Increasing possibly the number K1(s,N) introduced in Theorem 1.1, we may assume that C1(s,N)< K1(s,N). We define the stopping time Tε= sup{t∈[0,T0) such that H(t)< C2(s,N)M0}, whereC2(s,N) denotes a constant (to be specified later) satisfying C1(s,N)< C2(s,N)< K1(s,N). (6.3) We remark that Tε>0 by continuity of t/ma√sto→H(t). We next choose κ0(s,N) in such a way that κ0(s,N)C2(s,N)<K1(s,N)√ 2c(s,N). (6.4) By assumption on M0, this implies that for κ≤κ0(s,N) C2(s,N)M0<C2(s,N)νε K1(s,N)≤1√ 2c(s,N)ε.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 5 In particular, since /ba∇dbl(b,z)(t)/ba∇dblHs≤H(t), it follows that /ba∇dbl|Ψ|2(t)−1/ba∇dbl∞<1 2,∀t∈[0,Tε). (6.5) Our next purpose is to show that Tε=T0=T∗= +∞. First, mollifying possibly ( a0,u0) we may asume that ( b,z)∈C1([0,T0),Hs+1). By (6.5), Propositions 2.3 and 2.4 hold on [0 ,Tε), so that C(s,N)−1H≤H0+H2/parenleftig/parenleftbig 1+κH/parenrightbig κmax(1,ν−1 ε)2+/parenleftbig 1+εH/parenrightbig (κmax(1,ν−1 ε)2+ε+ν−1 ε)/parenrightig ≤H0+H2/parenleftbig 1+max( ε,κ)H/parenrightbig (κmax(1,ν−1 ε)2+ε+ν−1 ε). Observing that κmax(1,ν−1 ε)2+ε+ν−1 ε≤3(κ+ν−1 ε), we find C3(s,N)−1H≤H0+max(κ,ν−1 ε)H2/parenleftbig 1+max( ε,κ)H/parenrightbig . HereC3(s,N) is a constant depending only on sandN, which can be assumed to be larger than max( C1(s,N),1). On the other side, for t∈[0,Tε] we have according to (6.3) and by assumption on M0 max(ε,κ)H≤max(ε,κ)C2(s,N)M0≤1, so that H≤2C3(s,N)/parenleftig M0+max(κ,ν−1 ε)H2/parenrightig . (6.6) At this stage we may choose the constants C2(s,N) andK1(s,N) as follows: C2(s,N) = 4C3(s,N) and K1(s,N)>16C3(s,N)2max(√ 2c(s,N),1), so that all conditions (6.1), (6.3) and (6.4) are met. We now show that Tε=T0: otherwise Tεis finite. Hence, considering (6.6) at time Tεwe obtain 4C3(s,N)M0≤2C3(s,N)(M0+16max( κ,ν−1 ε)C3(s,N)2M2 0), whence 1≤16C3(s,N)2max(κ,ν−1 ε)M0≤16C3(s,N)2 K1(s,N). By definition of K1(s,N), this leads to a contradiction, therefore Tε=T0. Now, since (6.5) holds on [0 ,T0), Corollary 3.1 and a standard continuation argument imply thatT0=T∗. Invoking again (6.5) we easily show that /ba∇dbl∇Ψ(t)/ba∇dblHs≤C/parenleftbig 1+/ba∇dbl(b,v)(t)/ba∇dbl2 Hs+1×Hs/parenrightbig ,∀t∈[0,T∗) for a constant C. In view of the previous estimates we obtain limsup t→T∗/ba∇dbl∇Ψ(t)/ba∇dblHs≤limsup t→T∗C(1+H(t)2)<∞. We finally conclude that T∗= +∞thanks to Theorem 3.1. /square26 EVELYNE MIOT 6.2.Proof of Theorem 1.3. We present here the proof of Theorem 1.3. Here again, Calways stands for a constant depending only on sandN. We define ( bℓ,vℓ)(t,x) = (aℓ,uℓ)(ε−1t,x), where ( aℓ,uℓ) is the solution to the linear equation (1.6) with initial da tum (b0,v0) = (a0,u0). Introducing ( b,v) = (b−bℓ,v−vℓ), we have ∂tb+√ 2 εdivv+2νε εb−κ∆b=f(b,z) ∂tv+√ 2 ε∇b−κ∆v=g(b,z) +ε√ 2∇∆b. The proof of Theorem 1.3 relies on energy estimates, since th e method used in Section 5 is not convenient to establish uniform in time estimates. For 0 ≤k≤swe compute by integration by parts 1 2d dt/ba∇dbl(Dkb,Dkv)(t)/ba∇dbl2 L2=/integraldisplay RNDkbDk∂tb+Dkv·Dk∂tv =−2νε ε/integraldisplay RN|Dkb|2−κ/integraldisplay RN|∇Dkb|2−κ/integraldisplay RN|∇Dkv|2 +/integraldisplay RNDkbDkf(b,z)+/integraldisplay RNDkv·Dkg(b,z) +ε√ 2/integraldisplay RNDkv·Dk∇∆b. We recall the decompositions f=νεf0+f1andg=g1+εg2=∇h0+ε∇h1, where the fi,gi,hi,i= 0,1,2, which have been defined in Paragraph 5.2, are i-order derivatives of quadratic functions in ( b,z). We obtain 1 2d dt/ba∇dbl(Dkb,Dkv)(t)/ba∇dbl2 L2≤I+J+K, where I=−2νε ε/integraldisplay RN|Dkb|2+νε/integraldisplay RNDkbDkf0(b,z) J=/integraldisplay RNDkbDkf1(b,z)+/integraldisplay RNDkv·Dkg1(b,z), K=−κ/integraldisplay RN|∇Dkv|2+ε/integraldisplay RNDkv·Dkg2(b,z)+ε√ 2/integraldisplay RNDkv·Dk∇∆b. Estimates for IandJ. By virtue of Lemma 7.4 and by Sobolev embedding we find I≤ −νε ε/integraldisplay RN|Dkb|2+Cενε/integraldisplay RN|Dkf0|2≤Cκ/ba∇dblf0/ba∇dbl2 Hk≤Cκ/ba∇dbl(b,z)/ba∇dbl4 Hs. Next, Cauchy-Schwarz inequality yields J≤ /ba∇dbl(Dkb,Dkv)/ba∇dblL2/ba∇dbl(f1,g1)/ba∇dblHk≤C/ba∇dbl(Dkb,Dkv)/ba∇dblL2/ba∇dbl(b,z)/ba∇dbl2 Hk+1. Estimate for K. We perform an integration by parts in the last two integrals a nd insert the fact that g2= ∇h1to obtain K=−κ/integraldisplay RN|∇Dkv|2−ε/integraldisplay RNdivDkvDkh1−ε√ 2/integraldisplay RNdivDkvDk∆b ≤ −κ 4/integraldisplay RN|∇Dkv|2+Cε2 κ/integraldisplay RN|Dkh1|2+C κ/integraldisplay RN|ε∆Dkb|2.DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 7 First, by virtue of Cagliardo and Sobolev inequalities we ha ve /ba∇dblDkh1/ba∇dblL2≤C/ba∇dbl(b,z)/ba∇dbl∞/ba∇dbl(b,z)/ba∇dblHk+1≤C/ba∇dbl(b,z)/ba∇dblHs/ba∇dbl(b,z)/ba∇dblHk+1. Therefore: •If 0≤k≤s−2 we find K≤Cκ−1ε2/parenleftbig /ba∇dbl(b,z)/ba∇dbl4 Hs+/ba∇dblb/ba∇dbl2 Hs/parenrightbig . •Ifk=s−1 we observe that /ba∇dblε∆Dkb/ba∇dblL2≤C/ba∇dblc/ba∇dblHs,wherec= (1−ε2∆/2)1/2bis defined in the beginning of Section 5. So we find K≤Cκ−1/parenleftbig ε2/ba∇dbl(b,z)/ba∇dbl4 Hs+/ba∇dblc/ba∇dbl2 Hs/parenrightbig . •Ifk=s, similar arguments using that /ba∇dblε∆Dkb/ba∇dblL2≤C/ba∇dblc/ba∇dblHs+1≤C/ba∇dbl(b,z)/ba∇dblHs+1(see (5.4)) yield K≤Cκ−1/ba∇dbl(b,z)/ba∇dbl2 Hs+1/parenleftbig 1+ε2/ba∇dbl(b,z)/ba∇dbl2 Hs/parenrightbig . Integrating the previous estimates for I,JandKon [0,t] we find: •If 0≤k≤s−2, /ba∇dbl(Dkb,Dkv)(t)/ba∇dbl2 L2≤C/integraldisplayt 0/ba∇dbl(Dkb,Dkv)/ba∇dblL2/ba∇dbl(b,z)/ba∇dbl2 Hsdτ +C/integraldisplayt 0/parenleftig (κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl4 Hs+κ−1ε2/ba∇dbl(b,z)/ba∇dbl2 Hs/parenrightig dτ. Appyling Young inequality to the first term in the right-hand side we infer that C−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2 L∞ t(L2)≤ /ba∇dbl(b,z)/ba∇dbl4 L2 t(Hs) +(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs)/ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs)+κ−1ε2/ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs).(6.7) •Similarly, if k=s−1 we have C−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2 L∞ t(L2)≤ /ba∇dbl(b,z)/ba∇dbl4 L2 t(Hs) +(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs)/ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs)+κ−1/ba∇dblc/ba∇dbl2 L2 t(Hs).(6.8) •Ifk=sthen C−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2 L∞ t(L2)≤ /ba∇dbl(b,z)/ba∇dbl4 L2 t(Hs+1) +κ/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs)/ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs)+κ−1/parenleftbig 1+ε2/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs)/parenrightbig /ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs+1).(6.9) Proof of the uniform in time comparison estimates in Theorem 1.3. We control each term in the right-hand sides in (6.7), (6.8) a nd (6.9) by means of the various estimates established in the previous sections. We recall t hat the control function H(t), which is defined in (6.2), satisfies H(t)≤CM0. This controls the quantities /ba∇dbl(b,z)/ba∇dblL2 t(Hs) and/ba∇dbl(b,z)/ba∇dblL∞ t(Hs)in terms of M0. We use Proposition 5.1 to estimate /ba∇dblc/ba∇dblL2 t(Hs). Finally, to control/ba∇dbl(b,z)/ba∇dblL2 t(Hs+1)we rely on the second inequality in Proposition 2.3. Straigh tforward computations then lead to the uniform comparison estimates in Theorem 1.3. Proof of the time dependent comparison estimates in Theorem 1.3. We go back to the previous energy estimates. •If 0≤k≤s−2 we apply Cauchy-Schwarz inequality in (6.7) to obtain C−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2 L∞ t(L2)≤t/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs)/ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs) +t(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl4 L∞ t(Hs)+tκ−1ε2/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs).28 EVELYNE MIOT •Ifk=s−1 we similarly infer from (6.8) C−1/ba∇dbl(Dkb,Dkv)/ba∇dbl2 L∞ t(L2)≤t/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs)/ba∇dbl(b,z)/ba∇dbl2 L2 t(Hs) +t(κ+κ−1ε2)/ba∇dbl(b,z)/ba∇dbl4 L∞ t(Hs)+tκ−1/ba∇dbl(b,z)/ba∇dbl2 L∞ t(Hs). Using that H(t)≤CM0, the assumptions on M0as well as the fact that ( aε,uε)(t) = (bε,vε)(εt) we are led to the desired estimates. We omit the details. /square 7.Appendix. In this appendix we gather some helpful results. 7.1.Some parabolic estimates and useful tools. The following result is an immediate consequenceof maximal regularity fortheheat operator et∆. We referto[8] forfurtherdetails. Lemma 7.1. There exists C >0such that for all λ >0,a0∈L2(RN),a=a(s)∈L2(R+× RN)andT >0 /ba∇dbleλt∆a0/ba∇dblL2 T(˙H1)≤C√ λ/ba∇dbla0/ba∇dblL2 and /vextenddouble/vextenddouble/vextenddouble/vextenddouble∆/integraldisplayt 0eλ(t−s)∆a(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2 T(L2)≤C λ/ba∇dbla/ba∇dblL2 T(L2). We also have the following Lemma 7.2. There exists C >0such that for all λ >0andH∈L2(R+×RN) /vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0eλ(t−s)∆H(s)ds/vextenddouble/vextenddouble/vextenddouble L2 T(˙H1)≤C√ λ/integraldisplayT 0/ba∇dblH(t)/ba∇dblL2dt. Proof.We may assume that His smooth, compactly supported, and that the function u(t) =/integraltextt 0eλ(t−s)∆H(s)dsis the smooth solution to ∂tu−λ∆u=Handu(0) = 0. We infer that 1 2d dt/ba∇dblu(t)/ba∇dbl2 L2=/integraldisplay RNuH−λ/integraldisplay RN|∇u|2, so that λ/ba∇dbl∇u/ba∇dbl2 L2 T(L2)≤C/integraldisplayT 0/integraldisplay RN|u||H|dtdx≤Csup t∈[0,T]/ba∇dblu(t)/ba∇dblL2/ba∇dblH/ba∇dblL1 T(L2). Butu(0) = 0, therefore we also have /ba∇dblu(t)/ba∇dbl2 L2≤C/integraltextt 0/integraltext |uH|. This yields sup t∈[0,T]/ba∇dblu(t)/ba∇dblL2≤C/ba∇dblH/ba∇dblL1 T(L2) and the conclusion follows. /square Lemma 7.3. There exists C >0such that for all λ >0,a0∈L2,a∈L2(R+×RN)and T >0 /ba∇dble−λta0/ba∇dblL2 T≤C√ λ/ba∇dbla0/ba∇dblL2 and/vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−λ(t−s)a(s,·)ds/vextenddouble/vextenddouble/vextenddouble L2 T(L2)≤C λ/ba∇dbla/ba∇dblL2 T(L2).DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 2 9 Proof.We only establish the second estimate. We set ˜ a(s) =a(s) fors∈[0,T] and ˜a= 0 for s /∈[0,T], so that /vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−λ(t−s)a(s)ds/vextenddouble/vextenddouble/vextenddouble L2 T(L2)≤/vextenddouble/vextenddouble/vextenddouble/integraldisplayT 0e−λ(t−s)/ba∇dbl˜a(s)/ba∇dblL2(RN)ds/vextenddouble/vextenddouble/vextenddouble L2 T=/ba∇dble−λ·∗/ba∇dbl˜a(·)/ba∇dblL2(RN)/ba∇dblL2. By Young inequality for the convolution, we then have /vextenddouble/vextenddouble/vextenddouble/integraldisplayt 0e−λ(t−s)a(s)ds/vextenddouble/vextenddouble/vextenddouble L2 T(L2)≤C/ba∇dble−λ·/ba∇dblL1/ba∇dbl˜a/ba∇dblL2(R+,L2). We conclude by definition of ˜ a. /square We conclude this paragraph with the following result, which is a consequence of Gagliardo- Nirenberg inequality. Lemma 7.4 (see [2], Lemma 3) .Letk∈Nandj∈ {0,...,k}. There exists a constant C(k,N)such that /ba∇dblDjuDk−jv/ba∇dblL2≤C(k,N)/parenleftig /ba∇dblu/ba∇dbl∞/ba∇dblDkv/ba∇dblL2+/ba∇dblv/ba∇dbl∞/ba∇dblDku/ba∇dblL2/parenrightig and /ba∇dbluv/ba∇dblHk≤C(k,N)(/ba∇dblu/ba∇dbl∞/ba∇dblv/ba∇dblHk+/ba∇dblv/ba∇dbl∞/ba∇dblu/ba∇dblHk). 7.2.Proof of Lemma 5.2. In all the following Cdenotes a numerical constant. In order to simplify the notations we introduce the quantities ω=ε2|ξ|2andµ=1 νε|ξ|√ 2+ω, and we express Mas follows M=νε ε/parenleftbigg 2+ω µ −µ ω/parenrightbigg . First we compute the eigenvalues λ1andλ2ofM. Setting ∆ = 1−µ2, we have λ1=νε ε(ω+1−C√ ∆) and λ2=νε ε(ω+1+C√ ∆), whereC√ ∆is√ ∆if∆≥0andisi√ −∆if∆<0. Hence M=P−1DP, whereD= diag(λ1,λ2) and P−1=1 µ2−α2/parenleftbigg −µ α α−µ/parenrightbigg , P=/parenleftbigg −µ−α −α−µ/parenrightbigg ,withα= 1+C√ ∆. Finally for all ( a,b)∈C2we have e−tM/parenleftbigg a b/parenrightbigg =P−1e−tDP/parenleftbigg a b/parenrightbigg =1 µ2−α2/parenleftbigg (µ2a+αµb)e−λ1t−(α2a+αµb)e−λ2t (αµa+µ2b)e−λ2t−(αµa+α2b)e−λ1t/parenrightbigg =e−νε ε(1+ω)t µ2−α2/parenleftigg (µ2a+αµb)etνε εC√ ∆−(α2a+αµb)e−tνε εC√ ∆ (αµa+µ2b)e−tνε εC√ ∆−(αµa+α2b)etνε εC√ ∆/parenrightigg , or equivalently e−tM/parenleftbigg a b/parenrightbigg =e−νε ε(1+ω)t/bracketleftigg e−tνε εC√ ∆/parenleftbigg a b/parenrightbigg +etνε εC√ ∆−e−tνε εC√ ∆ µ2−α2/parenleftbigg αµb+µ2a −αµa−α2b/parenrightbigg/bracketrightigg .(7.1)30 EVELYNE MIOT First case |ξ|2≥3ν2 ε/8. Thenµ2≥3/4, hence ∆ ≤1/4. We need to examine the following subcases. •0≤∆≤1/4. It follows thatC√ ∆ =√ ∆ andµ2−α2=−2(∆+√ ∆), so that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleexp(tνε ε√ ∆)−exp(−tνε ε√ ∆) µ2−α2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sinh/parenleftig tνε ε√ ∆/parenrightig √ ∆≤Csinh/parenleftbiggνεt 2ε/parenrightbigg , where the second inequality is due to the fact that x/ma√sto→sinh(x)/xis an increasing function onR+. We infer that /vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig −νε 2εt/parenrightig exp/parenleftig −νεω εt/parenrightig/parenleftbig |a|+|b|/parenrightbig . (7.2) •−1≤∆<0. ThenC√ ∆ =i√ −∆ andµ2−α2=−2(∆+i√ −∆), therefore |µ2−α2|= 2/radicalbig ∆2−∆≥2√ −∆. It follows that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleexp(itνε ε√ −∆)−exp(−itνε ε√ −∆) µ2−α2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/vextendsingle/vextendsinglesin/parenleftbig tνε ε√ −∆/parenrightbig/vextendsingle/vextendsingle √ −∆≤Cνεt ε, where in the last inequality we have inserted that |sinx| ≤xfor allx≥0. Since |µ| ≤Cand |α| ≤Cthis yields /vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig −νε ε(1+ω)t/parenrightig/parenleftig 1+νε εt/parenrightig/parenleftbig |a|+|b|/parenrightbig , so finally/vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig −νε 2ε(1+ω)t/parenrightig/parenleftbig |a|+|b|/parenrightbig . (7.3) •∆≤ −1. We have |µ2−α2|= 2/radicalbig ∆2−∆≥2|∆| ≥Cµ2, while|α|=√ 1−∆ =µ. Hence we find /vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig −νε ε(1+ω)t/parenrightig/parenleftbig |a|+|b|/parenrightbig . (7.4) Second case |ξ|2≤3ν2 ε/8. We check that µ2≤3(2 + 3κ2/8)/8, therefore 1 /8≤∆≤1 whenever κ < κ 0=/radicalbig 8/9. Moreover C−1≤ |µ2−α2| ≤C, α≤C, µ≤Candµ≤C|ξ| νε. In addition, νε ε(−1+√ ∆) =−νε ε1−∆ 1+√ ∆=−νε εµ2 1+√ ∆≤ −Cνε εµ2. Therefore in view of (7.1) /vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig −νε ε(1+ω)t/parenrightig/parenleftbig |a|+|b|/parenrightbig +Cexp/parenleftig −νεω εt/parenrightig exp/parenleftbigg −Cνεµ2 εt/parenrightbigg/parenleftbigg|ξ| νε|a|+|b|/parenrightbigg .DAMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 3 1 Now, since C|ξ|2 ν2ε≥µ2=|ξ|2 ν2ε(2+ω)≥|ξ|2 ν2ε we obtain /vextendsingle/vextendsingle/vextendsinglee−tM(ξ)(a,b)/vextendsingle/vextendsingle/vextendsingle≤Cexp/parenleftig −νεω εt/parenrightig/parenleftbigg exp/parenleftig −νε εt/parenrightig +exp/parenleftbigg −C|ξ|2 νεεt/parenrightbigg/parenrightbigg/parenleftbigg|ξ| νε|a|+|b|/parenrightbigg .(7.5) Gathering estimates (7.2) to (7.5) and setting r=/radicalbig 3/8 we are led to the conclusion of the Lemma. /square Acknowlegments. I warmly thank Didier Smets for his help. This work was partly sup- ported by the grant JC05-51279 of the Agence Nationale de la R echerche. References [1] I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation , Rev. Mod. Phys. 74 (2002), 99-143. [2] F. Bethuel, R. Danchin and D. Smets, On the linear wave regime for the Gross-Pitaevskii equation , J. Anal. Math. , to appear. [3] A. Capella, C. Melcher and F. Otto, Wave-type dynamics in ferromagnetic thin films and the motio n of N´ eel walls , Nonlinearity 20(2007), 2519-2537. [4] C. Gallo, The Cauchy problem for defocusing nonlinear Schr¨ odinger e quations with non-vanishing initial data at infinity , Comm. Partial Differential Equations 33(2008), no. 4-6, 729-771. [5] P. G´ erard, The Cauchy problem for the Gross-Pitaevskii equation , Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 23(2006), no. 5, 765-779. [6] M. Kurzke, C. Melcher, R. Moser and D. Spirn, Dynamics of Ginzburg-Landau vortices in a mixed flow , Indiana Univ. Math. Jour. , to appear. [7] M. Kurzke, C. Melcher, R. Moser and D. Spirn, Ginzburg-Landau vortices driven by the Landau-Lifschitz- Gilbert equations , preprint. [8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uralceva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R.I. (1967), vol. 23. [9] E. Miot, Dynamics of vortices for the complex Ginzburg-Landau equat ion, Analysis and PDE 2(2009), no. 2, 159-186. (E. Miot) Dipartimento di Matematica G. Castelnuovo, Universit `a di Roma “La Sapienza”, Italy E-mail address :miot@ann.jussieu.fr

2010-03-28

We consider a complex Ginzburg-Landau equation, corresponding to a Gross-Pitaevskii equation with a small dissipation term. We study an asymptotic regime for long-wave perturbations of constant maps of modulus one. We show that such solutions never vanish and we derive a damped wave dynamics for the perturbation.

Damped wave dynamics for a complex Ginzburg-Landau equation with low dissipation

1003.5375v1

arXiv:1003.6092v1 [cond-mat.mes-hall] 31 Mar 2010APS/123-QED Magnonic Crystal with Two-Dimensional Periodicity as a Waveguide for Spin Waves Rakesh P. Tiwari and D. Stroud Department of Physics, Ohio State University, Columbus, OH 43210 (Dated: October 29, 2021) Abstract We describe a simple method of including dissipation in the s pin wave band structure of a periodic ferromagnetic composite, by solving the Landau-L ifshitz equation for the magnetization with the Gilbert damping term. We use this approach to calcul ate the band structure of square and triangular arrays of Ni nanocylinders embedded in an Fe h ost. The results show that there are certain bands and special directions in the Brillouin zone w here the spin wave lifetime is increased by more than an order of magnitude above its average value. Th us, it may be possible to generate spin waves in such composites decay especially slowly, and p ropagate especially large distances, for certain frequencies and directions in k-space. PACS numbers: 1Theexistenceofaperiodicsuperlatticestronglyaffectsmanytype sofexcitationsinsolids. For example, the electronic band structure of a conventional sem iconductor or semimetal[1], andthedispersionrelationsofelectromagneticwaves[2], elasticwav es[3–6], andspinwaves[7– 11] are all greatly influenced by a periodic superlattice potential. In many cases, such potentials can give rise to new, and even complete, electronic, phot onic, elastic, or magnonic band gaps which may have important implications for the properties o f these materials. These excitations have, by now, been extensively studied numerica lly and analytically, using a variety of methods, and have been probed in many experiments[12 –15]. Inthepresent paper, we consider aparticular classofsuch excita tions, namely, spinwaves in periodic magnetic materials. Such magnetic superlattices are ofte n called magnonic crys- tals. We go beyond previous work by calculating the spin wave lifetimes in such materials. Our most striking finding is that the “figure of merit” (FOM) of these spin waves (product of spin wave frequency and lifetime) is strongly dependent on the Blo ch wave vector k, even though, in our model, the same spin waves would have a k-independen t FOM in a homoge- neous magnetic material. This strong k-dependence suggests tha t magnetization in periodic magnetic materials may be transported most efficiently by spin waves propagating along special directions in k-space. Possibly this k-dependence could be t ested by experiments in which spin waves are launched in particular directions corresponding to the largest FOMs. This spin wave generation could be accomplished using real magnetic fi elds, or (via the spin torque effect[16]) using spin currents. Measurements of spin wave lifetimes might be carried out, e. g., by neutron spin-echo techniques/citebayrakci. Our calculations are carried out for an array of infinitely long circular cylinders made of a ferromagnetic material Aembedded in another infinite ferromagnetic material B. All the cylinders are taken to be parallel to the ˆ zaxis and their intersection with the xyplane forms a two-dimensional periodic lattice. We consider two arrangem ents of such cylinders: a triangular and a square superlattice. An external static magnetic fieldH0is applied parallel to the axis of the cylinders, and both ferromagnets are assumed t o be magnetized parallel toH0. TheequationofmotionforthisperiodiccompositeisgivenbytheLand au-Lifshitz-Gilbert (LLG) equation[18]: ∂ ∂tM(r,t) =γµ0M(r,t)×Heff(r,t)+α Ms(r)/parenleftbigg M(r,t)×∂ ∂tM(r,t)/parenrightbigg .(1) 2Hereγis the gyromagnetic ratio, which is assumed to be the same in both fer romagnets, Heffis the effective field acting on the magnetization M(r,t),ris the position vector, αis the Gilbert damping parameter and Msis the spontaneous magnetization. For this inho- mogenous composite Heffcan be written as Heff(r,t) =H0ˆz+h(r,t)+2 µ0Ms/parenleftbigg ∇·A Ms∇/parenrightbigg M(r,t), (2) whereh(r,t) is the dynamic dipolar field and Adenotes the exchange constant. The last term on the right-hand side of eq. (2) denotes the exchange fi eld. For the two- component composite we consider, the exchange constant, the s pontaneous magnetiza- tion and the Gilbert damping parameter take the forms A(r) =AB+ Θ(r)(AA−AB), Ms(r) =Ms,B+ Θ(r)(Ms,A−Ms,B), andα(r) =αB+ Θ(r)(αA−αB), where the step function Θ( r) = 1 ifris inside ferromagnet A, and Θ( r) = 0 otherwise. We separate the static and time-dependent parts of the magnetiz ation by writing M(r,t) =Msˆz+m(r,t), where m(r,t) =m(r)e−iωtis the time-dependent part of the magnetization. The time-dependent dipolar field h(r)e−iωt, whereh(r) =−∇Ψ(r) and Ψ(r) is the magnetostatic potential. Since ∇ ·(h(r) +m(r)) = 0, the magnetostatic potential Ψ(r) obeys∇2Ψ(r)−∇·m(r) = 0. Withinthelinear-magnonapproximation[19], thesmalltermsofsecon dorderin m(r)and h(r) are neglected in the equation of motion. This is equivalent to setting m(r)·ˆ z= 0[20]. Substituting the above equations into eqs. (1), we obtain iΩmx(r)+∇·[Q∇my(r)]−my(r)−Ms H0∂Ψ ∂y+iΩαmy(r) = 0, iΩmy(r)−∇·[Q∇mx(r)]+mx(r)+Ms H0∂Ψ ∂x−iΩαmx(r) = 0, (3) where Ω = ω/(|γ|µ0H0) andQ= 2A/(Msµ0H0). Next, usingtheperiodicityof Q,Msandαinthexyplane, wecanexpandthesequantities in Fourier series as Q(x)≡Q(x,y) =/summationtext GQ(G)eiG·x, with analogous expressions for Ms(x) andα(x). Here xandGare two-dimensional position and reciprocal lattice vectors in thexyplane. The vector r= (x,z), but none of the above quantities will have any z dependence for the composite we consider. The inverse Fourier tr ansforms are of the form Q(G) =1 S/integraltext /integraltext d2xQ(x)e−iG·x, whereSis the area of the unit cell; similar expressions hold forMs(G) andα(G). 3To calculate the band structure for spin waves propagating in the xyplane, we con- sider the two-dimensional Bloch vector, kand use Bloch’s theorem to write mx(x) = eik·x/summationtext Gmx,K(G)eiG·x,my(x) =eik·x/summationtext Gmy,K(G)eiG·x, and Ψ(x) =eik·x/summationtext GΨK(G)eiG·x. After some straightforward algebra, the equations of motion red uce to iΩ/summationdisplay G′˜A(G,G′) mx,K(G) my,K(G) =/summationdisplay G′˜M(G,G′) mx,K(G′) my,K(G′) ; (4) the 2×2 matrix ˜A(G,G′) = δGG′α(G−G′) −α(G−G′)δGG′, (5) whereδGG′is the Kronecker delta and the four components of the 2 ×2 matrix ˜M(G,G′) are given by ˜M(G,G′)xx=Ms(G−G′) H0(Kx+G′ x)(Ky+G′ y) (K+G′)2 ˜M(G,G′)xy=δGG′+Q(G−G′)(K+G)·(K+G′)+Ms(G−G′) H0(Ky+G′ y)2 (K+G′)2 ˜M(G,G′)yx=−δGG′−Q(G−G′)(K+G)·(K+G′)−Ms(G−G′) H0(Kx+G′ x)2 (K+G′)2 ˜M(G,G′)yy=−Ms(G−G′) H0(Kx+G′ x)(Ky+G′ y) (K+G′)2. (6) On left-multiplying eq. (4) by the inverse of the matrix ˜A, we reduce the band structure problem, including Gilbert damping, to that of finding the (complex) eig envalues of ˜A−1˜M. A similar plane wave expansion has been previously used to calculate th e magnonic band structure, for the case of zero damping, by several others (se e, e. g., Refs. [7] and [18]). We have used this formalism to calculate band structures for both a triangular Bravais lattice, with basis vectors a1=aˆx,a2=a/parenleftBig 1 2ˆx+√ 3 2ˆy/parenrightBig , and a square Bravais lattice, with a1=aˆx,a2=aˆy, whereais the edge of the magnonic crystal unit cell. Since Fourier transformsareavailableanalyticallyforcylinders ofcircular crosss ection, thebandstructure is easily calculated in this plane wave representation. In order to solve Eq. (4), we restrict the sum over G′to the first 625 reciprocal lattice vectors, which requires the diagonalization of a 1250 ×1250 complex matrix. The resulting eigenvalues of the matrix ˜B(G,G′) are all complex. For a given k, the imaginary part of the eigenvalue for gives the spin wave frequency, while the real part re presents the inverse spin 4wave lifetime. We have found that both the frequencies and lifetimes are well converged to within 0.1 % for this number of plane waves. For each eigenvalue, the figure of merit (FOM) mentioned above is th e ratio of the imaginary part to the real part of the eigenvalue. If the Gilbert dam ping parameters αA= αB, the FOM would be same for all k’s and all bands. By contrast, when αA/negationslash=αBwe find that the FOM varies from band to band and depends strongly on k. In particular, the FOM is particularly large in certain high symmetry directions. As a result, s pin waves will have a longer lifetime when they are launched at special kvalues and with special frequencies. We first consider the case of zero damping. In the left panel of Fig. 1, we plot the band structure of a composite of Fe cylinders arranged on a triangular la ttice and embedded in a Ni host, as calculated at an applied field µ0H0= 0.1T. The lattice constant a= 10 nm and the Fe filling fraction f= 0.5 (f is the area fraction occupied by the cylinders). The center-hand panel shows a similar composite, but for Fe cylinders a rranged on a square lattice, again with f= 0.5. The right-hand panel shows the Brillouin zones of the square and triangular lattices with symmetry points indicated. In calculating the band structure, we use an exchange constant and spontaneous magnetization at r oom temperature of 8.3 pJ/m and 1.71092 ×106A/m for Fe, and 3.4 pJ/m and 0.485423 ×106A/m for Ni[21]. We have not found band structures for exactly these materials in t he literature, but when we carry out analogous calculations for Co cylinders in a Permalloy mat rix (not shown), using the plane wave method, we obtain nearly identical results to th ose found by Vasseur et al[18], who also used a plane wave expansion. In Fig. 2, we show analogous calculations including damping for a squar e lattice. We use the same parameters, magnetic field, and value of fas in Fig. 1, except that the Gilbert damping parameters are αFe= 0.0019 and αNi=0.064, following Ref. [22]. In the left panel, the width of each cross-hatched region is proportional to the figu re of merit (FOM) for the given band and kvalue. The right panel shows the FOM for the fourth lowest spin wave band, as a function of magnonic crystal wave vector k, along specified directions in the superlattice (or magnonic crystal) Brillouin zone (SBZ), and at t hree different filling fractions f. The inset again shows the SBZ and symmetry points. We plot the firs t nine bands. The scales for the FOM and the real frequencies are differe nt, as indicated. In Fig. 3, we show the corresponding quantities for a triangular mag nonic crystal, again using a superlattice constant 10 nm and f= 0.5. The other parameters are the same as 5in Fig. 2, except that now the right hand panel shows the FOM for th e third lowest spin wave band. In Fig. 4, we show how the FOM for the optimal special sy mmetry points of Figs. 2 and 3 and bands depends on the superlattice filling fraction f. Note, in particular, that the FOM increases strongly near the close-packing values of ffor both the square and triangular lattices. The most striking feature of these plots is the strong dependence of the FOM on both k and band index. For example, in the square superlattice, the FOM is la rgest in the fourth band at the symmetry point M, and in the triangular superlattice, it is largest for the third band at K. The physics behind these strong maxima in the FOM is that, in both superlattices, the spin waves at these k-points propagate mainly through the Fe host, which isthelow-damping component. Thisresult suggests somepossible wa ys toincrease theFOM even further at these points: if we can arrange that a spin wave pr opagates entirely through the low-dissipation material, this should give an FOM close to the theor etical maximum, which is that of this material in its homogeneous form. Thus, a judicio us exploration of different periodic composites made of Fe and Ni, or other materials, c ould well lead to an even stronger dependence of spin lifetime on kvalue. We should add a few words of caution regarding the “spin waveguiding effect.” In prin- ciple, a measure of distance traveled by a propagating spin waves is g iven by the coherence length (or spin wave mean free path) lc[23]. This coherence length, for a given band nat wave vector k, is defined as lc(k,n) =|Vg kn|/γkn, whereVg knrepresents the groupvelocity and γknrepresents the imaginary part of the eigenfrequency, i. e., the inv erse lifetime. Since the group velocity may itself depend strongly on n and k, the behavior of lc(k,n) may be quite different from that of the lifetime. Nevertheless, we expect that lc(k,n), likeτ(k,n) and the FOMγkn, will depend strongly on both kand n, with sharp extrema near special symmetry points. Hence the waveguiding effect is likely to remain when one consid erslc(k,n) rather thanγkn. A full answer to this question would require a calculation of Vg knfor different k andn. Since single crystal Fe and Ni already have some intrinsic anisotropy , one might expect that this anisotropy could be exploited to obtain a strongly n and k-dependent FOM even in single crystals. However, in practice, most magnetic studies of Fe and Ni are carried out on polycrystalline samples, which no longer have this anisotropy. The present work provides a possible way of recovering this anisotropy, and even more, by use of a periodic lattice of 6inclusions. The present work can be generalized in various other ways. For exa mple, if a homoge- neous magnetic layer is perturbed by a periodic array of spin torque oscillators, this would generate an artificial magnetic superlattice, because the spin tor que would provide another contribution to Heff. Another possibility is to extend the present work to magnonic crys - talswith three-dimensional periodicity, thoughthismight beanexperimental challenge. The present work could conceivably have applications, e. g., in magnonic c ircuits which exploit the strong anisotropy in magnon lifetimes found in the present work . In summary, we have calculated the spin wave spectrum of a magnet ic superlattice with two-dimensional periodicity, including for the first time the effects o f dissipation. We find a striking anisotropy ofthespin wave figureofmerit, which fortypica l materials ismuch larger in certain bands near particular points of symmetry in the Brillouin zon e. This anisotropy implies that propagating spin waves will have much longer lifetimes at ce rtain frequencies and in certain directions in k-space , which could be interpreted as a w aveguiding effect for these excitations. We suggest that this anisotropy might be furth er increased with suitable tuning of the array parameters. Funding for this research was provided by the Center for Emergen t Materials at the Ohio State University, an NSF MRSEC (Award Number DMR-0820414). 7[1] R. Tsu, Superlattice to Nanoelectronics (Elsevier, Oxford, 2005). [2] E. Yablonovitch, J. Opt. Soc. Am. B 10, 283 (1993). [3] M. M. Sigalas and E. N. Economou, J. Sound Vib. 158,377 (1992). [4] M. M. Sigalas and E. N. Economou, Solid State Commun. 86, 141 (1993). [5] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari- Rouhani, Phys. Rev. Lett. 71, 2022 (1993). [6] M. S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, an d B. Djafari-Rouhani, Phys. Rev. B49, 2313 (1994). [7] M. Krawczyk and H. Puszkarski, Phys. Rev. B 77, 054437 (2008). [8] H. Puszkarski and M. Krawczyk, Solid State Phenom. 94, 125 (2003). [9] V. V. Kruglyak and R. J. Hicken, J. Magn. Magn. Mater. 306, 191 (2006). [10] S. A. Nikitov, P. Tailhades, and C. S. Tai, J. Magn. Magn. Mater.236, 320 (2001). [11] V. V. Kruglyak and A. N. Kuchko, Physica B 339, 130 (2003). [12] J. Cheon, J.-I. Park, J.-S. Choi, Y.-W. Jun, S. Kim, M. G. Kim, Y.-M. Kim, and Y. J. Kim, Proc. Natl. Acad. Sci. U.S.A. 103, 3023 (2006). [13] S. L. Vysotskii, S. A. Nikitov, and Yu. A. Filimonov, JET P101, 547 (2005). [14] Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Ja in, and A. O. Adeyeye, Appl. Phys. Lett. 94, 083112 (2009). [15] N. I. Polushkin, Phys. Rev. B 77, 180401(R) (2008). [16] See, e. g., S. I. Kisilev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003); S. Kaka. M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russak, and J. A. Katine, Nature 137, 389 (2005). [17] S. P. Bayrakci, T. Keller, K. Habicht, and B. Keimer, Sci ence312, 1926 (2006). [18] J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, and H . Puszkarski, Phys. Rev. B 541043 (1996). [19] M. G. Cottam and O. J. Lockwood, Light Scattering in Magnetic Solids (Wiley, New York, 1987). [20] M. Vohl, J. Barnas and P. Gr¨ unberg, Phys. Rev. B 39, 12003 (1989). [21] R. Skomski and D. J. Sellmyer, Handbook of Advanced Magnetic Materials, Nanostructural 8Effects, Vol. 1, edited by Yi Liu, D. J. Sellmyer, and Daisuke Shindo ( Springer, New York, 2006), p. 20. [22] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando and T. Miyazaki, Jpn. J. Appl. Phys. 45(2006), 3889. [23] M. P. Kostylev and A. A. Stashkevich, Phys. Rev. B 81, 054418 (2010). 9050100150200Re[iΩ]=Re[iω/γµ0H0] Γ MK Γ050100150Re[iΩ]=Re[iω/γµ0H0] Γ XM ΓXM Γ ΓΜK FIG. 1: (Color online) Left panel: band structure for a trian gular lattice of Fe cylinders in Ni, with lattice constant a= 10 nm, Fe filling fraction f= 0.5, and no Gilbert damping. Other parameters are given in the text. Center panel: same as left panel but for a square lattice. Right panel: Brillouin zone for square and triangular lattices with symm etry points indicated. 10050100150Re[iΩ]=Re[iω/γµ0H0] 0100200300400 FOMf=0.5 f=0.77 f=0.1 Γ XM Γ ΓMX Γ FIG. 2: (Color Online) Left panel: same as center panel of Fig . 1, but with Gilbert damping pa- rameters αFe= 0.0019 and αNi= 0.064. The widths of the cross-hatched regions are proportion al to the figure of merit (FOM) for the given band, as defined in the text. Right panel: FOM for the fourth lowest spin wave band, as a function of superlattice w ave vector k, along specified directions in the superlattice Brillouin zone (SBZ), and at three differe nt filling fractions f. 11050100150200Re[iΩ]=Re[iω/γµ0H0] 050100150200250300350400 FOMf=0.5 f=0.9 f=0.1 Γ MK Γ Γ MK Γ FIG. 3: (Color online.) Same as Fig. 2 but for a triangular lat tice of Fe cylinders in Ni, with lattice constant a= 10 and f= 0.5 (left panel) and f= 0.1, 0.5, and 0.9 (right panel). 120100200300400 FOM 0 0.2 0.4 0.6 0.8 filling fraction, fSquare superlattice Triangular superlattice FIG. 4: (Color Online) Same as Figs. 2 and 3, but showing the FO M as a function of filling fraction fforµ0H0= 0.1T,a= 10 nm. 13

2010-03-31

We describe a simple method of including dissipation in the spin wave band structure of a periodic ferromagnetic composite, by solving the Landau-Lifshitz equation for the magnetization with the Gilbert damping term. We use this approach to calculate the band structure of square and triangular arrays of Ni nanocylinders embedded in an Fe host. The results show that there are certain bands and special directions in the Brillouin zone where the spin wave lifetime is increased by more than an order of magnitude above its average value. Thus, it may be possible to generate spin waves in such composites decay especially slowly, and propagate especially large distances, for certain frequencies and directions in ${\bf k}$-space.

Magnonic Crystal with Two-Dimensional Periodicity as a Waveguide for Spin Waves

1003.6092v1

Concatenated quantum codes can attain the quantum Gilbert-Varshamov bound Yingkai Ouyang Department of Combinatorics and Optimization, Institute of Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada. y3ouyang@math.uwaterloo.ca Abstract A family of quantum codes of increasing block length with positive rate is asymptotically good if the ratio of its distance to its block length approaches a positive constant. The asymptotic quantum Gilbert-Varshamov (GV) bound states that there exist q-ary quantum codes of suciently long block length Nhaving xed rate Rwith distance at least NH1 q2((1R)=2), where Hq2is the q2-ary entropy function. For q < 7, only random quantum codes are known to asymptotically attain the quantum GV bound. However, random codes have little structure. In this paper, we generalize the classical result of Thommesen [1] to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specied feasible region. 1arXiv:1004.1127v6 [quant-ph] 16 Jan 2014I. INTRODUCTION A family of q-ary quantum codes [2] of increasing block length with positive rate is dened to be asymptotically good if the ratio of its distance to its block length approaches a positive constant. Designing good quantum codes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamov (GV) bound [3{8] is a lower bound on an achievable relative distance of a quantum code of a xed rate, and is attainable for various families of random quantum codes [3, 5, 7]. Explicit families of quantum codes, both unconcatenated [9, 10] and concatenated [11{14], have been studied, but do not attain the quantum GV bound forq <7 [15]. We show that concatenated quantum codes can attain the quantum GV bound. We are motivated by the historical development of the idea of concatenating a sequence of increasingly long classical Reed-Solomon (RS) outer codes with various types of classical inner codes. In both cases where the inner codes are all identical [16] or all distinct [17], the resultant sequence of concatenated codes while asymptotically good nonetheless fail to attain the GV bound. A special case of Thommesen's result [1] shows that even if the inner codes all have a rate of one, if they are chosen uniformly at random, the resultant sequence of concatenated codes almost surely attains the GV bound. Our work extends this classical observation to the quantum case. We show the quantum analog of Thommesen's result { the sequence of concatenated quantum codes with the outer code being a quantum generalized RS code [14, 18{20] and random inner stabilizer codes almost surely attains the quantum GV bound when the rates of the inner and outer codes lie in feasible region (III.1) with an example depicted in Figure 2. The property of the outer code that we need is that the normalizer of its stabilizer is classical maximal distance separable (MDS) code [20]. Our work is closest in spirit to that of Fujita [12], where quantum equivalents of the Zyablov and the Blokh-Zyablov bounds are obtained (not attaining the quantum GV bound) by choosing a quantum RS code with essentially random inner codes. In the proof of the classical result, Thommesen uses a random coding argument to compute the probability that any codeword of weight less than the target minimum distance belongs to the random code. Subsequently, he uses the union bound, the spectral property 2of the Reed-Solomon outer code, and properties of the q-ary entropy function (dened in II.1), to prove that the proposed random code almost surely does not contain any codeword of weight less than the prescribed minimum distance. The proof of our quantum result follows a similar strategy, with codewords replaced by elements of the normalizer not in the stabilizer. However the feasible region for the rates of the inner and outer codes for the classical and the quantum result are not analogous, because the monotonicity of the q-ary entropy function applies in a dierent feasible region from that of the classical case. The organization of this paper is as follows: Section II introduces the notation and preliminary material used in this paper. This section lays out the formalism of concatenating stabilizer codes, which is crucial to the proof of the main result. We state our main result in Theorem III.1 of Section III, and the remainder of the paper is dedicated to its proof. II. PRELIMINARIES LetL(Cq) denote the set of complex qbyqmatrices. Dene 1qto be a size qidentity matrix and !q:=e2i=qto be a primitive q-th root of unity, where q2 is an prime power. Dene 0 logq0:= 0. Dene the q-ary entropy function and its inverse to be Hq: [0;1]![0;1] andH1 q: [0;1]![0;q1 q] respectively where Hq(x):=xlogq(q1)xlogqx(1x) logq(1x): (II.1) Theq-ary entropy function is important here because it helps us to count the size of sets withqsymbols. The base- qlogarithm of the number of vectors from Fn qthat dier in at mostxncomponents from the zero-vector is dominated by nHq(x) asnbecomes large. For a ground set and n-tuples x2 n, denexjto bej-th element of the n-tuple x. Given tuples x2 nandy2 m, dene the pasting of the tuples xandyto be (xjy):= (x1;:::;xn;y1;:::;ym). WhenM1andM2are matrices with the same number of columns, dene ( M1;M2):=0 @M1 M21 A:For positive integer `, dene [`]:=f1;:::;`g. Dene the Hamming distance dH(x;y) between x2 nandy2 nas the number of indices on which xandydier. Dene the minimum distance of any subset C nto be mindist(C):= min x;y2CfdH(x;y) :x6=yg: 3A code over a vector eld Fn qisq-ary linear code of length nif it is a subspace of Fn q. An additive code is a subgroup of the eld under the eld addition operation. A classical q-ary linear code [16] of block length nandkgenerators with minimum distance of dis said to be an [n;k]qcode or an [ n;k;d ]qcode. A classical [ n;k;d ]qcode is maximally distance separated (MDS) if d=nk+ 1. A quantum q-ary stabilizer code [2] of block length n encodingkqudits is said to be an Jn;k Kqcode. The rates of an Jn;k Kqcode and an [ n;k]q code are both dened to bek n. A. Finite Fields and q-ary Error Bases We brie y review q-ary error bases [5]. Given a prime number p, letq=pkwherekis a positive integer. Let generalizations of the qubit Pauli matrices be X:=p1X j=0j(j+ 1) modpihjj Z:=p1X j=0(!p)jjjihjj (II.2) which satisfy the commutation property XaZb= (!p)abZbXafor non-negative integers a andb. We dene the matrix XaZb:=Xa1Zb1 ::: XakZbk (II.3) as a single qudit q-ary error basis element. We dene a q-ary error basis on a single qudit as the setEq:=fXaZb:a;b2Zk pg:Aq-ary error basis on nqudits is dened as E n qand its basis elements have the form Xa(1)Zb(1) ::: Xa(n)Zb(n)=X(a(1)j:::ja(n))Z(b(1)j:::jb(n)): Now lettbe any positive integer. Observe that for a;b;c;d2Zt p, the matrices XaZband XcZdsatisfy the commutation relation (XaZb)(XcZd) = (XcZd)(XaZb)(!p)Pt i=1aidibici: Hence the symplectic scalar product h(ajb);(cjd)is:=tX i=1aidibici=adTbcT 4quanties the commutation relation between the matrices XaZbandXcZd. When this scalar product is zero, we say that the vectors ( ajb) and ( cjd) ares-orthogonal, and the matrices XaZbandXcZdcommute under matrix mutiplication. We now elucidate the connection between q-ary error bases and nite elds. Dene the trace function from the eld FqtoFpto be Tr : x7!Pk1 i=0xpi. Also letf ; qgbe a basis of Fq2over Fq, where and qare the distinct roots of an irreducible degree-2 polynomial over Fq. Now let a:= (1;:::;k) and b:= (1;:::;k) be dual bases of Fq so that aTbis a sizekidentity matrix. Also let a;b;c, and dbe vectors from Zk p. Then Tr((aaT)(bbT)) = Tr( aaTbbT)) =abT;which implies that Tr((aaT)(dbT)(baT)(cbT)) =adTbcT: (II.4) Given the vectors xandyinFn q2, the Hermitian scalar product (see (28) of [5]) between x andyis hx;yih:=nX i=1(xi)qyi: When this Hermitian scalar product is zero, we say that xandyareh-orthogonal. This scalar product is called Hermitian because taking an element of Fq2to theq-th power is analogous to conjugation over the complex eld. For any subset CFn q2, we also dene its Hermitian dual to be C?h:=fy2Fn q2:hx;yih= 0;x2Cg. The following proposition shows that if two error basis elements are to commute, it suces for theirq2-ary nite eld counterparts to be h-orthogonal. Proposition II.1 ([5]).Letx;y2Fn q2, and suppose that hx;yih= 0. For alli2[n], letxi andyihave the decompositions xi=xi;1 +xi;2 q=a(i)aT +b(i)bT q; yi=yi;1 +yi;2 q=c(i)aT +d(i)bT q; wherexi;1;xi;2;yi;1;yi;22Fqand a(i);b(i);c(i);d(i)2Zk p. Then the matrices X(a(1)j:::ja(n))Z(b(1)j:::jb(n))andX(c(1)j:::jc(n))Z(d(1)j:::jd(n))from the setE n qcommute under matrix multiplication. Proof. Sincehx;yih= (hy;xih)qand 0q= 0, we havehx;yih= 0 implying that hy;xih= 0. 5W2E n q w2F2n q~w2Fn q2~'' FIG. 1: Equivalent representations of an n-qudit q-ary error basis element. Thushx;yihhy;xih= 0, which implies thatPn i=1xq iyiyq ixi= 0. Hence 0 =nX i=1((xi;1 q+xi;2 )(yi;1 +yi;2 q)(yi;1 q+yi;2 )(xi;1 +xi;2 q)) = ( 2)nX i=1(xi;1yi;2xi;2yi;1): (II.5) If = 2, then = qwhich is a contradiction. Hence 6= 2which implies that nX i=1(xi;1yi;2xi;2yi;1) = 0: Leta= (a(1)j:::ja(n)),b= (b(1)j:::jb(n)),c= (c(1)j:::jc(n)), and d= (d(1)j:::jd(n)). Tracing both sides of the above equation gives h(ajb);(cjd)is= 0, which implies that the matricesXaZbandXcZdcommute. In view of Proposition II.1 and (II.4), each element of a q-ary error basis over nqubits W=X(a(1)j:::ja(n))Z(b(1)j:::jb(n))can be represented by the codewords '(W):=w2F2n qand ~'(W):=~w2Fn q2, where for i2[n], wi=a(i)aT; wi+n=b(i)bT; ~wi=wi +wi+n q: We dene the map to take wto~w. Let the maps ;' and ~'act component-wise on sets and matrices. Consequently, elements of an error basis can be studied in their dierent nite eld representations, with the bijective maps ';~'and depicted in Figure 1. 6B. Stabilizer Codes Given a prime number p, letq=pkwherekis a positive integer. Given a subset S E n qwhere'(S) is an additive group with sindependent additive generators, the maximal subspace of ( Cq) nleft invariant under the action of all elements of Sis called an Jn;ns kKqstabilizer code. The sets S,'(S) and ~'(S) are the stabilizers of our stabilizer code in the matrix representation, the F2n q-representation and the Fn q2-representation respectively. We study stabilizer codes in the language of nite elds [3, 5]. Consider the full rank generator matrix G= (Gstb;Gx;Gz) over Fqwith (2kns) rows and 2ncolumns where the stabilizer generator Gstb= (s(1);:::;s(s)), the logical-X generator Gx= (x(1);:::;x(kns)), and the logical-Z generator Gz= (z(1);:::;z(kns)) are submatrices of G. We also require G= (Gstb;Gx;Gz) to have the properties: 1. Each row of Gstbiss-orthogonal to every row of G. 2. For alli;j2[kns],hx(i);z(i)is=i;j, wherei;jis the Kronecker delta. The error basis elements corresponding to the rows of GxandGzare generators for logical operations that can be applied on the stabilizer code. We denote the additive (not necessarily linear) classical codes generated by GstbandG under eld addition by CstbandCnrmrespectively. The set of all elements in F2n qthat are s-orthogonal to all elements in CstbisCnrm. The minimum distance of our stabilizer code is the minimum distance of the punctured code ~Cpnc:=fx2 (Cnrm) :x =2 (Cstb)g[5]. We denote an Jn;ns kKqstabilizer code with distance dasJn;ns k;dKq. The rate of the stabilizer code is 1 s knand its relative distance isd n. We dene a random Jn;ns kKqstabilizer code to be a stabilizer code corresponding to a generator matrix G= (Gstb;Gx;Gz) chosen uniformly at random from all possible generator matrices with (2 kns) rows and 2 ncolumns over the vector eld F2n q. Let the rates and relative distances of an innite code sequence of fJn;nrn;nnKqgn converge to the positive numbers randrespectively. If H1 q21r 2 ; (II.6) we say that the code sequence attains the asymptotic quantum q-ary GV bound. 7C. Concatenation of Stabilizer Codes Concatenation makes a longer code from an appropriately chosen set of shorter codes. We consider only the concatenation of stabilizer codes. Let q=pkwherepis prime. The quantum message that we wish to encode into a concatenated quantum code is aqK-dimension quantum state which is rst encoded into an JN;K Kqouter code . Let our JN;K Kqouter code be generated by G(out)= (G(out) stb;G(out) x;G(out) z). The outer code comprises of Nblocks of dimension qcomplex Euclidean spaces, with each of these N blocks further encoded as an Jn;k Kpinner code . Let thej-th Jn;k Kpinner code be generated byG(j)= (G(j) stb;G(j) x;G(j) z), withG(j) x= (x(j);1;:::;x(j);k) andG(j) z= (z(j);1;:::;z(j);k) for j2[N]. The resultant code is a concatenated code with parameters JnN;kK Kpgenerated byG(concat)= (G(concat) stb ;G(concat) x ;G(concat) z ). We now elucidate the construction of the generator of the concatenated code G(concat) using the generator of the outer code G(out)and the generators of the inner codes G(j)for j2[N]. Using the notation dened in Section II A, let the letter w2Fq2have the decomposition w=aaT +bbT qwhere a;b2Zk p. We dene the image of wover the smaller eld Fp2 with respect to the j-th inner code to be the (C(j) stb)-coset representative given by (j)(w):=kX `=1 a`x(j);`+b`z(j);`
: (II.7) Given vectors s2[N]mandw2Fm q2, we dene s(w):= ((s1)(w1)j:::j(sm)(wm)):As a shorthand we dene :=(1;:::;N ). Letalso act component-wise on both matrices and sets. Then the Fp2-representations of the stabilizer generator, the X-logical generator and the Z-logical generator of our concatenated code are given by (G(concat) stb ) =0 BBBBBB@( (G(out) stb));0 BBBBBB@ (G(1) stb)0 0 0 0 (G(2) stb)0 0 0 0...0 0 0 0 (G(N) stb)1 CCCCCCA1 CCCCCCA (G(concat) x ) =( (G(out) x)); (G(concat) z ) =( (G(out) z)) (II.8) 8respectively. The Fp2-representations of the stabilizer and the normalizer of the concatenated code are (C(concat) stb ):=( (C(out) stb)) + (C(1) stb::::C(N) stb) and (C(concat) nrm ):=( (C(out) nrm)) + (C(concat) stb ) respectively. In this paper, we use some of the q-ary quantum codes of Li, Xing and Wang [20] as the outer codes of our concatenated codes. The stabilizers and normalizers of these codes are classical MDS codes in the Fq2-representation, which is not necessarily the case for other quantum codes [19]. Theorem II.2 (Li, Xing, Wang [20] ) .LetNbe a prime power and Kbe an even integer in[0;N]such thatNK 2is also an integer. Then there exists a quantum generalized Reed- Solomon code with parameters JN;K;NK 2+ 1 KN. Moreover, the stabilizer (Cstb)and normalizer (Cnrm)of this code in the FN2-representation are classical generalized Reed- Solomon codes (are hence classical MDS codes), with (Cnrm) = (Cstb)?h. III. THE MAIN RESULT Our main result is that our sequence of concatenated p-ary quantum codes asymptotically attains the quantum GV bound. The outer code is a quantum generalized RS code with (Cnrm) = (Cstb)?hgiven by [20], and the inner codes are independently chosen random stabilizer codes. Theorem III.1 is our main result. Theorem III.1. Letr;R2Q\[0;1]be the rates of the inner and outer code respectively. Letpbe a prime number and nbe a positive integer such that rn,N=prn, and1R 2N2Z are also integers. Also suppose that R< min 12Hp2(1pr1);1 : (III.1) Let JnN;rRnN;d Kpbe a concatenated quantum code with a JN;RN KNouter code of given by Theorem II.2 concatenated with Nindependent and identically distributed random Jn;rn Kp inner quantum codes. Then with probability at least 11 p21p2N(1R 2), d nN>H1 p21rR 2 3c(p2;1+r 2) 2n wherec(p2;1+r 2)is a continuity constant as dened in the Appendix in equation (IV.4). 9Corollary III.2. Letpbe a prime and r;R2[0;1]such that the inequality (III.1) holds. For all positive integers n, letkn=dnre,Nn=pknandKn=Nn2d1R 2e. LetCnbe a code formed by concatenating an JNn;KnKNnouter code given by Theorem II.2 with Nn independent and identically distributed random Jn;knKpstabilizer codes. Then the code sequencefCngn2Z+asymptotically attains the quantum Gilbert-Varshamov bound. rFeasible Region: 0 ≤R < 1−2H4(1−2r−1)R 0.70.750.80.850.90.95 100.20.40.60.81 FIG. 2: When p= 2, the shaded region depicts the rates randRfor which Theorem III.1 applies. We proceed to introduce Proposition III.3 and Lemma III.4, which are used in the random coding aspects of the proof of Theorem III.1. Proposition III.3. Letwbe any nonzero element of Fn p2. Let (Cnrm)and (Cstb) be the normalizer and stabilizer over Fp2of a random Jn;k Kpstabilizer code, and 10let the corresponding punctured code be ~Cpnc:=fw2 (Cnrm) :w=2 (Cstb)g. Then Pr[w2~Cpnc]<p(n+k). Proof. LetUF2n pbe a set of independent mutually s-orthogonal vectors. Then the number of vectors in F2n pthat ares-orthogonal to all elements of Uisp2njUj. Hence Pr[w2 (Cnrm)] =Qnk1 i=0(p2n1ipi)Qnk1 i=0(p2nipi)<p(nk):The number of cosets of CstbinCnrmdistinct fromCstbisp2k1. Hence Pr[ w2~Cpnc]<p(nk)1 p2k1<pn+k. Lemma III.4. LetWbe any nonzero vector in FN q2of weightw, andhbe a positive integer no greater thanp21 p2nw. LetS=(W) + (C(1) stb:::C(N) stb)be a random coset. Then Pr [minwt(S)h]<(p2)nwHp2(h nw)p(n+k)w: Proof. The minimum weight of Sis equal to the minimum weight of the random coset S0=((W1;:::;Ww))+ (C(1) stb:::C(w) stb), whereW1;:::;Wware the nonzero letters of W. Whenhp21 p2nw, there are at most ( p2)nwHp2(h nw)members of Fn p2of weight no more than h(see [16]). Let w= (v1j:::jvw) be any such member of Fnw p2, where v1;:::;vw2Fn p2. If wis also an element of S0, each viis necessarily an element of the non-trivial random coset (i)(Wi) + (C(i) stb), the probability of which is less than p(n+k)by the Proposition III.3. Hence the probability that wis an element of the random set S0is less than p(n+k)w. Subsequently, applying the union bound on the number of wwith a weight no more than h gives the result. Now we proceed to prove our main result, Theorem III.1. Proof of Theorem III.1. To prove our main result, we have to nd a designed distance h>0 such that: 1. The probability that the distance of our concatenated quantum code is less that his negligible. 2. The designed relative distanceh nNasymptotically attains the quantum GV bound. We rst determine a sucient condition for Pr[ dh] to vanish as nbecomes large. Now our outer code's normalizer (C(out) nrm) is a classical MDS code [20] with parameters [N;NR nrm;D]q2whereD=N(1Rnrm) + 1 andRnrm:=1+R 2. The MDS property of 11our outer code's normalizer implies that the spectrum of the normalizer Aw, dened as the number of codewords in (C(out) nrm) with weight w2[D;N ], is at mostN w (p2k)wD+1(see the references [1, 16]). Let ~C(concat) pnc :=fW2 (C(concat) nrm ) :W=2 (C(concat) stb )g. Our upper bound on the spectrum Aw, the union bound and Lemma III.4 imply that Pr[dh] = Pr[minwt( ~C(concat) pnc )h] X W2 (C(out) nrm ) W6=0Prh minwt((W) + (C(1) stb):::C(N) stb))hi <NX w=D2N(p2k)wD0+1(p2)nwHp2(h nw)n+k 2w1X w=D(p2)nw; where :=N 2nwr 1D w+1 w Hp2h nw +1 +r 2: (III.2) Now let= 1D w+1 wand observe that 0 <R nrmfor our feasible values of w. If1 n for allw2[D;N ], then Pr[dh](p2)D 1 1p2:We will determine feasible values of the designed distance hfor which the inequality 1 nholds. Since the inverse entropy function is monotone increasing on the open unit interval, it suces to require that our choice of hsatises the inequality h nNw NH1 p21 +r 2rN 2nw1 n : (III.3) It suces to haveh nNequal to some lower bound on the right hand side of the inequality (III.3). Continuity of the inverse entropy (Lemma IV.1) and the substitutionw N=1Rnrm 1 gives 1Rnrm 1H1 p21 +r 2r1 nN 2w+ 1 1Rnrm 1H1 p21 +r 2r 1 2+w Nc(p2;1+r 2r) n: (III.4) The inequality (III.1) together with our restriction that r;R2[0;1] imply that r andRsatisfy the requirements of Lemma IV.1. Hence Lemma IV.1 implies that 1Rnrm 1H1 p21+r 2r
is a monotonic non-increasing function of . Sincec(p2;1+r 2r) nis also a monotonic non-increasing function of for feasible values of randR, the right hand side of (III.4) is at least H1 p21rR 2
3c(p2;1+r 2) 2nby settingto beRnrm. We seth nNto be this lower bound so that the inequality (III.3) holds, from which the result follows. 12IV. APPENDIX : THE Q-ARY ENTROPY AND ITS INVERSE In this section, we derive properties of the q-ary entropy function and its inverse. Since Hqis a strictly increasing concave function on (0 ;q1 q),H1 qis a strictly increasing convex function on the open interval (0 ;1). Observe that for x2(0;1), H0 q(x):=d dxHq(x) = logq(q1)logqx+ logq(1x); (IV.1) (1x)H0 q(1x) =Hq(1x) + logqx: (IV.2) SinceHq(y) is a continuously dierentiable function for y2(0;11 q), by the inverse function theorem, we have that (H1 q)0(y) =1 H0 q(H1 q(y))(IV.3) fory2(0;1), where ( H1 q)0(y):=d dyH1 q(y). These technical properties of the q-ary entropy function are used to obtain Lemma IV.1 which pertains to the monotonicity of 1 1H1 q1+r 2r
with respect to , and Lemma IV.2 which is about continuity. Now dene f:= 1H1 q(1+r 2r). Observe thatdf d=r(H1 q)0(1+r 2r) = r H0q(H1 q(1+r 2r))=r H0q(1f). We now introduce Lemma IV.1 which makes an assertion on the monotonicity of the function1f 1. Lemma IV.1. [Monotonicity] Let pbe prime,q=p2, andr;R2[0;1]such that (III.1) holds. Then1f 1is a non-increasing function with respect to 2[0;1+R 2]. Proof. Nowd d1f 1=1f (1)21 1df danddf d=r H0q(1f). Henced d1f 10 if and only if (1f)H0 q(1f)r(1). From (IV.2), we get (1f)H0 q(1f) =Hq(1f) + logqf=1 +r 2r + logqf: Thus (1f)H0 q(1f)r(1) holds if and only if r1 + 2 logqf, the latter inequality of which holds because of (III.1). Lemma IV.2. [Continuity] Let x;y2(0;q1 q)where the integer qis greater than 2 and x>y . ThenH1 q(y)H1 q(y)(xy)c(q;x);where our continuity constant is c(q;x):= logq(q1) + logq1 H1 q(x)11 : (IV.4) 13Proof. The convexity and continuous dierentiability of H1 qon the unit open interval imply thatH1 q(y)H1 q(x)(xy)(H1 q)0(x):Use of (IV.1) with (IV.3) then gives the result. [1] C. Thommesen, \The existence of binary linear concatenated codes with Reed-Solomon outer codes which asymptotically meet the Gilbert-Varshamov bound," IEEE Transactions on Information Theory , vol. 29, no. 6, pp. 850{853, 1983. [2] E. Rains, \Nonbinary quantum codes," IEEE Transactions on Information Theory , vol. 45, pp. 1827 {1832, Sep 1999. [3] D. Gottesman, Stabilizer Codes and Quantum Error Correction . PhD thesis, California Institute of Technology, 1997. quant-ph/9705052. [4] A. Ashikhmin, A. Barg, E. Knill, and S. Litsyn, \Quantum error detection .II. bounds," IEEE Transactions on Information Theory , vol. 46, pp. 789 {800, May 2000. [5] A. Ashikhmin and E. Knill, \Nonbinary quantum stabilizer codes," IEEE Transactions on Information Theory , vol. 47, pp. 3065 {3072, Nov 2001. [6] K. Feng and Z. Ma, \A nite Gilbert-Varshamov bound for pure stabilizer quantum codes," IEEE Transactions on Information Theory , vol. 50, no. 12, pp. 3323{3325, 2004. [7] Y. Ma, \The asymptotic probability distribution of the relative distance of additive quantum codes," Journal of Mathematical Analysis and Applications , vol. 340, pp. 550{557, 2008. [8] L. Jin and C. Xing, \Quantum Gilbert-Varshamov bound through symplectic self-orthogonal codes," in IEEE International Symposium on Information Theory Proceedings (ISIT) , pp. 455 {458, Aug 2011. [9] A. Ashikhmin, S. Litsyn, and M. A. Tsfasman, \Asymptotically good quantum codes," Phys. Rev. A , vol. 63, p. 032311, Feb 2001. [10] R. Matsumoto, \Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes," IEEE Transactions on Information Theory , vol. 48, pp. 2122 {2124, Jul 2002. [11] H. Chen, S. Ling, and C. Xing, \Asymptotically good quantum codes exceeding the Ashikhmin-Litsyn-Tsfasman bound," IEEE Transactions on Information Theory , vol. 47, pp. 2055 {2058, Jul 2001. 14[12] H. Fujita, \Several classes of concatenated quantum codes: Constructions and bounds," IEIC Technical Report (Institute of Electronics, Information and Communication Engineers) , vol. 105, no. 662, pp. 195{200, 2006. [13] M. Hamada, \Concatenated quantum codes constructible in polynomial time: Ecient decoding and error correction," IEEE Transactions on Information Theory , vol. 54, pp. 5689 {5704, Dec 2008. [14] Z. Li, L. Xing, and X. Wang, \A family of asymptotically good quantum codes based on code concatenation," IEEE Transactions on Information Theory , vol. 55, pp. 3821 {3824, Aug 2009. [15] A. Niehage, \Nonbinary quantum Goppa codes exceeding the quantum Gilbert-Varshamov bound," Quantum Information Processing , vol. 6, no. 3, pp. 143{158, 2007. [16] F. J. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes . North-Holland publishing company, rst ed., 1977. [17] J. Justesen, \Class of constructive asymptotically good algebraic codes," IEEE Transactions on Information Theory , vol. 18, pp. 652{656, Sep 1972. [18] M. Grassl, W. Geiselmann, and T. Beth, \Quantum Reed-Solomon codes," Proceedings Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-13), Springer Lecture Notes in Computer Science , p. 1719, 1999. [19] M. Grassl, T. Beth, and M. Roetteler, \On optimal quantum codes," International Journal of Quantum Information , vol. 2, no. 1, pp. 55{64, 2004. [20] Z. Li, L.-J. Xing, and X.-M. Wang, \Quantum generalized Reed-Solomon codes: Unied framework for quantum maximum-distance-separable codes," Phys. Rev. A , vol. 77, p. 012308, Jan 2008. 15

2010-04-07

A family of quantum codes of increasing block length with positive rate is asymptotically good if the ratio of its distance to its block length approaches a positive constant. The asymptotic quantum Gilbert-Varshamov (GV) bound states that there exist $q$-ary quantum codes of sufficiently long block length $N$ having fixed rate $R$ with distance at least $N H^{-1}_{q^2}((1-R)/2)$, where $H_{q^2}$ is the $q^2$-ary entropy function. For $q < 7$, only random quantum codes are known to asymptotically attain the quantum GV bound. However, random codes have little structure. In this paper, we generalize the classical result of Thommesen to the quantum case, thereby demonstrating the existence of concatenated quantum codes that can asymptotically attain the quantum GV bound. The outer codes are quantum generalized Reed-Solomon codes, and the inner codes are random independently chosen stabilizer codes, where the rates of the inner and outer codes lie in a specified feasible region.

Concatenated quantum codes can attain the quantum Gilbert-Varshamov bound

1004.1127v6

Limit theory for planar Gilbert tessellations Tomasz Schreiberand Natalia Soja, Faculty of Mathematics & Computer Science, Nicolaus Copernicus University, Toru n, Poland, e-mail: tomeks,natas at mat.umk.pl Abstract A Gilbert tessellation arises by letting linear segments (cracks) in R2unfold in time with constant speed, starting from a homogeneous Poisson point process of germs in randomly chosen directions. Whenever a growing edge hits an already existing one, it stops growing in this direction. The resulting process tessellates the plane. The purpose of the present paper is to establish law of large numbers, variance asymptotics and a central limit theorem for geometric functionals of such tessellations. The main tool applied is the stabilization theory for geometric functionals. keywords Gilbert crack tessellation, stabilizing geometric functionals, central limit the- orem, law of large numbers. MSC classication Primary: 60F05; Secondary: 60D05. 1 Introduction and main results LetXR2be a nite point set. Each x2X is independently marked with a unit length random vector ^ xmaking a uniformly distributed angle x2[0;) with thex-axis, which is referred to as the usual marking in the sequel. The collection X=f(x;x)gx2Xdetermines a crack growth process (tessellation) according to the following rules. Initially, at the time t= 0;the growth process consists of the points (seeds) in X. Subsequently, each point x2X gives rise to two segments growing linearly at constant unit rate in the directions of ^xand^xfromx:Thus, prior to any collisions, by the time t>0 the seed has developed into the edge with endpoints xt^xandx+t^x;consisting of two segments, say the upper one [x;x+t^x] and the lower one [x;xt^x]:Whenever a growing segment is blocked by Research supported by the Polish Minister of Science and Higher Education grant N N201 385234 (2008-2010) 1arXiv:1005.0023v1 [math.PR] 30 Apr 2010an existing edge, it stops growing in that direction, without aecting the behaviour of the second constituent segment though. Since the possible number of collisions is bounded, eventually we obtain a tessellation of the plane. The resulting random tessellation process is variously called the Gilbert model/tessellation, the crack growth process, the crack tessellation, and the random crack network, see e.g. [8, 12] and the references therein. LetG(X) denote the tessellation determined by X:We shall write +(x;X); x2X; for the total length covered by the upper segment emanating from xinG(X);and likewise we let(x;X) stand for the total length of the lower segment from x:Note that we use xfor marked version of x;according to our general convention of putting bars over marked objects. For future use we adopt the convention that if  xdoes not belong to X; we extend the denition of +=(x;X) by adding  xtoXand endowing it with a mark drawn according to the usual rules. Observe that for some xthe values of +=may be innite. However, in most cases in the sequel Xwill be a realization of the homogeneous Poisson point process P=Pof intensity  > 0 in growing windows of the plane. We shall use the so-called stabilization property of the functionals +and;as discussed in detail below, to show that the construction of G(X) above can be extended to the whole plane yielding a well dened process G(P);where, as usual, Pstands for a version of P marked as described above. This yields well dened and a.s. nite whole-plane functionals +(
;P) and(
;P): The conceptually somewhat similar growth process whereby seeds are the realization of a time marked Poisson point process in an expanding window of R2and which subse- quently grow radially in all directions until meeting another such growing seed, has received considerable attention [1, 3, 4, 5, 6, 11, 16], where it has been shown that the number of seeds satises a law of large numbers and central limit theorem as the window size in- creases. In this paper we wish to prove analogous limit results for natural functionals (total edge length, sum of power-weighted edge lengths, number of cracks with lengths ex- ceeding a given threshold etc.) of the crack tessellation process dened by Poisson points in expanding windows of R2:We will formulate this theory in terms of random measures keeping track not only the cumulative values of the afore-mentioned functionals but also their spatial proles. Another interesting class of model bearing conceptual resemblance to Gilbert tessella- tions are the so-called lilypond models which have recently attracted considerable attention [2, 7, 9, 10] and where the entire (rather than just directional) growth is blocked upon a collision of a growing object (a ball, a segment etc.) with another one. To proceed, consider a function : [R+[f+1g]2!Rwith at most polynomial growth, i.e. for some 0 <q< +1 (r1;r2) =O((r1+r2)q): (1) WithQ:= [0;p ]2standing for the square of area inR2;we consider the empirical measure  :=X x2P\Q +(x;P);(x;P)
x=p : (2) 2Thus, is a random (signed) measure on [0 ;1]2for all>0:The largeasymptotics of these measures is the principal object of study in this paper. Recalling that stands for the intensity ofP=P;we dene E() :=E +(0;P);(0;P)
: (3) The rst main result of this paper is the following law of large numbers Theorem 1 For any continuous function f: [0;1]2!Rwe have lim !11 Z [0;1]2fd =E()Z [0;1]2f(x)dx inLp; p>1: Note that this theorem can be interpreted as stating that E() is the asymptotic mass per point in ;since the expected cardinality of P\Qis:To characterize the second order asymptotics of random measures  we consider the pair-correlation functions c[x] :=E2 +(x;P);(x;P)
; x2R2(4) and c[x;y] :=E +(x;P[fyg);(x;P[fyg)

 +(y;P[fxg);(y;P[fxg)
[E()]2: (5) In fact, it easily follows by translation invariance that c[x] above does not depend on x whereasc[x;y] only depends on yx:In terms of these functions we dene the asymptotic variance per point V() =c[0] +Z R2c[0;x]dx: (6) Notice that in a special case when function (
;
) is homogeneous of degree k(i.e. forc2R we have(cr1;cr2) =ck(r1;r2)) one can simplify (3) and (6). Then the following remark is a direct consequence of standard scaling properties of Gilbert's tessellation construction and those of homogeneous Poisson point processes, whereby upon multiplying the intensity parameterby some factor we get all lengths in G(P) re-scaled by factor 1=2: Remark 1 For: [R+[f+1g]2!Rhomogeneous of degree kwe have E() =k=2E(1) V() =kV(1): (7) In other words, E(
)andV(
)are homogeneous of degree k=2andk, respectively. Our second theorem gives the variance asymptotics for  : 3Theorem 2 The integral in (6) converges and V()>0for all >0:Moreover, for each continuous f: [0;1]2!R lim !11 VarZ [0;1]2fd  =V()Z [0;1]2f2(x)dx: Our nal result is the central limit theorem Theorem 3 For each continuous f: [0;1]2!Rthe family of random variables 1p Z [0;1]2fd  >0 converges in law to N 0;V()R [0;1]2f2(x)dx as!1:Even more, we have sup t2RP8 >>< >>:R [0;1]2fd r VarhR [0;1]2fd i6t9 >>= >>;(t)6C(log)6 p (8) for all>1, whereCis a nite constant. Principal examples of functional where the above theory applies are 1.(l1;l2) =l1+l2:Then the total mass of  coincides with the total length of edges emitted in G(P) by points inP\Q:Clearly, the so-dened is homogeneous of order 1 and thus Remark 1 applies. 2. More generally, (l1;l2) = (l1+l2); 0:Again, the total mass of  is seen here to be the sum of power-weighted lengths of edges emitted in G(P) by points in P\Q:The so-dened is homogeneous of order : 3.(l1;l2) =1fl1+l2g;whereis some xed threshold parameter. In this set-up, the total mass of  is the number of edges in G(P) emitted from points in P\Qand of lengths exceeding threshold :This is not a homogeneous functional. The main tool used in our argument below is the concept of stabilization expressing in geometric terms the property of rapid decay of dependencies enjoyed by the functionals considered. The formal denition of this notion and the proof that it holds for Gilbert tessellations are given in Section 2 below. Next, in Section 3 the proofs of our Theorems 1, 2 and 3 are given. 42 Stabilization property for Gilbert tessellations 2.1 Concept of stabilization Consider a generic real-valued translation-invariant geometric functional dened on pairs (x;X) for nite point congurations XR2and withx2X:For notational convenience we extend this denition for x62X as well, by putting (x;X) :=(x;X[fxg) then. More generally, can also depend on i.i.d. marks attached to points of X;in which case the marked version of Xis denoted by X: For an input i.i.d. marked point process PonR2;in this paper always taken to be homogeneous Poisson of intensity ;we say that the functional stabilizes atx2R2on input Pi there exists an a.s. nite random variable R[x;P] with the property that (x;P\B(x;R[x;P])) =(x;(P\B(x;R[x;P]))[A) (9) for each nite AB(x;R[x;P])c;with Astanding for its marked version and with B(x;R) denoting ball of radius Rcentered at x:Note that here and henceforth we abuse the notation and refer to intersections of marked point sets with domains in the plane { these are to be understood as consisting of those marked points whose spatial locations fall into the domain considered. When (9) holds, we say that R[x;P] is a stabilization radius for Pat x:By translation invariance we see that if stabilizes at one point, it stabilizes at all points ofR2;in which case we say that stabilizes on (marked) point process P:In addition, we say thatstabilizes exponentially on input Pwith rateC > 0 i there exists a constant M > 0 such that PfR[x;P]>rg6MeCr(10) for allx2R2andr >0:Stabilizing functionals are ubiquitous in geometric probability, we refer the reader to [1, 13, 14, 15, 16, 17, 18, 19] for further details, where prominent ex- amples are discussed including random geometric graphs (nearest neighbor graphs, sphere of in uence graphs, Delaunay graphs), random sequential packing and variants thereof, Boolean models and functionals thereof, as well as many others. 2.2 Finite input Gilbert tessellations LetX  R2be a nite point set in the plane. As already mentioned in the intro- duction, each x2X is independently marked with a unit length random vector ^ x= [cos(x);sin(x)] making a uniformly distributed angle x2[0;) with thex-axis and the so marked conguration is denoted by X:In order to formally dene the Gilbert tessellation G(X) as already informally presented above, we consider an auxiliary partial tessellation mappingG(X) :R+!F (R2) whereF(R2) is the space of closed sets in R2and where, roughly speaking, G(X)(t) is to be interpreted as the portion of tessellation G(X);identi- ed with the set of its edges, constructed by the time tin the course of the construction sketched above. 5Figure 1 Finite input Gilbert tessellation. We proceed as follows. For each  x= (x;x)2Xat the time moment 0 the point xemits in directions ^ xand^xtwo segments, referred to as the  x+- and x-branches respectively. Each branch keeps growing with constant rate 1 in its xed direction until it meets on its way another branch already present, in which case we say it gets blocked , and it stops growing thereupon. The moment when this happen is called the collision time. Fort>0 byG(X)(t) we denote the union of all branches as grown by the time t: Note that, withX=fx1;::: ;xmg;the overall number of collisions admits a trivial bound given by the number of all intersection points of the family of straight lines ffxj+s^js2 Rg;j= 1;2;::: ;mgwhich ism(m1)=2. Thus, eventually there are no more collisions and all growth unfolds linearly. It is clear from the denition that G(X)(s)G(X)(t) for s < t: The limit set G(X)(+1) =S t2R+G(X)(t) is denoted by G(X) and referred to as theGilbert tessellation . Obviously, since the number of collisions is nite, the so-dened G(X) is a closed set arising as a nite union of (possibly innite) linear segments. For x2Xby+(x;X) we denote the length of the upper branch  x+emanating from xand, likewise, we write (x;X) for the length of the corresponding lower branch. For future reference it is convenient to consider for each x2X thebranch history functions x+(
);x(
) dened by requiring that  x+=(t) be the growth tip of the respective branch x+=at the time t2R+:Thus, prior to any collision in the system, we have just x+=(t) =x+=^xt;that is to say all branches grow linearly with their respective speeds +=^x:Next, when some  y+=; y2X gets blocked by some other  x+=; x2X at time t;i.e. y+=(t) = x+=(s) for somes6t;the blocked branch stops growing and its growth tip remains immobile ever since. Eventually, after all collisions have occured, the branches not yet blocked continue growing linearly to 1: 62.3 Stabilization for Gilbert tessellations We are now in a position to argue that the functionals +andarising in Gilbert tessellation are exponentially stabilizing on Poisson input P=Pwith i.i.d. marking according to the usual rules. The following is the main theorem of this subsection. Theorem 4 The functionals +andstabilize exponentially on input P: Before proceeding to the proof of Theorem 4 we formulate some auxiliary lemmas. Lemma 1 LetXbe a nite point set in R2and Xthe marked version thereof, according to the usual rules. Further, let y62X:Then for any t>0we have G(X)(t)4G(X[f yg)(t)B(y;t) with4standing for the symmetric dierence. Proof For a point setYR2andx2Y we will use the notation ( x;Y)+and (x;Y) to denote, respectively, the upper and lower branch outgrowing from  xinG(Y):Also, we use the standard extension of this notation for branch-history functions. Note rst that, by the construction of G(Y) and by the triangle inequality (x;Y)"(s0)2B(y;s0))8s>s0(x;Y)"(s)2B(y;s); s0>0;"2f 1;+1g: (11) This is a formal version of the obvious statement that, regardless of the collisions, each branch grows with speed at most one throughout its entire history. Next, writeX0=X[fygand
(t) =G(X)(t)4G(X0)(t) fort>0. Further, let t1< t2< t3< ::: < t nbe the joint collection of collision times for congurations Xand X0: Choose arbitrary p2
(t):Then there exist unique Y=Y(p)2fX;X0gandx2Yas well as"2f+;gwith the property that p= (x;Y)"(u) for some u6t:We also write Y0for the second element of fX;X0g;i.e.fY;Y0g=fX;X0g:With this notation, there is a uniquei=i(p) withtimarking the collision time in Y0where the branch ( x;Y0)"gets blocked inG(Y0);clearlyu>tithen and for s<tiwe have (x;Y)"(s)=2
(t). We should show that p2B(y;t). We proceed inductively with respect to i:Fori= 0 we havex=yandY=X0:Since (y;X0)"(0) =y2B(y;0), the observation (11) implies thatp= (y;X)"(u)2B(y;u)B(y;t). Further, consider the case i >0 and assume with no loss of generality that Y(p) =X, the argument in the converse case being fully symmetric. The fact that p2G(X)(t)4G(X0)(t) and that p= (x;X)"(u) implies the existence of a point z2X0such that a branch emitted from zdoes block  x"inG(X0) (by denition necessarily at the time ti) but does not block it in G(X). In particular, we see that ( z;X0)(s) = (x;X)"(ti) and (z;X0)(s0)2
(s0) for some ;s;s0such that 2f+;gands0< s6ti. By the inductive hypothesis we get ( z;X0)(s0)2B(y;s0). Using again observation (11) we conclude thus that ( x;X)"(ti) = (z;X0)(s)2B(y;s) and hencep= (x;X)"(u)2B(y;u)B(y;t). This shows that p2B(y;t) as required. Since pwas chosen arbitrary, this completes the proof of the lemma. 2 Our second auxiliary lemma is 7Lemma 2 For arbitrary nite point conguration XR2andx2Xwe have +(x;X) =+(x;X\B(x;2+(x;X))) (x;X) =(x;X\B(x;2(x;X))): (12) Proof We only show the rst equality in (12), the proof of the second one being fully analogous. Dene A(X;x) = XnB(x;2+(x;X)). Clearly, A(X;x) is nite and we will proceed by induction in its cardinality. IfjA(X;x)j= 0, our claim is trivial. Assume now that jA(X;x)j=nfor somen>1 and let y= (y;y)2A(X;x). Putt=+(x;X) and X0=Xnfyg. Applying Lemma 1 we see that G(X)(t)4G(X0)(t)B(y;t). We claim that +(x;X) =+(x;X0). Assume by contradiction that +(x;X)6=+(x;X0). Then for arbitrarily small  > 0 we have (G(X)(t)4G(X0)(t))\B(x;t+)6=;. On the other hand, since kxyk>2tasy =2B(x;2t); for"0>0 small enough we get B(x;t+0)\B(y;t) =;. Thus, we are led to ;6= (G(X)(t)4G(X0)(t))\B(x;t+"0)B(y;t)\B(x;t+"0) =; which is a contradiction. Consequently, we conclude that t=+(x;X) =+(x;X0) as required. SincejA(X0;x)j=n1, the inductive hypothesis yields +(x;X0) =+(x;X0\ B(x;2+(x;X0)) =+(x;X0\B(x;2t)). Moreover, X0\B(x;2t) = X\B(x;2t). Putting these together we obtain +(x;X) =+(x;X0) =+(x;X0\B(x;2t)) =+(x;X\B(x;2t)); which completes the proof. 2 In full analogy to Lemma 2 we obtain Lemma 3 For a nite point conguration XR2andx2X we have +(x;X) =+(x;X[ A1)and(x;X) =(x;X[ A2) for arbitrary A1B(x;2+(x;X))c,A2B(x;2(x;X))c. Combining Lemmas 2 and 3 we conclude Corollary 1 Assume that nite marked congurations XandYcoincide on B(x;2+(x;X)): Then +(x;X\B(x;2+(x;X))) =+(x;X) =+(x;Y): Analogous relations hold for : We are now ready to proceed with the proof of Theorem 4. 8Proof of Theorem 4 We are going to show that the functional +stabilizes expo- nentially on input process P:The corresponding statement for follows in full analogy. Consider auxiliary random variables + %; %> 0 given by + %=+(x;P\B(x;%)) which is clearly well dened in view of the a.s. niteness of P\B(x;%):We claim that there exist constants M;C > 0 such that for %>t>0 P(+ %>t)6MeCt: (13) Figure 2 Indeed, let %>0. Consider the branch  x+:= (x;P\B(x;%))+and planar regions Biand Di; i>1 along the branch as represented in gure 2. Say that the event Eioccurs i the regionBicontains exactly one point yofPand the angular mark ylies within (x+=2;x+=2 +); and there are no further points of Pfalling into Di; whereis chosen small enough so as to ensure that with probability one on Eithe branch x+does not extend past Bi;either getting blocked in Bior in an earlier stage of its growth, for instance = 0:01 will do. Let pstand for the common positive value of P(Ei); i>0: By standard properties of Poisson point process the events Eiare collectively independent. We conclude that, for N3n6%=3 P(+ %>3n)6P n\ i=1Ec i! = (1p)n 9which decays exponentially whence the desired relation (13) follows. Our next step is to dene a random variable R+=R+[x;P;] and to show it is a stabilization radius for +atxfor input process P:We shall also establish exponential decay of tails of R+:For%>0 we putR+ %= 2+ %:Further, we set ^ %= inffm2NjR+ m6mg: Since P(T m2NfR+ m>mg)6infm2NP(R+ m>m) which is 0 by (13), we readily conclude that so dened ^ %is a.s. nite. Take R+:=R+ ^%: (14) Then, using that by denition R+6^%;for any nite AB(x;R+)cwe get a.s. by Lemma 3 and Corollary 1 +(x;(P\B(x;R+))[A) =+(x;P\B(x;^%)\B(x;2+(x;P\B(x;^%)))[A) = =+(x;P\B(x;^%)\B(x;2+(x;P\B(x;^%)))) =+(x;(P\B(x;R+))): Thus,R+is a stabilization radius for +onPas required. Further, taking into account thatR+ k=R+for allk>^%by Corollary 1, we have for m2N P(R+>m) =P( lim k!1R+ k>m) = lim k!1P(R+ k>m) = = lim k!1P(+ k>m=2)6MeCm=2(15) whence the desired exponential stabilization follows. 2 Using the just proved stabilization property of +andwe can now dene +(x;P) =+(x;P\B(x;R+)) = lim %!1+(x;P\B(x;%)) =R+=2 (16) and likewise for :Clearly, the knowledge of these innite volume functionals allows us to dene the whole-plane Gilbert tessellation G(P): 3 Completing proofs Theorems 1,2 and 3 are now an easy consequence of the exponential stabilization Theorem 4. Indeed, observe rst that, by (1), (16) and (15) the geometric functional (x;X) :=(+(x;X);(x;X)) satises the p-th bounded moment condition [19, (4.6)] for all p >0:Hence, Theorem 1 follows by Theorem 4.1 in [19]. Further, Theorem 2 follows by Theorem 4.2 in [19]. Finally, Theorem 3 follows by Theorem 4.3 in [19] and Theorem 2.2 and Lemma 4.4 in [13]. Acknowledgements Tomasz Schreiber acknowledges support from the Polish Minister of Science and Higher Education grant N N201 385234 (2008-2010). He also wishes to express his gratitude to J.E. Yukich for helpful and inspiring discussions. 10References [1] Yu. Baryshnikov and J. E. Yukich, Gaussian limits for random measures in geometric probability. Annals Appl. Prob. 15, 1A (2005), pp. 213-253. [2] C. Cotar and S. Volkov, A note on the Lilypond model Adv. Appl. Prob. 36(2004), 325-339 [3] S. N. Chiu, A central limit theorem for linear Kolmogorov's birth-growth models. Stochastic Proc. and Applic. 66(1997), pp. 97-106. [4] S. N. Chiu and M. P. Quine, Central limit theory for the number of seeds in a growth model in Rdwith inhomogeneous Poisson arrivals. Annals of Appl. Prob. 7(1997), pp. 802-814. [5] S. N. Chiu and M. P. Quine , Central limit theorem for germination-growth models in Rdwith non-Poisson locations. Advances Appl. Prob. 33no. 4 (2001). [6] S. N. Chiu and H. Y. Lee, A regularity condition and strong limit theorems for linear birth growth processes. Math. Nachr. 241(2002), pp. 21 - 27. [7] D.J. Daley and G. Last, Descsnding chains, the lilypond model and mutual-nearest- neighbour matching. Adv. Appl. Probab. 37(2005), 604-628. [8] N.H. Gray, J.B. Anderson, J.D. Devine and J.M. Kwasnik, Topological Properties of Random Crack Networks , Mathematical Geology 8(1976), 617-626. [9] M. Heveling and G. Last, Existence, Uniqueness, and Algorithmics Computation of General Lilyponcd Systems Random Structures and Algorithms 29(2006), 338-350. [10] O. Haeggstroem and R. Meester, Nearest neighbour and hard sphere models in con- tinuum percolation , Random Structures and Algorithms 9(1996), 295-315. [11] L. Holst, M. P. Quine and J. Robinson, A general stochastic model for nucleation and linear growth. Annals Appl. Prob. 6(1996), pp. 903-921. [12] M.S. Makisack and R.E. Miles, Homogeneous rectangular tessellations. Adv. Appl. Probab. 28(1996), 993-1013. [13] M. D. Penrose, Gaussian limits for random geometric measures. European Journal of Probability 12(2007), pp. 989-1035. [14] M. D. Penrose, Laws of large numbers in stochastic geometry with statistical applica- tions. Bernoulli 13(2007), pp. 1124-1150. [15] M. D. Penrose and J. E. Yukich, Central limit theorems for some graphs in computa- tional geometry. Ann. Appl. Probab. 11(2001), pp. 1005-1041. 11[16] M. D. Penrose and J. E. Yukich, Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12(2002), pp. 272-301. [17] M. D. Penrose and J. E. Yukich, Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13(2004), pp. 277-303. [18] M. D. Penrose and J. E. Yukich, Normal approximation in geometric probability. In: Stein's Method and Applications, Lecture Note Series, Institute for Mathematical Sciences, National University of Singapore, 5, A. D. Barbour and Louis H. Y. Chen, Eds. (2005), pp. 37-58. [19] T. Schreiber, Limit theorems in stochastic geometry. In: New Perspectives in Stochas- tic Goemetry, W.S. Kendall and I. Molchanov, Eds., Oxford University Press, 2009, pp. 111-144. 12

2010-04-30

A Gilbert tessellation arises by letting linear segments (cracks) in the plane unfold in time with constant speed, starting from a homogeneous Poisson point process of germs in randomly chosen directions. Whenever a growing edge hits an already existing one, it stops growing in this direction. The resulting process tessellates the plane. The purpose of the present paper is to establish law of large numbers, variance asymptotics and a central limit theorem for geometric functionals of such tessellations. The main tool applied is the stabilization theory for geometric functionals.

Limit theory for planar Gilbert tessellations

1005.0023v1

Structural, static and dynamic magnetic properties of Co 2MnGe thin films on a sapphire a-plane substrate Mohamed Belmeguenai1, Fatih Zighem2, Thierry Chauveau1, Damien Faurie1, Yves Roussigné1, Salim Mourad Chérif1, Philippe Moch1, Kurt Westerholt3and Philippe Monod4 1LPMTM, Institut Galilée, UPR 9001 CNRS, Université Paris 13, 99 Avenue Jean-Baptiste Clément F-93430 Villetaneuse, France 2LLB (CEA CNRS UMR 12), Centre d’études de Saclay, 91191 Gif-Sur-Yvette, France 3Institut für Experimentalphysik/Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany and 4LPEM, UPR A0005 CNRS, ESPCI, 10 Rue Vauquelin, F-75231 Paris cedex 5, France Magnetic properties of Co 2MnGe thin films of different thicknesses (13, 34, 55, 83, 100 and 200 nm), grown by RF sputtering at 400C on single crystal sapphire substrates, were studied using vibrating sample magnetometry (VSM) and conventional or micro-strip line (MS) ferromagnetic resonance (FMR). Their behavior is described assuming a magnetic energy density showing twofold and fourfold in-plane anisotropies with some misalignment between their principal directions. For all the samples, the easy axis of the fourfold anisotropy is parallel to the c-axis of the substrate while the direction of the twofold anisotropy easy axis varies from sample to sample and seems to be strongly influenced by the growth conditions. Its direction is most probably monitored by the slight unavoidable miscut angle of the Al 2O3substrate. The twofold in-plane anisotropy field Hu is almost temperature independent, in contrast with the fourfold field H4which is a decreasing function of the temperature. Finally, we study the frequency dependence of the observed line-width of the resonant mode and we conclude to a typical Gilbert damping constant á value of 0.0065 for the 55-nm-thick film. PACS numbers: Keywords: I. Introduction Ferromagnetic Heusler half metals with full spin po- larization at the Fermi level are considered as potential candidates for injecting a spin-polarized current from a ferromagnetintoasemiconductorandfordevelopingsen- sitive spintronic devices [1]. Some Heusler alloys, like Co2MnGe, are especially promising for these applica- tions, due to their high Curie temperature (905 K) [2] and to their good lattice matching with some techno- logically important semiconductors [3]. Therefore, great attention was recently paid to this class of Heusler alloys [4-10]. In a previous work [11], we used conventional and micro-strip line (MS) ferromagnetic resonance (FMR), as well as Brillouin light scattering (BLS) to study magnetic properties of 34-nm-, 55-nm- and 83-nm-thick Co2MnGe films at room temperature. We showed that the in-plane anisotropy is described by the superposition of a twofold and of a fourfold term. The easy axes of the fourfold anisotropy were found parallel to the c-axis of the Al 2O3substrate (and, consequently, the hard axes lie at45ofc). The easy axes of the twofold anisotropy were found at 45ofcfor the 34-nm- and 55-nm-thick films and slightly misaligned with this ori- entation in the case of the 83-nm-thick sample. However, a detailed study of the in-plane anisotropy, involving temperature and thickness dependence, allowing for their physical interpretation is still missing. Therefore, it forms the aim of the present paper. Rather completex-rays diffraction (XRD) measurements over a large thickness range of Co 2MnGe films are reported below in an attempt to find correlations between in-plane anisotropies, thickness and crystallographic textures. The thickness- and the temperature-dependence of these anisotropies are investigated using vibrating sample magnetometry (VSM) and the above mentioned FMR techniques. In addition, we present intrinsic damping parametersdeducedfrombroadbandFMRdataobtained withthehelpofavectornetworkanalyzer(VNA)[12-14]. I. Sample properties and preparation Co2MnGe films with 13, 34, 55, 83, 100 and 200 nm thickness were grown on sapphire a-plane substrates (showing an in-plane c-axis) by RF-sputtering with a fi- nal 4 nm thick gold over layer. A more detailed descrip- tion of the sample preparation procedure can be found elsewhere [11, 15]. The static magnetic measurements were carried out at room temperature using a vibrating sample mag- netometer (VSM). The dynamic magnetic properties were investigated with the help of 9.5 GHz conven- tional FMR and of MS-FMR [11]. The conventional FMR set-up consists in a bipolar X-band Bruker ESR spectrometer equipped with a TE 102resonant cavity immersed is an Oxford cryostat, allowing for exploring the 4-300 K temperature interval. The MS-FMR set-up is home-made designed and, up to now, only works at room temperature. The resonance fields (conventionalarXiv:1005.4595v3 [cond-mat.mtrl-sci] 26 Aug 20102 FMR) and frequencies (MS-FMR) are obtained from a fit assuming a Lorentzian derivative shape of the recorded data. The experimental results are analyzed in the frame of the model presented in [11]. XRD experiments were performed using four circles diffractometers in Bragg-Brentano geometry in order to determine 2patterns and pole figures. The diffractometer devoted to the 2patterns was equipped with a point detector (providing a precision of 0:015in2scale). The instrument used for recording pole figures was equipped with an InelTMcurved linear detector ( 120aperture with a precision of 0:015in2 scale). The X-rays beams (Cobalt line focussource at = 1:78897) were emitted by a BrukerTMrotating an- ode. define a direct macroscopic ortho-normal reference (1,2,3), where the 3axis stands for the direction normal to the film. 'and are the so-called diffraction angles used for pole figure measurements. is the declination angle between the scattering vector and the 3-axis, ' is the rotational angle around the 3-axis. The 2 patterns (not shown here) indicate that, for all the Co2MnGe thin films, the <110>axis can be taken along the3-axis. The Co 2MnGe deduced lattice constant (a= 5:755is in good agreement with the previously published ones [6, 16]. Due to the [111] preferred orien- tation of the gold over layer along the 3-axis, only partial {110} pole figures could be efficiently exploited. They behave as {110} fiber textures containing well defined zones showing significantly higher intensities (Figure 1 (a) and (b)). These regions correspond to orientation variants which can be grouped into two families (see Figure 1). The first one, where the threefold [111]or the[111]axis is oriented along the crhombohedral direction, consists of two kinds of distinct domains with the [001] axis at 54:5from the c-axis. The second family, which is rotated around the 3-axis by 90from the first one, also contains two variants. This peculiar in-plane domain structure is presumably induced by the underlying vanadium seed layer. As illustrated in Figure 1b, which represents '-scans at = 60, we do not observe major differences between the crystallographic textures of the 55-nm and of the 100-nm-thick samples : the first family shows a concentration twice larger than the second one ; at least for the first family, which allows for quantitative evaluations, the concentrations of the two variants do not appreciably differ from each other; finally, about 50% of the total scattered intensity arises from domains belonging to these oriented parts of the scans. In the 200-nm-thick sample the anisotropy of the fiber is less marked but the two families remain present. III. Results and discussion 1- Static magnetic measurements In order to study the magnetic anisotropy at room temperature, the hysteresis loops were measured for all 0 25 50 75 100 125 150 175 0.0 0.2 0.4 0.6 0.8 1.0 (b)Normalized Intensity φ (degrees) 55 nm 100 nm a) 13 nm 100 nm 55 nm 200 nm Figure 1: (Color online) (a) Partial {110} X-rays pole figures (around 60) of 13, 55, 100 and 200-nm-thick films. (b) Dis- play of the angular variations of the intensity derived from the above figures for the 55 and 100-nm-thick samples (the blue and pink vertical dashed lines respectively refer to the two expected positions of the diffraction peak relative to the two variants belonging to family 1). the studied films with an in-plane applied magnetic field along various orientations as shown in Figure2 ( 'His the in-plane angle between the magnetic applied field Hand thec-axis of the substrate). The variations of the coer- cive field ( Hc) and of the reduced remanent magnetiza- tion ( Mr/Ms) were then investigated as function of 'H. Thetypicalbehaviorisillustratedbelowthroughtworep- resentative films which present different anisotropies. Figure 2a shows the loops along four orientations for the 100-nm-thick sample. One observes differences in shape of the normalized hysteresis loops depending upon the field orientation. For Halong c-axis ('H=0) we observe a typical easy axis square-shaped loop with a nearly full normalized remanence ( Mr/Ms=0.9), a co- ercive field of about 20 Oe and a saturation field of 100 Oe. As'Hincreases away from the c-axis, the coerciv-3 -100 -50 0 50 100 -1.0 -0.5 0.0 0.5 1.0 Normalized magnetization (M/M s) Applied magnetic field (Oe) ϕΗ=0° ϕΗ=45° ϕΗ=90° ϕΗ=135° 100 nm (a) -100 -50 0 50 100 -1.0 -0.5 0.0 0.5 1.0 (b) Normalized magnetization (M/M s) Applied magnetic field (Oe) ϕH=0° ϕH=45° ϕH=90° ϕH=135° 55 nm Figure 2: (Color online) VSM magnetization loops of the (a) 100-nm-thick and the (b) 55-nm-thick samples. The magnetic field is applied parallel to the film surface, at various angles ('H)with the c-axis of the sapphire substrate. ity increases and the hysteresis loop tends to transform into a hard axis loop. When 'Hslightly overpasses 90 (90< 'H<100) the loop evolves into a more com- plicated shape: it becomes composed of three (or two) open smaller loops. Further increasing the in-plane rota- tion angle, it changes from such a split-open curve up to an almost rectangular shape. The results for 'H= 45 and'H= 135are different: they show a rounded loop withMr=Msequal to 0.75 and 0.63 and with saturation fields of about 170 Oe and 200 Oe, respectively. This result qualitatively agrees with a description of the in- plane anisotropy in terms of four-fold and two-fold con- tributions with slightly misaligned easy axes. The variations of HcandMr=Msversus'Hare il- lustrated in Figures 3a and 3b for the 100-nm-thick film. The presence of a fourfold anisotropy contribution is sup- ported by the behavior of Hc(Figure 3a), since two min-ima appear within each period ( 180, as expected), as shown in Figure 3a. The minimum minimorum is mainly related to the uniaxial anisotropy term. In the same way, asdisplayedinFigure3b,thebehaviorof Mr/Msisdom- inated by the uniaxial anisotropy. It is worth to notice that the minimum minimorum position slightly differs from 90(lying around 96), thus arguing for a misalign- ment between the twofold and the fourfold anisotropy axes. Figure 2b shows a series of hysteresis loops, recorded withanin-planeappliedfield, forthe55-nm-thickfilm. A careful examination suggests that the fourfold anisotropy contribution is the dominant one and that the related easy axis lies along c-axis. The Mr/Msvariation versus 'H, reported in Figure 3c, is consistent with an easy uniaxial axis oriented at 45of this last direction. Both fourfold and uniaxial terms are smaller than for the 100- nm-thick sample. 2- Dynamic magnetic properties As previously published [11], the dynamic properties are tentatively interpreted assuming a magnetic energy density which, in addition to Zeeman, demagnetizing and exchange terms, is characterized by the following anisotropy contribution: Eanis: =K?sin2M1 2(1 +cos(2('M 'u))Kusin2M1 8(3 + cos 4( 'M'4))K4sin4M (1) In the above expression, Mand'Mrespectively represent the out-of-plane and the in-plane (referring to the c-axis of the substrate) angles defining the direction of the magnetization Ms;'uand'4stand for the angles of the uniaxial axis and of the easy fourfold axis, respectively, with this c-axis. With these definitions KuandK4are necessarily positive. As done in ref. [11], it is often convenient to introduce the effective magnetization 4Meff= 4Ms2K=Ms, the uniaxial in-plane anisotropy field Hu= 2Ku=Ms and the fourfold in-plane anisotropy field H4= 4K4=Ms. For an in-plane applied magnetic field H, the studied model provides the following expression for the frequen- cies of the experimentally observable magnetic modes: Fn:= 2(Hcos('H'M) +2K4 Mscos 4('M'4) + 2Ku Mscos 2('M'u) +2Aex: Msn d
2) (Hcos('H'M)+4Meff+K4 2Ms(3+cos 4('M'4))+ Ku Ms(1 + cos 2('M'u)) +2Aex: Ms(n d)2)(2) In the above expression is the gyromagnetic factor: ( =2) =g1:397106Hz/Oe. The uniform mode corresponds to n=0. The other modes to be considered (perpendicular standing modes) are connected to integer values of n: their frequencies depend upon the exchange4 0 50 100 150 200 250 300 3501520253035 VSM measurements (a)Coercive field (Oe) ϕΗ (degrees)100 nm 0 50 100 150 200 250 300 350 0.2 0.4 0.6 0.8 (b) Reduced remanent magnetization (M r/M s) ϕΗ(degrees) VSM measurements 100 nm 0 50 100 150 200 250 300 350 0.75 0.80 0.85 0.90 0.95 Reduced remanent magnetization (M r/M s) ϕΗ(degrees) VSM measurements 55 nm (c) Figure 3: (a) Coercive field and (b) reduced remanent mag- netization of the 100-nm-thick sample as a function of the in-plane field orientation ( 'H). (c) Reduced remanent mag- netization of the 55-nm-thick-film stiffness constant Aexand upon the film thickness d. For all the films the magnetic parameters at room temperature were derived from MS-FMR measurements. The deduced gfactor is equal to 2.17, as previously published [11]. The in-plane MS-FMR spectrum of the 100 nm-thick 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 -400 -300 -200 -100 0100 200 300 Amplitude (arb. units) Frequency (GHz) (a) H=520 Oe ϕΗ=0° Mode 2 Mode 1 0 300 600 900 1200 1500 246810 12 14 (b) Frequency (GHz) Applied magnetic field (Oe) Mode 2 Mode 1 Fit mode 2 Fit mode 1 ϕΗ=0° Figure 4: (Color online) (a) MS-FMR spectrum under a mag- neticfieldapplied(H=520Oe)parallelthec-axisand(b)field- dependence of the resonance frequency of the uniform excited modes, in the 100-nm-thick thin film. The fits are obtained using equation (2) with the parameters indicated in Table I. sample (Figure 4a) submitted to a field of 520 Oe shows two distinct modes: a main one (mode 2), with a wide line-width (about 0.6 GHz) and a second weaker one (mode 1) at lower frequency with a narrower line-width (0.2 GHz)). Their field-dependences are presented in Figure 4b. In contrast with mode 2, which presents significant in-plane anisotropy, the measured resonance frequency of mode 1 does not vary versus the in-plane angular orientation of the applied magnetic field: such a different behavior prevents from attributing mode 1 to a perpendicular standing excitation. Consequently, mode 1 is presumably a uniform mode arising from the presence of an additional magnetic phase in the film, possessing a lower effective demagnetizing field. In the following, we focus on mode 2 which is assumed to be the uniform mode arising from the main phase. As previously published, only one uniform mode is observed with the 55-nm-thick sample. Figures 5b and 5d illustrate the experimental in-plane angular-dependencies of the resonance frequency of the5 uniform mode for the 100- and for the 55-nm-thick samples, compared to the obtained fits using equation (2). As expected from the VSM measurements, in the 100-nm sample the fourfold and uniaxial axes of anisotropy are misaligned: it results an absence of symmetry of the representative graphs around 'H= 90. The best fit is obtained for the following values of the magnetic parameters: 4M e= 9800Oe,Hu= 55 Oe,H4= 110Oe,'4=0,'u= 12. As previously published, in the case of the 55-nm sample the direction of the easy uniaxial axis does not coincide with the observed one for the fourfold axis. The best fit for this film corresponds to: 4M e= 9800Oe,Hu= 10Oe, H4= 54Oe,'4= 0,'u=45. In both samples, the fourfold anisotropy easy direction is parallel to the c axis of the substrate: this presumably results from an averaging effect of the above described distribution of the crystallographic orientations, in spite of the facts that such a conclusion requires equal concentrations of the two main variants, a condition which, strictly speaking, is not fully realized, and that the observed value of'4does not derive from probably oversimplified averaging model that we attempted to use, based on individual domain contributions showing their principal axis of anisotropy along their cubic direction. As usual, attempts to interpret the in-plane hysteresis loops using the coherent rotation model do not provide a quantitative evaluation of the anisotropy terms involved in the expression of magnetic energy density. However, the experimentally measured Mr/Msangular variation, which, with this model, is given by cos('M-'H)) in zero-applied field and is easily calculated knowing ' 'u,'4andHu/H4, is in agreement with the values of these coefficients fitted from resonance data, as shown in Figures 5a and 5c. t (nm) 4M e (kG)Hu (Oe)H4 (Oe)'u (deg.)'4 (deg.) 13 8000 45 40 12 0 34 9000 6 20 45 0 55 9800 10 54 45 0 89 9200 15 22 -5 0 100 9800 60 110 12 0 200 9900 24 0 Table I : Magnetic parameters obtained from the best fits to our experimental results. 'uand'4are the angles of in-plane uniaxial and of fourfold anisotropy easy axes, respectively The magnetic parameters deduced from our resonance measurements are given in Table I for the complete set of the studied films. In contrast with the direction of the fourfold axis which does not vary, the orientation of the uniaxial axis is sample dependent: for some of them (34 and 55nm) the easy uniaxial direction lies at 0 50 100 150 200 250 300 350 6.8 7.2 7.6 8.0 0.0 0.3 0.6 0.9 100 nm (b) Frequency (GHz) ϕH (degrees) MS-FMR measurements Fit 100 nm (a) Mr/M s Fit VSM measurements 0 50 100 150 200 250 300 350 2.7 3.0 3.3 3.6 0.6 0.8 1.0 (d) 55 nm H=130 Oe Frequency (GHz) ϕH (degrees) MS-FMR Fit (c) 55 nm Mr/M s VSM measurements Fit Figure5: Reducedremanentmagnetizationofthe(a)100-nm- andofthe(c)55-nm-thickfilms. Thesimulationsareobtained from the energy minimization using the parameters reported in Table I. (b) and (d) show the compared in-plane angular- dependencesoftheresonancefrequencyoftheuniformmodes. The fit is obtained using equation (2) with the parameters indicated in Table I. 45from the c-axis of the substrate (thus coinciding with the hard fourfold direction); for other ones (13, 83, 100 nm) it shows a variable misalignment; finally, the uniaxial anisotropy field vanishes for the thickest sample (200 nm). We tentatively attribute at least a fraction of the uniaxial contribution as originating from a slight misorientation of the surface of the substrate. The amplitudes of both in-plane anisotropies are sample dependent and cannot be simply related to the film thickness. It should be mentioned that some authors [17] have reported on strain-dependent uniaxial and fourfold anisotropies in Co 2MnGa. This suggests a forthcoming experimental X-rays study of the strains present in our films. In addition, it is useful to get information about the damping terms involved in the dynamics of magnetic ex- citations in the above samples. Notice that in order to6 4 5 6 7 8 9 10 25 30 35 40 45 50 Field linewidth ∆H (Oe) Frequency (GHz) VNA-FMR measurements Fit Figure 6: Line-width
Has a function of the resonance fre- quency for 55-nm-thick film.
His derived from the experi- mental VNA-FMR frequency-swept line-width. integrate these films in application devices like, for in- stance, MRAM, it is important to make sure that their damping constant is small enough. The damping of the 55-nm-thick film was studied by VNA-FMR [12-14]: it is analyzedintermsofaGilbertcoefficient intheLandau- Lifschitz-Gilbert equation of motion. The frequency line- width
fof the resonant signal around frobserved us- ing this technique is related to the field line-width
H measured with conventional FMR excited with a radio- frequency equal to frthrough the equation [18]:
H=
@H(f) @fjf=fr(3)
His given by:
H=
H0+4fr j j(4) (where
H0stands for a small contribution arising from inhomogeneous broadening). The measured linear dependence of
His shown versus frin Figure 6. We then obtain the damping coefficient: =0.0065. This value lies in the range observed in the Co 2MnSi thin films [19-21]. Finally, the temperature dependence was studied for the 55-nm-thick sample using conventional FMR. The fits of the magnetic parameters were performed assum- ing that gpractically does not vary versus the tempera- ture T, as generally expected. We then take: g= 2:17. The results for the uniaxial and for the fourfold in-plane anisotropy fields are reported in Figure 7. Huis temper- ature independent while H4is a significantly decreasing function of T. This behavior of H4is presumably related to the magneto-crystalline origin of this anisotropy term. 0 50 100 150 200 250 300 10 20 30 40 50 60 70 80 90 100 Anisotropy fields (Oe) Temperature (K) H u H 4Figure7: (Coloronline)Temperature-dependenceofthefour- foldanisotropyfield( H4)andtheunixialanisotropyfield( Hu) of the 55-nm-thick film, measured by FMR at 9.5 GHz. V Summary The static and dynamic magnetic properties of Co2MnGe films of various thicknesses sputtered on a-plane sapphire substrates have been studied. The present work focused on the dependence of the pa- rameters describing the magnetic anisotropy upon the crystallographic texture and upon the thickness of the films. The crystallographic characteristics were obtained through X-ray diffraction which reveals the presence of a majority of two distinct (110) domains. Magnetometric measurements were performed by VSM and magnetization dynamics was analyzed using conven- tional and micro-strip resonances (FMR and MS-FMR). The main results concern the in-plane anisotropy which contributes to the magnetic energy density through two terms: a uniaxial one and a fourfold one. The easy axis related to the fourfold term is always parallel to the c-axis of the substrate while the easy twofold axis shows a variable misalignment with the c-axis. The fourfold anisotropy is a decreasing function of the temperature: it is presumably of magneto-crystalline nature and its orientation is related to the above noticed domains. The observed misalignment of the two-fold axis is tentatively interpreted as induced by random slight miscuts affect- ing the orientation of the surface of the substrate. The two-fold anisotropy does not significantly depend on the temperature. There is no evidence of a well-defined dependence of the anisotropy versus the thickness of the films. Finally, we show that the damping of the magne- tization dynamics can be interpreted as arising from a Gilbert term in the equation of motion, that we evaluate. References [1] S. Tsunegi, Y. Sakuraba, M. Oogane, K. Takanashi, Y. Ando, Appl. Phys. Lett. 93, 112506 (2008)7 [2] S. Picozzi, A. Continenza, and A. J. Freeman, Phys. Rev. B 66, 094421 (2002). [3] S. Picozzi, A. Continenza, and A. J. Freeman, J. Phys. Chem. Solids 64, 1697 (2003). [4] T. Ambrose, J. J. Krebs, and G. A. Prinz, J. Appl. Phys. 89, 7522 (2001). [5] T. Ishikawa, T. Marukame, K. Matsuda, T. Uemura, M. Arita, and M. Yamamoto, J. Appl. Phys. 99, 08J110 (2006) [6] F. Y. Yang, C. H. Shang, C. L. Chien, T. Ambrose, J. J. Krebs, G. A. Prinz, V. I. Nikitenko, V. S. Gornakov, A. J. Shapiro, and R. D. Shull, Phys. Rev. B 65, 174410 (2002). [7] H. Wang, A. Sato, K. Saito, S. Mitani, K. Takanashi, and K. Yakushiji, Appl. Phys. Lett. 90, 142510 (2007) [8] Y. Sakuraba, M. Hattori, M. Oogane, Y. Ando, H. Kato, A. Sakuma, T. Miyazaki, and H. Kubota, Appl. Phys. Lett. 88, 192508 (2006). [9] T. Marukame, T. Ishikawa, K. Matsuda, T. Uemura, and M. Yamamoto, Appl. Phys. Lett. 88, 262503 (2006). [10] D. Ebke, J. Schmalhorst, N.-N. Liu, A. Thomas, G. Reiss, and A. Hütten, Appl. Phys. Lett. 89, 162506 (2006). [11] M. Belmeguenai, F. Zighem, Y. Roussigné, S-M. Chérif, P. Moch, K. Westerholt, G. Woltersdorf, and G. Bayreuther Phys. Rev. B 79, 024419 (2009).[12] M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and G. Bayreuther, Phys. Rev. B 76, 104414 (2007). [13] T. Martin, M. Belmeguenai, M. Maier, K. Perzlmaier, and G. Bayreuther, J Appl. Phys. 101, 09C101 (2007) [14] M. Belmeguenai, T. Martin, G. Woltersdorf, G. Bayreuther, V. Baltz, A. K; Suszka and B. J. Hickey, J. Phys.: Condens. Matter 20, 345206 (2008). [15] U. Geiersbach, K.Westerholt and H. Back J. Magn. Magn. Mater. 240, 546 (2002). [16] T. Ambrose, J. J. Krebs, and G. A. Prinz, J. Appl. Phys. 87, 5463 (2000) [17] M. J. Pechana, C. Yua, D. Carrb, C. J. Palmstrøm, J. Mag. Mag. Mat. 286, 340 (2005) [18] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 (2006) [19] R. Yilgin, M. Oogane, Y. Ando, T. Miyazaki, J. Mag. Mag. Mat. 310, 2322 (2007) [20] R. Yilgin, Y. Sakuraba, M. Oogane, S. Mizumaki, Y. Ando, and T. Miyazaki, Japan. J. Appl. Phys. 46, L205 (2007) [21] S. Trudel, O. Gaier, J. Hamrle, and B.Hillebrands, J. Phys. D: Appl. Phys. 43, 193001 (2010)

2010-05-25

Magnetic properties of CoMnGe thin films of different thicknesses (13, 34, 55, 83, 100 and 200 nm), grown by RF sputtering at 400{\deg}C on single crystal sapphire substrates, were studied using vibrating sample magnetometry (VSM) and conventional or micro-strip line (MS) ferromagnetic resonance (FMR). Their behavior is described assuming a magnetic energy density showing twofold and fourfold in-plane anisotropies with some misalignment between their principal directions. For all the samples, the easy axis of the fourfold anisotropy is parallel to the c-axis of the substrate while the direction of the twofold anisotropy easy axis varies from sample to sample and seems to be strongly influenced by the growth conditions. Its direction is most probably monitored by the slight unavoidable angle of miscut the Al2O3 substrate. The twofold in-plane anisotropy field is almost temperature independent, in contrast with the fourfold field which is a decreasing function of the temperature. Finally, we study the frequency dependence of the observed line-width of the resonant mode and we conclude to a typical Gilbert damping constant of 0.0065 for the 55-nm-thick film.

Structural, static and dynamic magnetic properties of CoMnGe thin films on a sapphire a-plane substrate

1005.4595v3

Precessing vortices and antivortices in ferromagnetic elements A. Lyberatos,1S. Komineas,2and N. Papanicolaou3, 4 1Department of Materials Science,University of Crete,PO BOX 2208,71003 Heraklion,Greece 2Department of Applied Mathematics, University of Crete, 71409 Heraklion, Crete, Greece 3Department of Physics, University of Crete, 71003 Heraklion, Crete, Greece 4Institute for Theoretical and Computational Physics, University of Crete, Heraklion, Greece A micromagnetic numerical study of the precessional motion of the vortex and antivortex states in soft ferromagnetic circular nanodots is presented using Landau-Lifshitz-Gilbert dynamics. For suciently small dot thickness and diameter, the vortex state is metastable and spirals toward the center of the dot when its initial displacement is smaller than a critical value. Otherwise, the vor- tex spirals away from the center and eventually exits the dot which remains in a state of in-plane magnetization (ground state). In contrast, the antivortex is always unstable and performs damped precession resulting in annihilation at the dot circumference. The vortex and antivortex frequencies of precession are compared with the response expected on the basis of Thiele's theory of collective coordinates. We also calculate the vortex restoring force with an explicit account of the magne- tostatic and exchange interaction on the basis of the 'rigid' vortex and 'two-vortices side charges free' models and show that neither model explains the vortex translation mode eigenfrequency for nanodots of suciently small size. PACS numbers: 75.70.Kw,75.75.Fk,75.75.Jn,75.78.Cd,75.78.Fg I. INTRODUCTION The vortex state is one of the equilibrium states of thin soft ferromagnetic elements of micrometer size and below (magnetic dots). The interplay between the magneto- static and exchange energy favours an in-plane, closed ux domain structure with a 10 20 nm central core, where the magnetization turns out of plane to avoid the high energetic cost of anti-aligned moments. Core rever- sal can be triggered by application of an in-plane pulsed eld or pulsed current allowing the possibility of applica- tion of patterned thin lm elements in data storage and magnetic and magneto-electronic random access mem- ory [1]. Core reversal is usually assumed to arise from the spontaneous creation of a vortex-antivortex (VA) pair (vortex dipole) of opposite polarity with respect to the original vortex, followed by collision of the pair with the original vortex. A fundamental understanding of the dy- namics of vortices and antivortices is therefore necessary to control the switching of the magnetization. The basic excitation mode of the vortex or antivortex state from its equilibrium position is in-plane gyrotropic motion. It is a low frequency (GHz) mode corresponding to the displacement of the whole structure. The gener- alized dynamic force can be determined using Thiele's collective-variable approach [2]-[3]. Theoretical [4]-[5] and experimental [6] studies of the dynamics of magnetic vortices in 2D lms have shown a connection with the topology of the magnetization structure . Magnetic vor- tices conned in circular dots can be described by ana- lytical models based on dierent methods for accounting for the magnetostatic interaction [7]-[9]. The vortex may be 'rigid' or deform so that no magnetic charges appear at the side of the cylinder. The latter (two-vortices side charge free model) provides a good description of the dy- namic behavior of vortices in submicron-sized permalloydots, in particular the increase of vortex eigenfrequency with dot aspect ratio L=R, whereLis the dot thickness andRis the dot radius. The basic assumption in these calculations is that the vortex displacement lfrom equi- librium, at the dot center, is small l<<R . The main objective here is to consider the response of the vortex state to large perturbations, induced for in- stance by thermal activation. We focus our attention to dots of suciently small radius and thickness so that the vortex state is metastable . Using micromagnetic calcu- lations it is shown that vortex stability can be dened in terms of a critical displacement lcleading to an irre- versible transition to the ground state characterized by in-plane magnetization. We test the accuracy of the col- lective coordinate representation for vortex and antivor- tex dynamics and discuss the limitations of the approxi- mate analytical models [8],[9]. II. VORTEX AND ANTIVORTEX PRECESSION The dynamical behavior of a single vortex or antivor- tex trapped in a ferromagnetic nanodot was studied using a nite-dierence Landau-Lifshitz-Gilbert micro- magnetic model. The material parameters for permal- loy (Ni 80Fe20) were used in the calculations: saturation magnetization Ms= 800 emu/cc and exchange sti- ness coecient A= 1:3106erg/cm. The mag- netic anisotropy was neglected and the exchange length islex=p A=(2M2s) = 5:7 nm. The disk thickness is in the rangeL= 510 nm , comparable to the exchange length, so the magnetization dependence along the dot normal can be neglected to a rst approximation. The disk radius is not large compared to the vortex core and is initially taken as R= 30 nm. Inspection of the phase diagram of magnetic ground states for this material [10]arXiv:1007.3508v1 [cond-mat.mtrl-sci] 20 Jul 20102 FIG. 1: a) Vortex snd b) antivortex structure in permalloy dot of thickness L= 5 nm and radius R= 30 nm. indicates that for the particular choice of disk dimensions, in-plane magnetization is the ground state of the system. The integration of the LLG equation is carried out using a damping parameter = 0:01, which is appropriate for permalloy [11]. The initial vortex or antivortex structure is dened as follows: Using the coordinate system with the z axis par- allel to the dot cylindrical axis, the magnetization compo- nents aremz=cos ();mx+imy= sin  exp( i( o)), where m(r) =M(r)=Msis the reduced magneti- zation,;are cylindrical coordinates, =1 is the vortex polarity, and ( ) is the magnetization angle to the normal (z) direction. The vortex number is = 1 for a vortex and =1 for an antivortex structure. The constant o==2 denes the chirality i.e. the direction of the curl of the vortex. The specic choice of chirality is largely insignicant for vortex dynamics, in contrast to the vortex number and polarity which play an important role. The initial prole ( ) is assumed to have the modelform cos  = (cosh( c=lex))1which is appropriate for easy-plane ferromagnets [12]. The parameter cdenes the core radius b. The core radius is determined by the dot thickness Land the exchange length lexbut is inde- pendent of the dot radius [13]. In the following, dot thick- ness isL= 5 nm, unless specied otherwise. Numerical calculations of relaxed vortex and antivortex structures at the dot center were found to be in good agreement with the model prole for c'1. Typical examples of a vortex and antivortex structure are shown in Fig. 1. It should be noted that other choices for the vortex pro- le are possible [14], for instance tan( =2) ==bwas obtained by Usov et al. [15] using a variational proce- dure to minimize the exchange energy whereas the vor- tex core radius bwas determined by minimization also of the magnetostatic energy. This form is employed in analytical calculations of vortex states in ferromagnetic disks [7]-[9], however, micromagnetic modelling by Scholz et al. [10], conrmed by our calculations, has shown that it underestimates the core radius b, dened in Ref.[10] as the rstmz= 0 crossover from the vortex center. The dynamical behavior of the magnetic vortex ( = 1) depends on the initial displacement l(t= 0) =lofrom the disk center. If the displacement is equal or smaller than some critical value lc= 0:52R, the damped preces- sion of the vortex leads to relaxation to the disk center. For positive vortex polarity = 1, an anticlockwise pre- cession is observed as shown in Fig. 2a. At the start of the simulation the precession is not smooth as a result of the internal relaxation (deformation) of the model vortex to minimize the energy during precession. If lo>lc, the damped precession is clockwise and the distance from the dot center increases, as shown in Fig. 2b, until the vor- tex is annihilated and the magnetization is aligned along the in-plane direction with quasi-uniform magnetization, the so-called 'leaf' state [16] which is the ground state of the system. Irregularities in the precessional motion arise from the uncertainty on the position of the vortex. The antivortex instead is always unstable. For any choice of initial displacement lo, the antivortex performs damped precession to the edge of the disk and is an- nililated. For positive polarity, anticlockwise precession is observed. The sense of gyrotropic motion of a vortex or antivortex is switched on reversal of the polarity. III. THE COLLECTIVE COORDINATE APPROACH The damped precession of the vortex or antivortex can be described using Thiele's equation [2],[3] augmented by a dissipative term. Gdl dt+ 2QGdl dt@E(l) @l= 0 (1) where l= (lx;ly) is the position of the vortex center andE(lx;ly) is the potential energy of the shifted vor-3 -0.6-0.4-0.20.00.20.40.6-0.6-0.4-0.20.00.20.40.6(a) y/Rx/R -0.8-0.6-0.4-0.20.00.20.40.60.8-0.8-0.6-0.4-0.20.00.20.40.60.8 y/Rx/R(b) FIG. 2: Trajectory of a vortex of positive polarity = 1 in zero eld for a time interval 0 < t < 2104. The initial position vector of the vortex is a) lo= (0:52R;0) and b) (0:53R;0). The anisotropy is neglected and the damping is = 0:01. tex. The gyroforce Gdl=dtdepends on the topolog- ical structure of the magnetization and is proportional to the gyrovector G=G^z, where the gyroconstant is G= 2LMs= and = 1:76107rad Oe1s1is the gyromagnetic ratio. Q=1 2is the skyrmion number andis the dissipation constant. For axially symmetric energy potential E=E(l) wherel=q l2x+l2y,@E=@lx=E0lx=land it is straightforward to show that _lx2Q_ly=!ly (2) 2Q_lx+_ly=!lx (3) where the angular frequency is !=1 Gl@E @l(4) The vortex motion in complex form is given by (1 + 2Qi) _lx+i_ly =i!(lx+ily) (5) Introducing polar coordinates lx+ily=lei _l=2Q 1 +2!(l)l (6) _=!(l) 1 +2(7) Dividing and integrating over the time interval of the damped precession, the time dependence of the preces- sion angle is (t) =2Q lnl lo (8) wherelois the initial displacement of the vortex center. The clockwise or anticlockwise sense of gyration is there- fore dependent on the skyrmion number Q. Micromagnetic simulations of vortex motion were per- formed and the position of the vortex center ( lx;ly) was determined by a method of interpolation for the position of maximum mz. The precession angle = arctan(ly=lx) was found to vary linearly with the logarithm of the vor- tex shift, in agreement with Eq.(8). Fig. 3 shows numeri- cal data for damped precession of a vortex of positive po- larity (Q=1=2) with initial displacement lo= 0:52R. The relaxation to the disk center involves many revolu- tions (Fig. 2a) and the gradient 1=provides an accu- rate estimate of the dissipation constant = 0:013. It is evident that for permalloy nanodots, the damping in the vortex motion is weak and the angular frequency of pre- cession in Eq.(7) can be approximated using !'d=dt . Fitting the numerical (t) curve to a 4th order poly- nomial, the time variation of the angular frequency !(t) can be determined. The vortex shift l(t) exhibits oscilla- tions that are neglected by tting to a 4th order polyno- mial and the !(l) dependence is obtained using the !(t) curve. The same procedure is employed for damped pre- cession leading to vortex annihilation ( lo= 0:53R). The4 FIG. 3: Azimuthal angle of the vortex position as a function of the logarithm of the reduced vortex displacement s=l=R. The initial radial position is lo= 0:52R. FIG. 4: Precession frequency of a vortex !=2as a function of the reduced diplacement s=l=Rfrom the center of the dot. combined results for the dependence of the frequency of precession!=2on reduced vortex shift s=l=Rare il- lustrated in Fig. 4. For small displacement of the vortex center from its equilibrium position ( l= 0), the potential energy is E(l) =E(0) + (1=2)l2, whereis the stiness coe- cient and the eigenfrequency is !o==G [9]. At the critical displacement lc, corresponding to a maximum in the potential energy E(l) the precession frequency van- ishes. 0.00.10.20.30.40.50.60.70.80.1000.1020.1040.1060.1080.1100.1120.1140.116 Reduced energy density e Reduced vortex displacement seMAX sc(a) 0.00.10.20.30.40.50.60.70.80.90.000.020.040.060.080.100.12eeexReduced energy density Reduced vortex shift sed(b)FIG. 5: (a) Reduced energy density of a permalloy dot =E=4M2 sVas a function of normalized vortex displace- ments. Results are shown for two sets of micromagnetic simu- lations (markers) and analytical (solid line) calculations using Thiele's collective variable theory (Eq.10). The maximum en- ergy density maxoccurs at vortex displacement sc. (b) The contribution of the magnetostatic and exchange terms to the total energy, obtained from micromagnetic calculations. The motion of vortices and antivortices is driven by the restoring force @E=@ l(Eq.1). The potential energy of the shifted vortex is axially symmetric E=E(l) and can be written E(l) =E(0) +GZl 0!()d (9) A simpler form in terms of the reduced energy density =E=4M2 sVover the dot volume Vis (l) =(0)4QZs 0!(s0)s0ds0(10)5 wheres0==R,s=l=Ris the reduced displacement of the vortex and the time associated with !is scaled byo= 1=4 Ms. The function !(s0) was determined from micromagnetic simulations of the vortex precession and using Eq(10) the curve (l) expected from appli- cation of the theory of collective coordinates, was ob- tained, as shown in Fig. 5a. Superimposed is the energy evaluated directly from simulations of vortex relaxation. Thiele's theory appears to provide a good description of the vortex precession despite its limitations, for instance, a) Thiele's approach is known to be an approximate de- scription of vortex dynamics in innite lms b) the vortex is here conned in a nanodot ( R= 5:3lex) and c) the vor- tex structure does not remain rigid during the relaxation process but is modied as a result of the change in the distribution of the demagnetizing elds. The magneto- static and exchange contribution to the energy variation (l) was obtained from micromagnetic calculations and is shown in Fig. 5b. It should be noted that incorporat- ing the demagnetizing energy to the total anisotropy, is strictly valid for innite thin lms and results in mono- tonically decreasing energy (l), as reported in Ref. [17]. The oscillations in the energy variation during precession are related to the nite micromagnetic grid. The potential energy of the vortex attains a maximum value at some critical value of the displacement sc= 0:52 corresponding to a zero crossover of the precession fre- quency!(Fig. 4). The stability of the vortex at the dot center arises from the magnetostatic energy, in particular the volume magnetic charges resulting from vortex defor- mation and the surface charges at the side of the cylinder [9]. The face charges do not depend on ssince the charge distribution on the top and bottom surfaces of the disk is unchanged with the vortex displacement. The exchange energy decreases with increasing vortex shift s[9]. The magnetostatic and exchange contributions to the restor- ing force are in exact balance at the point of maximum energy. A similar analysis was carried out for an antivortex structure in a dot of identical dimensions. The potential energy decreases monotonically with increasing displace- ment s, as shown in Fig. 6. Application of the collec- tive coordinates treatment results in the solid curve in Fig. 6 of slightly smaller curvature. The vortex insta- bility arises from the uncompensated magnetic charge distribution within the antivortex core (Fig. 1b), so the magnetostatic energy is reduced upon motion away from the dot center. The precession frequency increases dur- ing the relaxation process as shown in Fig. 7, as a result of the steeper energy gradient for large displacement s. Assuming identical position, it is evident that the an- tivortex precesses faster than the vortex as a result of the larger magnetostatic energy gradient. FIG. 6: Energy density =E=4M2 sVvs antivortex displace- ments. The notation is similar to Fig. 5a. FIG. 7: Antivortex precession frequency !=2as a function of the reduced diplacement s=l=Rfrom unstable equilibrium position at the dot center. IV. DEPENDENCE OF VORTEX PRECESSION ON DISK SIZE The maximum potential energy Emax of the shifted magnetic vortex, evaluated from plots such as Fig. 5a, depends on the radius of the dot, where it is conned. In Fig. 8, micromagnetic calculations of the reduced dot energy density are shown as function of dot radius R, scaled by the exchange length. The curves correspond to the maximum vortex energy Emax and the minima associated with the three equilibrium states of the mag- netization (in-plane,perpendicular,vortex). The vortex6 FIG. 8: Micromagnetic calculations of scaled dot energy den- sityvs dot radius R(in units of lex) for the three equilibrium states of the magnetization and the vortex state of maximum energy. FIG. 9: Barrier to vortex escape ( B), dened by the relation B=maxvortex as a function of dot radius (in units of lex). state is unstable for small dots R < Rs, metastable for Rs< R < R eqand a ground state for R > Reqwhere the values Rs= 2:5lexandReq= 16lexare obtained for the absolute and equilibrium single domain radius re- spectively. The corresponding variation of vortex barrier energyB=EmaxEvortex and displacement lc=scR for vortex escape are shown in Figs. 9 and 10 respectively. For small dots R= 3:5lex, the vortex is unstable and any shift from equilibrium at the dot center results in relax- ation to the ground state (in-plane magnetization). The FIG. 10: Reduced displacement for maximum energy of the vortex state scas a function of dot radius (in units of lex). critical size for vortex instability is larger than Rssince the latter is dened assuming a random perturbation dif- ferent than shifting the whole vortex. The vortex barrier energy increases for larger dots and attains a maximum value, forR > 10lex, related to the vortex annihilation eld [7]. The displacement lcfor vortex escape increases with disk size to the maximum value imposed by the disk perimeter ( lc=R!1), attained for sub-micron dots R>>lex. For suciently large dots, the vortex is within the domain of attraction of the dot center, irrespective of the initial position. Micromagnetic calculations of the dependence of the fundamental vortex eigenfrequency !o=2, obtained for small perturbation s<< 1 as in Fig. 4, on the dot aspect ratio=L=R are shown in Fig. 11. The dot thickness was xed (L= 10 nm) and the radius Rwas allowed to vary. The eigenfrequency attains a maximum value at = 0:35 and vanishes for smaller radius '0:5 when the vortex becomes unstable. The maximum value arises from the change in the relative contribution of the magne- tostatic and exchange energy to the stiness coecient . For instance, the eigenfrequency assuming a 'rigid' vortex is [9] !o= Ms Q F1()1 (R=lex)2 (11) whereF1(x) =R dtt1f(xt)J2 1(t) corresponds to the av- eraged in-plane dot demagnetizing factor, f(x) = 1 [1exp(x)]=xandJ1is the Bessel function. Using the approximation F1()'(=2)[ln(8=)]1=2], valid for << 1, it can be shown that the eigenfrequency is maximum at radius R= 4=L[ln(8=)3=2]1where all lengths are in units of lex. Previous studies were re- stricted to sub-micron sized dots where the second term7 FIG. 11: Micromagnetic (markers) and analytical (solid lines) calculations of vortex eigenfrequency !o=2as a function of dot aspect ratio =L=R a) the rigid vortex model and b) the two-vortices model. The dot thickness is here L= 10 nm. of Eq.(11) could be neglected [9], so a monotonically in- creasing eigenfrequency !o() arising from the magneto- static energy only was reported. The micromagnetic calculations are compared in Fig. 11 with the curves obtained using the 'rigid' vor- tex and 'two-vortices' approximations for the magneto- static energy. For sub-micron sized disks with aspect ratio < 0:05 corresponding to a radius R > 200 nm, the micromagnetic calculations are in good quantitative agreement with the 'two vortices' model, as reported in Ref.[9]. In this regime, the eigenfrequency is determined primarily by the magnetostatic energy. The rigid vor- tex approximation fails to describe the dynamic behav- ior since the magnetostatic energy can be decreased by elimination of the surface charges at the disk perime- ter at the expense of some contribution from volume magnetic charges arising from vortex deformation. Forsmaller disks R < 200 nm, the vortex eigenfrequency is between the predictions of the two models. A reduction in side charges occurs but is not complete as a result of the large expense in exchange energy arising from vortex deformation. Similar results can be obtained in principle for thinner disks, however, the 'two vortices' approxima- tion is then valid for larger cylinders which are not easily amenable to micromagnetic simulations. V. CONCLUSIONS Micromagnetic calculations were carried out of the pre- cessional behaviour of a single magnetic vortex or an- tivortex conned in a permalloy circular nanodot. The existence of two domains of attraction for the vortex state are identied arising from a maximum in the potential energy of the shifted vortex. This eect is atrributed to the competition between the magnetostatic attractive and exchange repulsive forces on the shifted vortex. An- tivortices instead are always unstable and trace a spi- ral trajectory of increasing distance from the dot center followed by annihilation at the dot envelope. The pre- cessional behaviour of vortices and antivortices is satis- factorily described by Thiele's theory of collective coor- dinates, relating the angular frequency of precession to the gradient of the potential energy. For small nano- sized dots, however, the 'rigid' vortex and 'two-vortices' approximation for the magnetostatic energy is not sat- isfactory and the exchange forces have a signicant ef- fect on the translation mode vortex eigenfrequency. For antivortices, the development of an analytical model to account for the magnetostatic interaction in circular dots is clearly needed to provide further insight on the results of our micromagnetic calculations. Vortex stability is necessary in applications of nanos- tructured patterned media for data storage. Microfabri- cation downscaling implies that the displacement lcand associated energy barrier may become useful characteri- zation parameters of the thermal stability of the recorded information. [1] B. Van Waeyenberge, A. Puzic, H. Stoll, K.W. Chou, T. Tyliszczak, R. Hertel, M. F ahnle, H. Br uckl, K. Rott, G. Weiss, I. Neudecher, D. Weiss, C.H. Back and G. Sch utz, Nature (London) 444, 461 (2006). [2] A.A. Thiele, PHys. Rev. Lett. 30, 230 (1973). [3] D.L. Huber, Phys. Rev. B 26, 3758 (1982). [4] N. Papanicolaou and T.N. Tomaras, Nucl. Phys. B 360, 425 (1991). [5] S. Komineas and N. Papanicolaou, Physica (Amsterdam) 99D , 81 (1996). [6] S.-B Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J. St ohr and H.A. Padmore, Nature 304, 420 (2004). [7] K. Yu Guslienko, V. Novosad, Y. Otani, H. Shima andK. Fukamichi, Appl. Phys. Lett. 78, 3848 (2001). [8] K.L. Metlov and K. Yu Guslienko, J. Magn. Magn. Mater. 242-245 , 1015 (2002). [9] K. Yu Guslienko, B.A. Ivanov, V. Novosad, Y. Otani, H. Shima and K. Fukamichi, J. Appl. Phys. 91, 8037 (2002). [10] W. Scholz, K. Yu Guslienko, V. Novosad, D. Suess, T.Schre , R.W. Chantrell and J. Fidler, J. Magn. Magn. Mater. 266, 155 (2003). [11] D.V. Berkov and J. Miltat, J. Magn. Magn. Mater. 320, 1238 (2008). [12] S. Komineas and N. Papanicolaou, in Electromagnetic, magnetostatic and exchange-interaction vortices in con- ned magnetic structures , edited by E.O. Kamenetskii,8 (Transworld Research Network, Kerala,2008). [13] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto and T. Ono, Science 289, 034318 (2005). [14] P. Landeros, J. Escrig, D. Altbir, D. Laroze, J. d' Albu- querque e Castro and P. Vargas, Phys. Rev. B 71, 094435 (2005) [15] N.A. Usov and S.E. Peschany, J. Magn. Magn. Mater.118, L290 (1993). [16] K.L. Metlov and K. Yu Guslienko, Phys. Rev. B 70, 052406 (2004) [17] D.D. Sheka, J.P. Zagorodny, J. Caputo, Y. Gaididei and F.G. Mertens, Phys. Rev. B 71, 134420 (2005)

2010-07-20

A micromagnetic numerical study of the precessional motion of the vortex and antivortex states in soft ferromagnetic circular nanodots is presented using Landau-Lifshitz-Gilbert dynamics. For sufficiently small dot thickness and diameter, the vortex state is metastable and spirals toward the center of the dot when its initial displacement is smaller than a critical value. Otherwise, the vortex spirals away from the center and eventually exits the dot which remains in a state of in-plane magnetization (ground state). In contrast, the antivortex is always unstable and performs damped precession resulting in annihilation at the dot circumference. The vortex and antivortex frequencies of precession are compared with the response expected on the basis of Thiele's theory of collective coordinates. We also calculate the vortex restoring force with an explicit account of the magnetostatic and exchange interaction on the basis of the 'rigid' vortex and 'two-vortices side charges free' models and show that neither model explains the vortex translation mode eigenfrequency for nanodots of sufficiently small size.

Precessing vortices and antivortices in ferromagnetic elements

1007.3508v1

arXiv:1008.2177v1 [cond-mat.mes-hall] 12 Aug 2010Magnetization dynamics in the inertial regime: nutation pr edicted at short time scales M.-C. Ciornei and J.-E. Wegrowe Ecole Polytechnique, LSI, CNRS and CEA/DSM/IRAMIS, Palais eau F-91128, France. J. M. Rub´ ı Departement de Fisica Fonamental, Universitat de Barcelon a, Diagonal 647, Barcelona 08028, Spain. (Dated: April 23, 2022) The dynamical equation of the magnetization has been recons idered with enlarging the phase space of the ferromagnetic degrees of freedom to the angular momentum. The generalized Landau- Lifshitz-Gilbert equation that includes inertial terms, a nd the corresponding Fokker-Planck equa- tion, are then derived in the framework of mesoscopic non-eq uilibrium thermodynamics theory. A typical relaxation time τis introduced describing the relaxation of the magnetizati on acceleration from the inertial regime towards the precession regime defin ed by a constant Larmor frequency. For time scales larger than τ, the usual Gilbert equation is recovered. For time scales be lowτ, nutation and related inertial effects are predicted. The ine rtial regime offers new opportunities for the implementation of ultrafast magnetization switching i n magnetic devices. PACS numbers: 75.78.jp, 05.70.Ln In 1935 Landau and Lifshitz proposed an equation for the dynamicsofthe magnetization M(ofconstantmodu- lus), composedofaprecessionterm M×Handa longitu- dinalrelaxationterm M×(M×H)thatdrivesthemagne- tization towards equilibrium along the magnetic field H [1]. Two decades later T. L. Gilbert derived the equation that bearshisname in whichthe relaxationtowardsequi- librium is described by a damping term η[2] through the dynamic equation dM/dt=γM×(H−ηdM/dt), withγ as the gyromagnetic ratio. The two equations (Landau- Lifshitz and Gilbert) are mathematically equivalent. The range of validity of the Landau-Lifshitz-Gilbert (LLG) equation was established one decade later by W. F. Brown, with a description of a magnetic moment cou- pled to a heat bath (”thermal fluctuations of a single- domain particle”, 1963 [3]). The magnetic moment is treated as a Brownian particle described by the slow de- grees of freedom (10−9s), the angles {θ,φ}. The remain- ing degrees of freedom of the system relax in a much shorter time scale ( <10−12s). The time scale separa- tion between the rapidly relaxing environmental degrees of freedom and the slow magnetic degrees of freedom al- lows the coupling between the magnetization and the en- vironment to be reduced to a single phenomenological damping parameter η, whatever the complexity of the microscopic relaxation involved [4, 5]. However, important experimental advances towards very short time-resolved response of the magnetization (sub-picosecondsresolution,i.e. belowthelimitproposed by Brown) have been reported in the last decade [6]. In parallel, industrial needs for very fast memory storage technologies are approaching the limits imposed by the precessional switching [7]. In these experiments and in the corresponding applications, time scale separation be- tween the conserved degrees of freedom {θ,φ}and the other degrees of freedom, assumed by Brown [3], finds its limit. The purpose of this Letter is to investigate the dynam-ics of the magnetization beyond this limit by extending thephasespacetoadditionaldegreesoffreedomexpected to be also out-of-equilibrium at short time scales [5, 9]. According to the well-known gyromagnetic relation [10], the next relevant degree of freedom of the ferromagnetic system (beyond the coordinates of position; i.e. the an- gles{θ,φ}) is the angular momentum L. As will be shown below, the consequence of considering also the conservation of the angular momentum is that inertial terms, i.e. acceleration terms proportional to d2M/dt2, appear in the equation of motion. The existence of in- ertial terms in the dynamics of the magnetization opens the way to deterministic ultrafast magnetization switch- ing strategies, beyond the limitations of the precessional regime [11]. We assume however that the microscopic re- laxationchannelsareinactiveat the time consideredhere (e.g. by choosing the adequate materials and excitations of the magnetization). Otherwise, a non-deterministic regime would take place [7, 12, 13]. We derive below the generalized Gilbert equation and the corresponding Fokker-Planck equation that includes the inertial effects for a uniform magnetic moment. The derivation is performed in the framework of mesoscopic nonequilibrium thermodynamics (MNET) [14–16], and is based on the expression of the conservation laws, ther- modynamic laws, and symmetry properties. It is convenient to model the dynamics of a magnetic moment M=Mse(submitted to an applied magnetic fieldH=−1 Ms∂VF ∂eand coupled to a heat bath) with a statistical ensemble composed by non-interacting identi- cal uniform magnetic moments found in the same given conditions (ergodic property). Here, e,MsandVFare respectively the radial unit vector of angles {θ,φ}, the magnetization at saturation, and the ferromagnetic po- tential energy. The ensemble of magnetic moments of constant modulus Msdefines a sphere surface Σ and the number of magnetic moments oriented within ( e,e+de) defines the density n(e) of magnetic moments over Σ.2 We have shown in previous works that associating two degrees of freedom {θ,φ}to a magnetic moment is suf- ficient to derive both the Gilbert equation and the cor- responding rotational Fokker-Planck equation from non- equilibrium thermodynamics principles only [13, 17]. Extending the configuration space to the magnetic an- gular momentum L, the space Σ is extended from a two dimensions space, to a priori five dimensions space {θ,φ,L}[18]. A distribution function f(e,L) of mag- netic momentswith the magnetizationorientationwithin (e,e+de) and the angularmomentum within ( L,L+dL) should then be defined, where fis assumed to vanish for infinite values of Las: lim L→±∞f(e,L) = 0. The an- gular momentum Lassociated to a magnetic moment is either changed by an applied torque N=M×Has/parenleftbigdL dt/parenrightbig s=N, either by the interaction with the heat bath. When considering the statistical ensemble, the interac- tion with the bath is modeled through a phase space flux JL(defined below) which vanishes for large values of L: limL→±∞JL= 0. The kinetic energy expression that Gilbert associated to the magnetization [2] is written as: K=LL:¯¯I−1/2, where the magnetic inertial tensor¯¯I, is related to the magnetic moment (and not to the inertia of matter). It is assumed that¯¯Ikeeps the symmetry of the magnetic moment, i.e. is axial symmetric of symmetry axis e:¯¯I= I1/parenleftBig¯¯U−ee/parenrightBig +I3ee, with¯¯Uthedyadicunit(where I1=I2 andI3arethe diagonalcoefficients ofthe inertial tensor). In the space-fixed reference frame denoted by the sub- scripts, the conservation law for the number of axial symmetric moments f(e,L) writes [19]: ∂ ∂tf(e,Ls) =−/braceleftbigg∂(f˙ e) ∂e/bracerightbigg Ls−Ns·∂f ∂Ls−∂JL ∂Ls(1) where the derivatives with respect to the angles are made while holding the Cartesian components of Lsconstant: /braceleftbigg∂(f˙ e) ∂e/bracerightbigg Ls=1 sinθ/braceleftBigg ∂(fsinθ˙θ) ∂θ/bracerightBigg Ls+/braceleftBigg ∂(f˙φ) ∂φ/bracerightBigg Ls(2) The density n(e) of magnetic moments in the space Σ is recovered by integrating over the angular momentum degree of freedom n(e)=/integraltextf(e,L)d3L. The conservation law for the magnetic moments in the Σ space is hence deduced from (1): ∂n ∂t=/integraldisplay∂f ∂td3Ls=−∂ ∂e·/integraldisplay f˙ ed3Ls=−∂(n˙e) ∂e(3) Beyond, the conservationlaw for the mean value of themagneticangularmomentum /an}bracketle{tLs/an}bracketri}htisalsoderived[19,20]: ∂n/an}bracketle{tLs/an}bracketri}ht ∂t=/integraldisplay∂f ∂tLsd3Ls (1)=−∂ ∂e·/integraldisplay f˙ eLsd3Ls+nNs(e)+/integraldisplay JLd3Ls nd/an}bracketle{tLs/an}bracketri}ht dt=−∂ ∂e·/parenleftBig e×Ps/parenrightBig +nNs(e)+/integraldisplay JLd3Ls(4) where the magnetic pressure tensor is defined as Ps= I−1/integraltext (L− /an}bracketle{tLs/an}bracketri}ht)(L− /an}bracketle{tLs/an}bracketri}ht)f d3Ls. The conservation equation (4) states that the rate of the average angular momentum /an}bracketle{tLs/an}bracketri}htis due to three con- tributions: an applied torque Ns, an average interaction with the bath/integraltext JLd3Ls(i.e. damping), and a torque due to pressure (i.e. rotational diffusion). The expressionfor JLis deduced fromthe entropypro- duction expression σ(e) [14, 15, 20]. Defining the fer- romagnetic chemical potential µ(e,L), the power Tσ(e) dissipated by the magnetic system is the product of the generalized flux by the generalized force: Tσ(e) =−/integraldisplay JL·∂µ ∂Lsd3Ls (5) where the chemical potential takes the canonical form [14, 16]: µ(e,L) =kT ln[f(L,e)]+K(e,L)+VF(e) (6) The application of the second law of thermodynam- ics, together with the local equilibrium hypothesis in the (e,L) space, lead us to the introduce the Onsager matrix Lsuch that: JL=−L ·∂µ ∂Ls. As the Onsager coeffi- cients are a reflection of the system’s symmetry [15], the relaxation tensor defined as τ−1=1 fLI−1 is also axial symmetric: τ−1=τ−1 1(U−ee)+τ−1 3ee(whereτ1=τ2 andτ3are the diagonal coefficients), and is related to damping. Moreover, as eis an axis of symmetry for the ferromagnetic potential VF(θ,φ), the relaxation tensor τ−1is not expected to have any components in the e direction [19], leading to τ−1 3= 0. The dynamic equation (4) can be rewritten: d/an}bracketle{tLs/an}bracketri}ht dt=Ns−τ−1 s·/an}bracketle{tLs/an}bracketri}ht−1 n∂ ∂e·/parenleftBig e×Ps/parenrightBig (7) As the inertial tensor Iand the relaxation tensor τ−1 are time independent in the rotating frame (or magne- tization frame), a simpler expression of Eq. (7) can be obtained in this frame. After introducing the averagean- gular velocity Ωsuch that /an}bracketle{tL/an}bracketri}ht=I·Ω, Eq. (7) rewrites as: dΩr dt=I−1 r·/bracketleftbigg Nr−1 n∂ ∂e·/parenleftBig e×P/parenrightBig/bracketrightbigg −τ−1 rot·Ωr(8)3 The rotating frame is denoted by the subscript rand τ−1 rot=τ−1 r−/parenleftBig I3 I1−1/parenrightBig Ω3e×U, or τ−1 rot= (τ1α∗)−1 α∗1 0 −1α∗0 0 0 0 (9) whereα∗=α(I3/I1−1)−1withα= (Ω3τ1)−1. The three components of Eq. (8) read: ˙Ω1=−Ω1 τ1−/parenleftbiggI3 I1−1/parenrightbigg Ω3Ω2−MsH2 I1−/bracketleftBigg 1 I1n∂(e×P) ∂e/bracketrightBigg 1 ˙Ω2=−Ω2 τ1+/parenleftbiggI3 I1−1/parenrightbigg Ω3Ω1+MsH1 I1−/bracketleftBigg 1 I1n∂(e×P) ∂e/bracketrightBigg 2 ˙Ω3=−τ−1 3Ω3= 0 (10) Since the quantity L3=I3Ω3is a constant of motion, the well-known gyromagnetic ratio γcan be defined as the ratio of the magnetization at saturation Msby the axial angular momentum γ=Ms /angbracketleftL3/angbracketright[2, 10]. Also,theaverageddynamicEq. (10)introducesachar- acteristic time scale τ1, which separates the behavior of the magnetic system of particles in two regimes: the dif- fusion regime or the long time scale limit t≫τ1, and the inertial regime or the short time scale limit t≪τ1. Since the modulus of the magnetization Mis conserved, the relationdM dt=Ω×Mholds. Cross-multiplying by M and using the above definition of γleads to the identity Ω=M M2s×dM dt+M I3γ. In a diffusive regime, i.e. for t≫τ1, the inertial terms dΩ1 dtanddΩ2 dtare negligible with respect toΩ1 τ1andΩ2 τ1. Eq. (8) then rewrites as the Gilbert equation with an inertial correction performed on the previously defined gyromagnetic coefficient γ∗=γ 1−I1/I3: dM dt=γ∗M×/parenleftbigg Heff−ηdM dt/parenrightbigg (11) The Gilbert damping coefficient ηis now defined as: η=I1 τ1M2s(so that α∗=γ∗ηMsis the corresponding dimensionless coefficient), and Heffis an effective field that includes the diffusion term. At the diffusive limit, the magnetic moments follow a distribution function f(e,L) close to a Maxwellian cen- tered on the average angular momentum /an}bracketle{tL/an}bracketri}ht[9]. This leads to a diagonal form for the pressure tensor: P= nkT/UandHeff=H−kT n1 Ms∂n ∂e[13, 17]. Eq. (11) containsthe density n(e) so that the equation is not closed. However, inserting Eq. (11) into the con- servation law (3) leads to the rotational Fokker-Planck equation of n(e), derived by Brown [3]:∂n ∂t=∂(ne×Ω) ∂e For short enough time scales t≈τ1, the inertial terms cannot be neglected and the Gilbert approximation is no longer valid. The dynamic equation (10) takes the following generalized form:dM dt=γM×/bracketleftbigg H−η/parenleftbiggdM dt+τ1d2M dt2/parenrightbigg/bracketrightbigg −γ n∂(M×P) ∂M (12) The corresponding generalized rotational Fokker- Planck equation for the statistical distribution fis ob- tained with replacing JLby the Onsager relation derived earlier into the conservation law (1) and rewriting the law in the rotating frame [20]: ∂f(e,Lr) ∂t= ∂(fe×I−1 rLr) ∂e Lr+ ∂ ∂Lr·/bracketleftbigg fτ−1 rot·Lr−fNr+kTτ−1 rIr·∂f ∂Lr/bracketrightbigg (13) At short time scales t≈τ1and due to the inertial effect, the usual precessional behavior is enriched by a nutation effect. The simplest way to understand nuta- tion is to imagine that the effective field is switched off suddenly with zero damping: the precession stops sud- denly because the Larmore frequency ωL=γ∗Hdrops to zero at the same time. However, in the absence of inertial terms, the magnetic moment also stops at this position within an arbitrarily short time scale. But if the kinetic energy is not zero (and this is the case for magne- tomechanical measurements of the magnetization [10]), the movement cannot be stopped suddenly: the preces- sion (around the magnetic field) stops but the magnetic moment starts to rotate around the angular momentum vector in order to conserve the energy: the precession is transformed into nutation. Fig. 1 shows the numerical resolution of Eq. (12) (ne- glecting thermal fluctuations) with a field along z axis and for a parameters τ1fixed to 2 pswith|α|= 0.05. The trajectories are plotted on the sphere Σ. The usual trajectory deduced from the LLG equation ( τ1≪ps) is also plotted for comparison. The motion of the mag- netic moment displays the familiar curve due to Larmor precession,withsuperimposedloopsgeneratedbythenu- tation effect. Fig.1(b) shows a trajectory starting with- out initial velocity under an effective field of 1 MA/m, changed suddenly to 3 MA/mand once again down to zero. Four curves are represented, two for the Eq. (12) with|α|= 0.05 (red continuous line) and α= 0 (red dashed line), and two for the usual LLG equation with and without damping (blue). At the end of the motion (left), the field is set to zero and the precession is de- stroyed, with the nutation effect shaping a circle (with- out damping) or a spiral (with damping). Note that the profile of the nutation loops depends on the initial condi- tions (the cusp presented in Fig.1 instead of loops is due to zero initial velocity). Fig. 1 (c) shows the time deriva- tive of the angle φas a function of time for the trajectory displayedinFig. 1(b). Thehorizontallinesrepresentthe4 01 -110 20dφ/dt(1012 s-1) H = 1 MA/mH = 3 MA/m t(ps)H = 0 (c)(b) LLG2 psτ 1 = (a) FIG. 1: Numerical resolution of Eq. (12) with τ1= 2ps. (a) Trajectories at two different fields with |α|= 0.05 (red) and curves deduced from the LLG equation (blue). (b) Trajec- toryofthemagnetization withchangingsuddenlytheeffecti ve fields from H=1 MA/m to H=3 MA/m and H=0, with damp- ing (continuous line) and without (dotted line). (c) Time derivative of the azimuth angle φplotted as a function of time for the trajectory of Fig. 1(b). constant Larmorfrequencies, and the oscillations are dueto nutation (for |α|= 0.05 andα= 0). Inthe caseoftwostablestatesseparatedbyapotential barrier(e.g. for magnetic memory units), efficient strate- giesbased on the inertial mechanismcan be perormed for ultrafast magnetization reversal. Such a strategy has al- ready been implemented in the case of antiferromagnets [11], in which inertial terms are present due to the energy storedbydeformationinthemagneticdomainwalls. The inertia has been used to overcomean energy barrier after having push the magnetization with a very short optical impulsion. The novelty of our results is that any kind of ferromagnets could in principle be used for ultra-short inertial magnetization switching. In conclusion, we have shown that extending the phase space of the magnetization to the degrees of freedom of the magnetic angular momentum leads to considere a generalized Landau-Lifshitz-Gilbert equation that con- tains inertial terms. This extension is justified by the well-known gyromagnetic relation that relates the mag- netization to the angularmomentum. It is predicted that inertial effects should be observed at short enough time scales (typicaly below the picosecond), e.g. by measur- ing nutation loops superimposed to the usual precession motion of a magnetic moment. The inertial regime at short time scales would also offer possibilities for new ex- periments and devices based on ultrafast magnetization switching. [1] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). [2] T. L. Gilbert, Phys. Rev. 100, 1243 (1955) (Abstract only), reprint in IEEE Trans. Mag. 40, 3443 (2004). [3] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963). [4] W. T. Coffey, Yu. P. Kalmykov and J. T. Waldron, The Langevin equation , World Scientific Series in contempo- rary Chemical Physics Vol. 11, 1996. [5] E. Fick and S. Sauermann, the quantum statistics of Dy- namic Processes , Springer Series in Solid-States Sciences 86, 1990. [6] E. Beaurepaire etal., Phys.Rev.Lett. 76, 4250 (1996), J. Hohlfeld etal. Phys.Rev.Lett. 78, 4861 (1997), A.Scholl et al. Phys. Rev. Lett. 79, 5146 (1997), M. Aeschlimann et al. Phys. Rev. Lett. 79, 5158 (1997), C. H. Back et al. Phys. Rev. Lett. 81, 3251 (1998 ), H. S. Rhie et al. Phys. Rev. Lett. 90, 247201 (2003), A. V. Kimel et al. Nature435, 655 (2005). C. D. Stanciu et al. Phys. Rev. Lett.99, 047601 (2007), [7] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann, J. Steohr, G. Ju, B. Lu and D. Weller, Nature 428, 831 (2004). [8] R. Kikuchi, J. of Appl. Phys. 27, 1352 (1956). [9] J. M. Rub´ ı, A. P´ erez-Madrid, Physica A, 264(1999) 492. [10] The gyromagnetic relation M=γLis observed through magnetomechanical measurements (in isolated systems),as shown by S. J. Barnett (see Rev. Mod. Phys. 7, 129 (1935)), and A. Einstein and W. J. de Haas (Verh. d. D. Phys. Ges. 17, 152 (1915)). Typically |γ| ≈2 105m/As. [11] A. V. Kimel et al., Nature Physics, 5, 727 (2009). [12] J.-Y. Bigot et al., Nature Physics 5, 515 (2009), B. Koop- mans et al., Nature Materials, 9, 259 (2009) and K. H. Bennemann, Ann. Phys. 18, 480 (2009). [13] J.-E. Wegrowe et al. Phys. Rev. B 77, 174408 (2008), and J.-E. Wegrowe, Solid State Com. 150519 (2010). [14] D. Reguera, J. M. G. Vilar and J. M. Rub´ ı, J. Phys. Chem. B, 109, 21502 (2005). [15] P. Mazur, Physica A 261, 451 (1998). [16] S. R. De Groot and P. Mazur, non-equilibrium thermo- dynamics Amsterdam : North-Holland, 1962. [17] J.-E. Wegrowe, M. C. Ciornei, H.-J. Drouhin, J. Phys.: Condens. Matter 19, 165213 (2007). [18] The symmetry of the ferromagnetic potential VF(θ) im- poses a constraint on the axial component of the angular momentum, fixing one degree of freedom. [19] D. W. Condiff and J. S. Dahler, J. Chem. Phys. 10, 3988 (1966). [20] M.-C. Ciornei, Role of magnetic inertia in damped macrospin dynamics , PhD Thesis, Ecole Polytechnique, France, January 2010.

2010-08-12

The dynamical equation of the magnetization has been reconsidered with enlarging the phase space of the ferromagnetic degrees of freedom to the angular momentum. The generalized Landau-Lifshitz-Gilbert equation that includes inertial terms, and the corresponding Fokker-Planck equation, are then derived in the framework of mesoscopic non-equilibrium thermodynamics theory. A typical relaxation time $\tau$ is introduced describing the relaxation of the magnetization acceleration from the inertial regime towards the precession regime defined by a constant Larmor frequency. For time scales larger than $\tau$, the usual Gilbert equation is recovered. For time scales below $\tau$, nutation and related inertial effects are predicted. The inertial regime offers new opportunities for the implementation of ultrafast magnetization switching in magnetic devices.

Magnetization dynamics in the inertial regime: nutation predicted at short time scales

1008.2177v1

arXiv:1010.1537v1 [cond-mat.mes-hall] 7 Oct 2010Power optimization for domain wall motion in ferromagnetic nanowires O. A. Tretiakov,1Y. Liu,1and Ar. Abanov1 Department of Physics & Astronomy, Texas A&M University, Co llege Station, Texas 77843-4242, USA (Dated: October 7, 2010) The current mediated domain-wall dynamics in a thin ferromagnetic w ire is investigated. We derive the effective equations of motion of the domain wall. They are used to stu dy the possibility to optimize the power supplied by electric current for the motion of domain walls in a nanowire . We show that a certain resonant time-dependent current moving a domain wall can significantly reduc e the Joule heating in the wire, and thus it can lead to a novel proposal for the most energy efficient me mory devices. We discuss how Gilbert damping, non-adiabatic spin transfer torque, and the presence o f Dzyaloshinskii-Moriya interaction can effect this power optimization. Introduction. Due to its direct relevance to future memory and logic devices, the dynamics of domain walls (DW) in magnetic nanowires has become recently a very populartopic.1–3Therearemainlytwogoalswhichscien- tists try to achieve in this field. One goal is to move the domain walls with higher velocity in order to make faster memory or computer logic. The other one is inspired by the modern trend of energy conservation and concerns a power optimization of the domain-wall devices. Generally, the domain walls can be manipulated whether by a magnetic field3,4or electric current.1,5Al- though the latter method is preferred for industrial ap- plications due to the difficulty with the application of magnetic fields locally to small wires. For this reason, we consider in this paper the current induced domain-wall dynamics. We make a proposal on how to optimize the powerfor the DWmotionby meansofreducingthe losses on Joule heating in ferromagnetic nanowires.6Moreover, because the averaged over time (often called drift) veloc- ity of a DW generally increases with applied current, we also address the first goal. Namely, our proposal allows to move the DWs with higher current densities without burning the wire by the excessive heat and thus archive higher drift velocities of DWs. The central idea of this proposal is to employ resonant time-dependent current to move DWs, where the period of the current pulses is related to the periodic motion of DW internal degrees of freedom. The schematic view of a domain wall in a narrow fer- romagnetic wire is shown in Fig. 1. These DWs are char- FIG. 1. (color online) A schematic view of a current-driven domain wall in a ferromagnetic wire. The DW width is ∆.acterized by their width ∆ which is mainly determined by exchange interaction and anisotropy along the wire λ. Another important quantity is the transverse anisotropy acrossthewire K, whichgovernsthepinningofthetrans- verse component of the DW magnetization. When no current is applied to the wire it leads to two degenerate positions of the transverse magnetization component of the wall: as shown in Fig. 1 and anti-parallel to it. To describe the dynamics of DW in a thin wire we derived the effective equations of motion from general- ized Landau-Lifshitz-Gilbert7,8(LLG) equation with the currentJ, ˙S=S×Heff−J∂S ∂z+βJS×∂S ∂z+αS×˙S,(1) whereSis magnetization unit vector, Heff=δH/δSis the effective magnetic field given by the Hamiltonian H of the system, βis non-adiabatic spin torque constant, andαis Gilbert damping constant. The derivation of the effective equations of motion is based on the fact that in thin ferromagnetic wires the static DWs are rigid topologically constrained spin-textures. Therefore, for not too strong drive, their dynamics can be described in terms of only a few collective coordinates associated with the DW degrees of freedom.9In very thin wires, there are two collective coordinates corresponding to two softest modes of the DW motion: the DW position along the wire z0and the magnetization angle φin the DW around the wire axis. All other degrees of freedom are gapped by strong anisotropic energy along the wire. By applying the orthogonality condition to LLG, one can obtain the equations of motion for the two DW soft- est modes, z0(t) andφ(t),10 ˙z0=AJ+B[J−jcsin(2φ)], (2) ˙φ=C[J−jcsin(2φ)], (3) whereJ(t) is a time-dependent current. The co- efficients A,B,C, and critical current jccan be evaluated for a particular model in terms of α,β and other microscopic parameters. Following Ref. 10, for the model with Dzyaloshinskii-Moriya interaction (DMI) one can find A=β/α,B= (α−β)(1 + αΓ∆)/[α(1 +α2)],C= (α−β)∆/[(1 +α2)∆2 0], and2 FIG. 2. (color online) DW motion characteristics for dc cur- rents. (a) Drift velocity Vdof DW as a function of current J forB >0 andB <0, see Eq. (2). The slope at J < jcis given byA, whereas at J≫jcit isA+B. (b) Power of Ohmic lossespdc(Vd/Vc) =J2/j2 cas a function of drift velocity Vd. ForB <0 the power has a discontinuity at Vd/Vc= 1. jc= (αK∆/|α−β|)[πΓ∆/sinh(πΓ∆)], where Jexis ex- change constant, Dis DMI constant, and Γ = D/Jex. Also, ∆ = ∆ 0//radicalbig 1−Γ2∆2 0where ∆ 0is the DW width in the absence of DMI. Alternatively, Eqs. (2) and (3) can be obtained in a more general framework by means of symmetry argu- ments. We note that because of the translational invari- ance ˙z0and˙φcannot depend on z0. Furthermore, to the first order in small transverse anisotropy K,˙φand ˙z0are proportional to the first harmonic sin(2 φ). Then the ex- pansion in small current Jup to a linear in Jorder gives Eqs. (2) and (3). In this case the coefficients A,B,C, andjchave to be determined directly from experimental measurements.11,12 For the dc current applied to the wire the DW dy- namics governed by Eqs. (2) and (3) can be obtained explicitly.10ForJ < j candA/negationslash= 0 the DW only moves along the wire and is tilted on angle φ0from the transverse-anisotropy easy axis given by condition sin(2φ0) =J/jc. Thedriftvelocityis Vd=/angbracketleft˙z0(J)/angbracketright=AJ, see Eq. (2). Therefore, the linear slope of Vd(J) belowjc gives constant A, see Fig. 2 (a). The value of jcis deter- mined as the endpoint of this linear regime. At J=jc the magnetization angle becomes perpendicular to the easy axis, φ0=π/2. ForJ > jcthe DW both moves and rotates, and Eqs. (2) and (3) give Vd=AJ+B/radicalbig J2−j2c, so that the slope of Vd(J) at large JgivesA+B. Power optimization. The largestlossesin the nanowire with a DW are the Ohmic losses of the current. In gen- eral, the influence of the DW on the resistance is negli- gible and therefore we can assume that the resistance of the wire is constant with time. Then the time-averaged power of Ohmic losses is proportional to /angbracketleftJ2(t)/angbracketright. Since the resistance is almost constant, in this paper we will calculate P=/angbracketleftJ2(t)/angbracketrightand loosely call it the power of Ohmic losses. Our goal is to minimize the Ohmic losses while keeping the DW moving with a given constant drift velocity. For the following it will be convenient to introduce the dimensionless variables for time, drift velocity, current, power, and the ratio of slopes of Vd(J) at large and smallcurrents, τ=Cjct, vd=Vd Vc, j=J jc, p=P j2c, a=A+B A. (4) Although we note that in the special case of α=β, it can be shown that C=B= 0 and one cannot use dimensionless variables (4). However, in this case the DW dynamics is trivial:13the DW does not rotate φ= 0,πand moves with the velocity ˙ z0=J. First, we consider the case of dc current and the power as a function of drift velocity. For vd<1 we find pdc= v2 d. For currents above jcthe power pdc(vd) =j2is given intermsofdriftvelocity vd=j+(B/A)/radicalbig j2−1asshown in Fig. 2 (b). The poweris quadraticin vd, and for B <0 it has a discontinuity at vd= 1. In general, the DW motion has some period Tand currentj(τ) must be a periodic function with the same Tto minimize the Ohmic losses. Measuring the angle from the hard axis instead of easy axis and scaling it by 2, i.e, 2 φ=θ−π/2, we can write the dimensionless current drift velocity as6 j(τ) =˙θ/2−cosθ, vd=a 2/angbracketleft˙θ/angbracketright−/angbracketleftcosθ/angbracketright,(5) where˙θ=∂θ/∂τ. To minimize the power of Ohmic losses we need to find the minimum of /angbracketleftj2(τ)/angbracketrightat fixedvd, p=/angbracketleftBig (˙θ/2−cosθ)2−2ρ(a˙θ/2−cosθ−vd)/angbracketrightBig ,(6) where we use a Lagrange multiplier 2 ρto account for the constraint given by vdfrom Eq. (5). Power (6) can be considered as an effective action for a particle in a peri- odic potential U, and its minimization gives the equation of motion ¨θ/2 =−∂U/∂θwhich in turn can be reduced to ˙θ=±2/radicalbig d−U(θ,ρ), U(θ,ρ) =−cos2θ−2ρcosθ. (7) wheredis an arbitrary constant. Since changing ρ→ −ρ inUof Eq. (7) is equivalent to changing θ→π+θ, below we can consider only positive ρ. Eq. (7) shows that there are two different regimes: 1) the bounded regime where d <max[U(θ,ρ)] in which caseθis bounded, and the particle oscillates in potential wellU(θ), see inset of Fig. 3 (a); and 2) the rotational regime where d >max[U(θ,ρ)] with freely rotating mag- netization in the DW. In the bounded regime the particle moves between the two turning points −θ0andθ0given by d=U(±θ0,ρ). Sinceθis a bounded function /angbracketleft˙θ/angbracketright= 0 and vd=−/angbracketleftcosθ/angbracketright. One can show6that in this regime the power of Ohmic losses is minimal for dc current, i.e., p=v2 d. In the rotational regime the term in Eq. (5) with /angbracketleft˙θ/angbracketright should be kept because θis not bounded. The equation of motion is the same as for a nonlinear oscillator.6Using3 0 00 0.2 0.4 0.6 0.8 1 1.2 1.402468 1-2- -- 01 10 FIG. 3. (color online) (a) Minimal power of Ohmic losses p=/angbracketleftJ2/angbracketright/j2 cas a function of drift velocity Vdshown by solid line fora= 0.5. The dashed line depicts pfor dc current. The inset shows the potential U(θ) in which a “particle” is moving in the bounded (pendulum-like) and unbounded (rotational) regimes. A sketch of /angbracketleftJ2/angbracketright(Vd) shown by solid line in (b) for β≫α(a≪1) and (c) for β≪α(a≫1). the minimization condition ∂p/∂ρ|vd= 0 one finds /integraldisplayπ −π/radicalbig d−U(θ,ρ)dθ= 2πaρ. (8) This equation defines the relationship between dandρ. The results for the minimal power of Ohmic losses p(vd) are presented in Fig. 3. For a >1 there is a crit- ical velocity vrc<1, such that at vd< vrcthe power of Ohmic losses is p=v2 d=pdc. Above vrcone can minimize the Ohmic losses by moving DW with resonant current pulses. Right above vrcthere is a certain rangeof vdwherep= 2ρ0vd−ρ2 0withρ0(a)<1 given by Eq. (8) withd=ρ2. The critical velocity is found as vrc=ρ0(a). Fora <1, see e.g. Fig. 3 (a), we find that vrc= 1, whereas at vd>1 minimal power pis significantly lower thanpdc. Immediately above vd= 1 we find that there is a range of vdwherepis linear in vd. At large vdthe minimal power is always smaller than pdc, the difference between them then approaches pdc−p= (1−1/a)2/2. We note that even in the limiting cases of the systems with weak ( β≪α) or strong ( β≫α) non-adiabatic spin transfer torque, see Fig. 3 (b) and (c), where the power of Ohmic losses is high for dc currents, the optimized ac current gives dramatic reduction in heating power thus greatly expanding the range of materials which can be used for spintronic devices.1,3We also note that DMI suppresses critical current jcand affects parameter a. Forvd< vrcthe optimal current coincides with the dc current, above vrcthe resonant current j(t) is plotted in Fig. 4 for a= 2 and two different velocities vd. Atvd> vrcthe current’s maximum jmaxincreases from 2 −vrc at small enough vd<∼1 up to jmax≈vd/aatvd≫ 1. The current’s minimum increases monotonically from0 10 20 30 40 50 600123 FIG. 4. (color online) Resonant time-dependent current J(τ) withτ=Cjctfor drift velocities vd= 0.5 (dashed line) and vd= 4.5 (solid line) for a= 2. small positive values jmin=vrcatvd∼1 up tojmin= jmax−2|1−a|/aatvd≫1. Atvd<∼1 (fora >1) the time between the current picks decreases with increasing velocity as T≃(πa−2arcsinvrc)/(vd−vrc), whereas the pick’s width is given by ≈1.3//radicalbig (1−vrc). Therefore, at smallvd−vrcthe picks are widely separated, then as vd increases the time between the picks decreases. At vd≫ 1 the optimal current has a large constant component and small-amplitude ac modulations on top of it. Conclusions. We have studied the current driven DW dynamicsinthinferromagneticwires. Theultimatelower bound for the Ohmic losses in the wire has been found for any DW drift velocity Vd. We have obtained the ex- plicit time-dependence of the current which minimizes the Ohmic losses. We believe that the use of these res- onant current pulses instead of dc current can help to dramatically reduce heating of the wire for any Vd. We thank Jairo Sinova for valuable discussions. This work was supported by the NSF Grant No. 0757992 and Welch Foundation (A-1678). 1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008); M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, ibid.320, 209 (2008). 2D.A. Allwood, G. Xiong, M. D.Cooke, C. C. Faulkner, D.Atkin- son, N. Vernier, and R. P. Cowburn, Science 296, 2003 (2002). 3D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit , and R. P. Cowburn, Science 309, 1688 (2005). 4T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and T. Shinjo, Science 284, 468 (1999). 5A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). 6O. A. Tretiakov, Y. Liu, and Ar. Abanov, Phys. Rev. Lett., in press; arXiv:1006.0725. 7Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004). 8A. Thiaville et al., Europhys. Lett. 69, 990 (2005). 9O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Bazaliy, and O. Tchernyshyov, Phys. Rev. Lett. 100, 127204 (2008); D. J. Clarke, O. A. Tretiakov, G.-W. Chern, Y. B. Bazaliy, and O. Tchernyshyov, Phys. Rev. B 78, 134412 (2008). 10O. A. Tretiakov and Ar. Abanov, Phys. Rev. Lett. 105, 157201 (2010). 11S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102, 067201 (2009). 12Y. Liu, O. Tretiakov, and Ar. Abanov, (unpublished). 13S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005).

2010-10-07

The current mediated domain-wall dynamics in a thin ferromagnetic wire is investigated. We derive the effective equations of motion of the domain wall. They are used to study the possibility to optimize the power supplied by electric current for the motion of domain walls in a nanowire. We show that a certain resonant time-dependent current moving a domain wall can significantly reduce the Joule heating in the wire, and thus it can lead to a novel proposal for the most energy efficient memory devices. We discuss how Gilbert damping, non-adiabatic spin transfer torque, and the presence of Dzyaloshinskii-Moriya interaction can effect this power optimization.

Power optimization for domain wall motion in ferromagnetic nanowires

1010.1537v1

arXiv:1101.3144v1 [math.MG] 17 Jan 2011Steiner Ratio for Riemannian Manifolds D. Cieslik, A. O. Ivanov, A. A. Tuzhilin Abstract For a metric space ( X,ρ) and any finite subset N⊂Xbyρ(SMT N) andρ(MST N) we denote respectively the lengths of a Steiner minimal tree and a minimal spanning tree with the boundary N. TheSteiner ratiom(X,ρ) of the metric space is the value inf {N:N⊂X}ρ(SMTN) ρ(MSTN). In this paper we prove the following results describing the Ste iner ratio of some manifolds: (1) the Steiner ratio of an arbitrary n-dimensional connected Rieman- nian manifold Mdoes not exceed the Steiner ratio of Rn; (2) the Steiner ratio of the base of a locally isometric cover ing is more or equal than the Steiner ratio of the total space; (3) the Steiner ratio of a flat two-dimensional torus, a flat Kl ein bottle, a projective plain having constant positive curvature is eq ual to√ 3/2; (4) the Steiner ratio of the curvature −1 Lobachevsky space does not exceed 3 /4; (5) the Steiner ratio of an arbitrary surface of constant neg ative cur- vature−1 is strictly less than√ 3/2. Keywords: Steinerminimaltree(SMT),minimalspanningtree(MST), the Steiner problem, the Steiner ratio, metric space, Riema nnian mani- fold. 1 Introduction and main results LetVbe an arbitrary finite set. Recall that a graphGonVis the pair (V,E), where Eis a finite set that consists of some pairs of elements from V. Notice that Ecan contain several copies of some pair, and also Ecan contain the pairs ofthe form {v,v}, wherev∈V. Elements from Vare called vertices of G, and the elements from Eare called edges of G. The edges of the form ( v,v) are called loops, and ifEcontains several copies of an edge e={v,v′} ∈E, then the edge eis called a multiple edge . For a given graph Gwe denote the set of all its vertices by V(G), and the set of all its edges by E(G). For convenience, we shall often denote the edge e={x,y} ∈E(G) byxy. Sometimes it is useful to consider graphs as topological spaces glue d from segments each of which corresponds to an edge of the graph. Suc h graphs are A. Ivanov and A. Tuzhilin were partially supported by RFBR (g rants 96–15–96142 and 98–01–00240) and INTAS (grant 97–0808). 1Introduction and main results. 2 calledtopological graphs . A continuous mapping Γ from a topological graph G into a topological space is called a network; the topological graph G, and also the standard graph corresponding to G, are called the typeof Γ or the topology of Γ. Thus, the edges of a network are continuous curves in the am bient space. Moreover,all the terminology of the Topological Spaces Theory is t ransferred to the topological graphsand networks. If the ambient space is a smo oth manifold, then a network in such space is called smooth(piecewise-smooth ), if all its edges are smooth (piecewise-smooth). Agraph Gis calledweighted ifit isgivena non-negativefunction ω:E(G)→ Rcalled the weight function . The number ω(e) is called the weight of the edge e∈E(G). The sum of the weights over all edges of Gis called the weight of the graph Gand it is denoted by ω(G). IfGis a connected weighted graph, then the set of all connected spanning subgraphs of Ghaving the least weight contains a tree. Each such tree is called a minimal spanning tree and is denoted by MST G. Notice that if all the weights are strictly greater than zero, then any connected spanning subgraph of Gof the least weight is a tree. LetXbe a set, ρbe a metric on X, andNbe an arbitraryfinite subset of X. LetGbe a complete graph on N. The metric ρgenerates the weight function that assigns to each edge xy∈E(G) the number ρ(x,y). This weight function will be denoted by the same letter ρ. Minimal spanning tree in the graph Gis denoted by MST N. Aminimal Steiner tree on the set Nor aminimal Steiner tree spanning the set Nis defined to be a tree Γ, N⊂V(Γ), such that ρ(Γ) = inf {¯N:¯N⊂N}ρ(MST ¯N), (1) where the least upper bound is taken over all finite subsets ¯NinXthat contain N. A minimal Steiner tree on the set Nis denoted by SMT N. Note that, generally speaking, an SMT Nexists not for any N(one of the reasonsof that can be the incompleteness of the metric space ( X,ρ)). Neverthe- less, the greatest lower bound from the definition of SMT Ndoes always exist. In what follows, the greatest lower bound from (1) is always d enoted byρ(SMTN), irrespective of the existence of SMTN. The novelty of Steiner’s Problem is that new points, the Steiner point s, may be introduced so that an interconnecting network of all these poin ts will be shorter. Given a set of points, it is a priori unclear how many Steiner points one has to add in order to construct an SMT. Whereas Steiner’s Problem is very hard as well in combinatorial as in computational sense, the determ ination of a Minimum Spanning Tree is simple. Consequently, we are interested in Definition. TheSteiner ratio m(X,ρ)of a metric space (X,ρ) is defined as the following value: m(X,ρ) = inf {N:N⊂X}ρ(SMTN) ρ(MSTN). It is clear that the Steiner ratio of any metric space is always a nonne gative number with m(X,ρ)≤1. The Steiner ratio is a parameter of the consideredIntroduction and main results. 3 spaceanddescribestheapproximationratioforSteiner’sProblem. Thequantity m(X,ρ)·ρ(MSTN) would be a convenient lowerbound for the length of an SMT forNin (X,ρ); that means, roughly speaking, m(X,ρ) says how much the total length of an MST can be decreased by allowing Steiner points. Proposition 1.1 (E.F.Moore, in [3]) For the Steiner ratio of any metric space(X,ρ)the inequalities 1 2≤m(X,ρ)≤1 hold. It is also shown that these inequalities are the best possible ones ove r the class of metric spaces.1 As an introductory example consider three points which form the no des of an equilateral triangle of unit side length in the Euclidean plane. An MST for these points has length 2. An SMT uses one Steiner point, which is uniq uely determined by the condition that the three angles at this point are e qual, and consequently equal 120◦. Consequently, we find the length of the SMT in 3 ·/radicalbig 1/3 =√ 3. So we have an upper bound for the Steiner ratio of the Euclidean plane: m≤√ 3 2= 0.86602.... (2) A long-standing conjecture, given by Gilbert and Pollak [3] in 1968, sa id that in the above inequality equality holds. This was the most important con jecture in the area of Steiner’s Problem in the following years. Finally, in 1990, D u and Hwang [2] created many new methods and succeeded in proving the G ilbert- Pollak conjecture completely: The Steiner Ratio of the Euclidean plan e equals√ 3/2 = 0.86602....2 For each dimension n >2, at present, exact values for the Steiner ratios of the Euclidean spaces are not yet known. In particular, this is true f orn= 3. SMT’s have been the subject of extensive investigations during the past 30 years or so. Most of this research has dealt with the Euclidean metr ic, with much of the remaining work concerned with the L1-metric, or more generally, the usual Lp-metric or with two-dimensional Banach spaces. An overview for the Steiner ratios of these metric spaces is given in [1]. The first results concerning the Steiner ratios of Riemannian manifo lds dif- ferent from Euclidean spaces were obtained by J. H. Rubinstein and J. F. Weng in 1997, see [7]. They have shown that the Steiner ratio for the stan dard two- dimensional spheres is the same as for the Euclidean plane, that is,√ 3/2. Now we list the main results of the present article. These results wer e obtainedbymeansofthetechniqueworkedoutin[1], [5],and[6]. Letus mention that in [5] and [6] the authors investigateso called local minimal netwo rkswhich turn out to be useful in the subject. 1And, indeed, there are metric spaces with Steiner ratios equ als 1 and equals 0 .5. 2This mathematical fact went in The New York Times, October 30 , 1990 under the title ”Solution to Old Puzzle: How Short a Shortcut?”Proofs of the theorems. 4 Theorem 1.1 The Steiner ratio of an arbitrary n-dimensional connected Rie- mannian manifold Mdoes not exceed the Steiner ratio of Rn. Theorem 1.2 Letπ:W→Mbe a locally isometric covering of connected Riemannian manifolds. Then the Steiner ratio of the base Mof the covering is more or equal than the Steiner ratio of the total space W. Corollary 1.1 The Steiner ratio for a flat two-dimensional torus, a flat Klei n bottle, a projective plain having constant positive curvat ure is equal to√ 3/2. Thus, taking into account the results of J. H. Rubinstein and J. F. W eng [7], the Steiner ratio is computed now for all closed surfaces having non -negative curvature. Theorem 1.3 The Steiner ratio of the curvature −1Lobachevsky space does not exceed 3/4. Theorem 1.4 The Steiner ratio of an arbitrary surface of constant negati ve curvature −1is strictly less than√ 3/2. Theauthorswanttothankthe Ernst–Moritz–ArndtUniversityof Greifswald for the opportunity to work together in Greifswald in March 2000. A . Ivanov and A. Tuzhilin are grateful to academic A. T. Fomenko for his kind int erest to our work. 2 Proofs of the theorems In the present section we give the proofs of the theorems stated above. We need the following two Lemmas proved in [1] (notice that Lemma 2.1 is proved in [1] for the case of normalized spaces only, but the proof in the general case of metric spaces is just the same.) Lemma 2.1 LetXbe a set, and ρ1andρ2be two metrics on X. We assume that for some numbers c2≥c1>0and for arbitrary points xandyfromXthe following inequality holds: c1ρ2(x,y)≤ρ1(x,y)≤c2ρ2(x,y). Then c1 c2m(X,ρ2)≤m(X,ρ1)≤c2 c1m(X,ρ2). Lemma 2.2 Let(X,ρ)be a metric space, and Y⊂Xbe some its subspace. Then m(Y,ρ)≥m(X,ρ). The following Proposition holds.Proofs of the theorems. 5 Proposition 2.1 Letf:X→Ybe some mapping of a metric space (X,ρX) onto a metric space (Y,ρY). We assume that fdoes not increase the distances, that is, for arbitrary points xandyfromXthe following inequality holds: ρY/parenleftbig f(x),f(y)/parenrightbig ≤ρX(x,y). Then for arbitrary finite set N⊂Ywe have: ρX/parenleftbig MSTN/parenrightbig ≥ρY/parenleftbig MSTf(N)/parenrightbig , ρX/parenleftbig SMTN/parenrightbig ≥ρY/parenleftbig SMTf(N)/parenrightbig . Proof.LetGbe an arbitrary connected graph constructed on N. We consider two weight functions on Gdefined on the edges xyofGas follows: ρX(xy) = ρX(x,y), andωY(xy) =ρY/parenleftbig f(x),f(y)/parenrightbig . Sincefdoes not increasethe distances, thenρX(G)≥ωY(G). LetG′be a graph on N′=f(N), such that the number of edges joining the vertices x′andy′fromN′=V(G′) is equal to the number of edges from G joining the vertices from f−1(x′)∩Nwith the vertices from f−1(y′)∩N. It is clear that G′is connected, and ρY(G′) =ωY(G). Conversely, it is easy to see that for an arbitrary connected grap hG′con- structed on f(N) there exists a connected graph GXonN, such that ρY(G′) = ωY(GX). (To construct GXit suffices to span each set N∩f−1(x′),x′∈N′, by a connected graph, and then to join each pair of the construct ed graphs cor- responding to some adjacent vertices G′bykedges, where kis the multiplicity of the corresponding edge in G′). Therefore, ρX(MSTN) = inf {G:V(G)=N}ρX(G)≥inf {G:V(G)=N}ωY(G) = inf {G′:V(G′)=f(N)}ρY(G′) =ρY/parenleftbig MSTf(N)/parenrightbig . Thereby, the first inequality is proved. Now let us prove the second inequality. We have: ρX(SMTN) = inf {¯N:¯N⊃N}ρX(MST ¯N)≥inf {¯N:¯N⊃N}ρY(MSTf(¯N))≥ inf {¯N′:¯N′⊃f(N)}ρY(MST ¯N′) =ρY(SMTf(N)). The proof is complete. Proposition 2.2 Letf:X→Ybe a mapping of a metric space (X,ρX)to a metric space (Y,ρY), and let fdo not increase the distances. We assume that for each finite subset N′⊂Ythere exists a finite subset N⊂X, such that f(N) =N′and ρX(SMTN)≤ρY(SMTN′). (3) Then m(X,ρX)≤m(Y,ρY).Proofs of the theorems. 6 Proof.LetN⊂Xbe an arbitrary finite set. We have m(X,ρX) = inf {N:N⊂X}ρX(SMTN) ρX(MSTN)= inf {N′:N′⊂Y}inf {N:f(N)=N′}ρX(SMTN) ρX(MSTN)≤ inf {N′:N′⊂Y}ρY(SMTN′) ρY(MSTN′)=m(Y,ρY), where the inequality follows from both condition (3) and the first ineq uality of Proposition 2.1. The proof is complete. Proposition 2.2 can be slightly reinforced as follows. Proposition 2.3 Letf:X→Ybe a mapping of a metric space (X,ρX)to a metric space (Y,ρY), and let fdo not increase the distances. We assume that for each finite subset N′⊂Ythe following inequality holds: inf {N:f(N)=N′}ρX(SMTN)≤ρY(SMTN′). (4) Then m(X,ρX)≤m(Y,ρY). Proof.LetN⊂Xbe an arbitraryfinite set. As in the proofofProposition2.2, we have: m(X,ρX) = inf {N:N⊂X}ρX(SMTN) ρX(MSTN)= inf {N′:N′⊂Y}inf {N:f(N)=N′}ρX(SMTN) ρX(MSTN). Sincefdoes not increase distances, then ρX(MSTN)≥ρY(MSTf(N)) (see Proposition 2.1); on the other hand, due to our assumption, there exists a se- quenceoffinitesets Ni⊂X,f(Ni) =N′, suchthat ρX(SMTNi)≤ρY(SMTN′)+ εi, where the sequence of positive numbers εitends to 0 as i→ ∞, and the se- quence of positive numbers ρX(SMTNi) tends to inf {N:f(N)=N′}ρX(SMTN). Therefore, ρX(SMTNi) ρX(MSTNi)≤ρY(SMTN′)+εi ρY(MSTN′), and, taking in account that {Ni} ⊂ {N:f(N) =N′}, we get: inf {N′:N′⊂Y}inf {N:f(N)=N′}ρX(SMTN) ρX(MSTN)≤ inf {N′:N′⊂Y}inf {Ni}ρX(SMTNi) ρX(MSTNi)≤inf {N′:N′⊂Y}inf iρY(SMTN′)+εi ρY(MSTN′)= inf {N′:N′⊂Y}ρY(SMTN′) ρY(MSTN′)=m(Y,ρY). The proof is complete.Proofs of the theorems. 7 LetMbe an arbitrary connected n-dimensional Riemannian manifold. For each piecewise-smooth curve γby len(γ) we denote the length of γwith respect to the Riemannian metric. By ρwe denote the intrinsic metric generated by the Riemannian metric. We recall that ρ(x,y) = inf γlen(γ), where the greatest lower bound is taken over all piecewise-smooth curvesγ joining the points xandy. LetPbe a point from M. We consider the normal coordinates ( x1,...,xn) centered at P, such that the Riemannian metric gij(x) calculated at Pcoincides withδij. LetU(δ) be the open convex ball centered at Pand having the radius δ. Any two points xandyfrom the ball are joined by a unique geodesic γ lying inU(δ). At that time, ρ(x,y) = len(γ). Thus, the ball U(δ) is a metric space with intrinsic metric, that is, the distance between the points equals to the greatest lower bound of the curves‘ lengths over all the meas urable curves joining the points. Notice that in terms of the coordinates ( xi) the ball U(δ) is defined as follows: U(δ) =/braceleftbig (x1)2+···+(xn)2< δ2/bracerightbig . Therefore, if we define the Euclidean distance ρeinU(δ) (in terms of the normal coordinates( xi)), thenthemetricspace/parenleftbig U(δ),ρe/parenrightbig alsoisthespacewithintrinsic metric generated by the Euclidean metric δij. Since the Riemannian metric gij(x) depends on x∈U(ε) smoothly, then for anyε, 1/n2> ε >0, there exists a δ >0, such that |gij(x)−δij|< ε (5) for all points x∈U(δ). The latter implies the following Proposition. Proposition 2.4 Let/bardblv/bardblgbe the length of the tangent vector v∈TxMwith respect to the Riemannian metric gij, and let /bardblv/bardblebe the length of vwith respect to the Euclidean metric δij. If for any iandjthe inequality (5) holds, then /radicalbig 1−n2ε/bardblv/bardble≤ /bardblv/bardblg≤/radicalbig 1+n2ε/bardblv/bardble. Proof.Consider an orthogonal transformation (with respect to the Euc lidean metricδij) reducing the matrix ( gij) to the diagonal form diag( λ1,...,λ n), and let (ci j) be the matrix of this transformation. Then λk=/summationtext i,jci kcj kgij, therefore, using that |ci j| ≤1 due to orthogonality of ( ci j), we get: |λk−1|=/vextendsingle/vextendsingle/vextendsingle/summationdisplay i,j(ci kcj kgij−ci kcj kδij)/vextendsingle/vextendsingle/vextendsingle≤ /summationdisplay i,j|ci k|·|cj k|·|gij−δij| ≤/summationdisplay i,j|gij−δij| ≤n2ε.Proofs of the theorems. 8 So we have: /bardblv/bardblg=/radicalBigg/summationdisplay kλkvkvk≤/radicalBigg max kλk/summationdisplay kvkvk≤/radicalbig 1+n2ε/bardblv/bardble. Similarly, we get /bardblv/bardblg≥/radicalbig 1−n2ε/bardblv/bardble. The proof is complete. Using the definition of the distance between a pair of points of a conn ected Riemannian manifold, we obtain the following result. Corollary 2.1 LetMbe an arbitrary connected n-dimensional Riemannian manifold, and let U(δ),ρ, andρebe as above. Then for an arbitrary ε,1/n2> ε >0, there exists a δ >0, such that /radicalbig 1−n2ερe(x,y)≤ρ(x,y)≤/radicalbig 1+n2ερe(x,y) for all points x,y∈U(δ). Since the Steiner ratio is evidently the same for any convex open sub sets of Rn, Corollary 2.1 and Lemma 2.1 lead to the following result. Corollary 2.2 LetMbe an arbitrary n-dimensional Riemannian manifold, let U(ε)⊂Mbe an open convex ball of a small radius ε, and let Pbe the center of U(ε). Byρwe denote the metric on Mgenerated by the Riemannian metric. Then /radicalbigg 1−n2ε 1+n2εm(Rn)≤m/parenleftbig U(ε),ρ/parenrightbig ≤/radicalbigg 1+n2ε 1−n2εm(Rn), wherem(Rn)stands for the Steiner ratio of the Euclidean space Rn. Now let us prove the main theorems stated in Introduction. Proof of Theorem 1.1. LetMbe an arbitrary connected n-dimensional Rie- mannian manifold, and let ρbe the metric generated by the Riemannian metric ofM. LetP∈Mbe an arbitrary point from M, and let U(ε) be an open convex ball centered at Pand having radius ε <1/n2. As above, let ( xi) be normal coordinates on U(ε), and let ρebe the metric on U(ε) generated by the Euclidean metric δij(with respect to ( xi)). For some decreasing sequence {εi}of positive numbers with εi< εfor any i, whereεi→0 asi→ ∞, we consider a family of nested subsets Xi=U(εi). Notice that due to convexity of Euclidean balls/parenleftbig U(ε),ρe/parenrightbig we have: m/parenleftbig U(ε),ρe/parenrightbig =m(Rn). Besides, due to convexity of the balls U(ε) with respect to the intrinsic metric ρ′ generated by the Riemannian metric gij, this intrinsic metric ρ′coincides withProofs of the theorems. 9 the restriction of the metric ρ. Thus, the ball U(ε) with the intrinsic metric ρ′ is a subspace in ( M,ρ). Corollary 2.2 implies that m(Xi,ρ)≤/radicalbigg 1+n2ε 1−n2εm(Rn). Since/radicalBig 1+n2ε 1−n2ε→1 asi→ ∞due to the choice of {εi}, we get inf im(Xi,ρ)≤m(Rn). But, due to Lemma 2.2 we have: m(M,ρ)≤inf im(Xi,ρ). The proof is complete. Proof of Theorem 1.2. Letπ:W→Mbealocallyisometriccovering,where WandMare connected Riemannian manifolds. By ρWandρMwe denote the metrics generatedby the Riemannian metrics on WandM, respectively. Notice that a locally isometric covering does not increase distances, since t he image of a measurable curve γhas the same length as γhas. We consider an arbitrary finite set N′⊂M. LetG′ ibe a family of trees on finite sets ¯N′ i⊃N′such that ρM(G′ i)→ρM(SMTN′) asi→ ∞. Foreach G′ ibyΓ′ iwedenoteanembeddednetworkofthetype G′ ionMsuchthat the vertex set of Γ′ iisV(G′ i) and the length of Γ′ idiffers from ρM(G′ i) at most by 1/i. Let Γ ibe a connected component of π−1(Γ′ i), andNi=π−1(N)∩Γi. Since the network Γ′ iis contractible, then the restriction of the fibration π onto Γ′ iis trivial. Therefore the restriction of the projection πonto Γ iis a homeomorphism. Since the projection πis locally isometric, then the length of the network Γ iinWcoincides with the length of the network Γ′ iinM. But ρW(SMTNi) does not exceed the length of Γ i, therefore ρW(SMTNi)≤ρM(SMTN′)+εi, where the sequence {εi}of positive numbers tends to 0 as i→ ∞. So, inf {N:f(N)=N′}ρW(SMTN)≤ρM(SMTN′). It remains to apply Proposition 2.3. The proof is complete. Proof of Corollary 1.1. ItfollowsfromTheorems1.1, and1.2; DuandHwang theorem [2] saying that the Steiner ratio of the Euclidean plane equa ls√ 3/2; and also from Rubinstein and Weng theorem [7] saying that the Steine r ratio of the standard two dimensional sphere with constant positive cur vature metric equals√ 3/2.Proofs of the theorems. 10 Proof of Theorem 1.3. LetusconsiderthePoincar´ emodeloftheLobachevsky planeL2(−1) with constant curvature −1. We recall that this model is a ra- dius 1 flat disk centered at the origin of the Euclidean plane with Carte sian coordinates ( x,y), and the metric ds2in the disk is defined as follows: ds2= 4dx2+dy2 (1−x2−y2)2. It is well known that for each regular triangle in the Lobachevsky pla ne the circumscribed circle exists. The radii emitted out of the center of t he circle to the vertices of the triangle forms the angles of 120◦. Letrbe the radius of the circumscribed circle. The cosine rule implies that the length aof the side of the regular triangle can be calculated as follows: cosha= cosh2r−sinh2rcos2π 3= 1+3 2sinh2r. It is easy to verify that for such triangle the length of MST equals 2 a, and the length of SMT equals 3 r. Therefore, the Steiner ratio m(r) for the regular triangle inscribed into the circle of radius rin the Lobachevsky plane L2(−1) has the form m(r) =3 2·r arccosh/parenleftbig 1+3 2sinh2(r)/parenrightbig. It is easy to calculate that limit of the function m(r) asr→ ∞is equal to 3 /4. The proof is complete. Proof of Theorem 1.4. It is easy to see that the Taylorseriesfor the function m(r) atr= 0 has the following form: √ 3 2−r2 16√ 3+O(r4). Therefore, m(r) is strictly less than√ 3/2 in some interval (0 ,ε). The latter means that for sufficiently small regular triangles on the surfaces o f constant curvature −1, the relation of the lengths of SMT and MST is strictly less than√ 3/2. The proof is complete. References [1]D. Cieslik , Steiner minimal trees. — Dordrecht, Boston, London, Kluwer Academic Publishers, 1998. [2]D. Z. Du and F. K. Hwang , A proof of Gilbert–Pollak Conjecture on the Steiner ratio. — Algorithmica, v. 7 (1992) pp. 121–135. [3]E.N. Gilbert and H.O.Pollak , SteinerMinimalTrees.SIAMJ.Appl.Math., 16:1–29, 1968.Proofs of the theorems. 11 [4]F. K. Hwang, D. Richards, and P. Winter , The Steiner Tree Problem. — Elsevier Science Publishers, 1992. [5]A. O. Ivanov and A. A. Tuzhilin , Minimal Networks. The Steiner Problem and Its Generalizations. — N.W., Boca Raton, Florida, CRC Press, 199 4. [6]A. O. Ivanov and A. A. Tuzhilin , Branching Solutions of One-Dimensional Variational Problems. — World Publisher Press, 2000, to appear. [7]J. H. Rubinstein and J. F. Weng , Compression theorems and Steiner ratios on spheres. — J. Combin. Optimization, v. 1 (1997) pp. 67–78.

2011-01-17

The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed. Steiner ratio - Steiner problem - Gilbert--Pollack conjecture - surfaces of constant curvature

Steiner Ratio for Manifolds

1101.3144v1

arXiv:1103.1455v1 [math-ph] 8 Mar 2011Applicationof ExplicitSymplecticAlgorithmsto IntegrationofDampingOscillators TianshuLuoc, YimuGuod aInstituteofAppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou, Zhejiang,310027,P.R.China bInstituteofAppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou, Zhejiang,310027,P.R.China URL:ltsmechanic@zju.edu.cn (YimuGuo), guoyimu@zju.edu.cn (YimuGuo) PreprintsubmittedtoCommunicationsinNonlinearScience andNumericalSimulationOctober23,2018Applicationof ExplicitSymplecticAlgorithmsto IntegrationofDampingOscillators TianshuLuoc, YimuGuod cInstituteof AppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou, Zhejiang,310027,P.R.China dInstituteofAppliedMechanics,DepartmentofMechanics,Z hejiangUniversity, Hangzhou, Zhejiang,310027,P.R.China Abstract Inthispaperanapproachisoutlined. Withthisapproachsom eexplicitalgorithms canbeappliedtosolvetheinitialvalueproblemof n−dimensionaldampedoscil- lators. Thisapproach isbaseduponfollowingstructure: fo ranynon-conservative classical mechanical system and arbitrary initial conditi ons, there exists a con- servative system; both systems share one and only one common phase curve; and, the value of the Hamiltonianof the conservativesystem is, up to an additive constant, equal to the total energy of the non-conservative system on the afore- mentioned phase curve, the constant depending on the initia l conditions. A key way applying explicit symplecticalgorithms to damping osc illators is that by the Newton-Laplace principle the nonconservative force can be reasonably assumed to be equal to a function of a component of generalized coordi natesqialong a phase curve, such that the damping force can be represented a s a function analo- gous to an elastic restoring force numerically in advance. T wo numerical exam- plesare giventodemonstratethegoodcharacteristics ofth ealgorithms. Keywords: Hamiltonian,dissipation,non-conservativesystem,damp ing,explicit symplecticalgorithm 1. Introduction Feng[1, 2, 3, 4],Marsden[5],Neri[6] and Yoshida[7]had dev eloped a series of symplectic algorithms for Hamiltonian systems. These algo rithms possess some URL:ltsmechanic@zju.edu.cn (YimuGuo), guoyimu@zju.edu.cn (YimuGuo) PreprintsubmittedtoCommunicationsinNonlinearScience andNumericalSimulationOctober23,2018advantages. But it is difficult to apply these algorithms to d amping dynamical systems, because it has been stated in most classical textbo oks that the Hamilto- nian formalism focuses on solving conservative problems. D amping phenomena is very important in the modeling of dynamical systems, and c an not be avoided. Our aim is to apply someexplicit canonical algorithms to non linear damping dy- namical systems, which is generated generally by FE-method . These canonical algorithmsreportedinthispapercanbereadilyutilizedfo rcomputinglarge-scale nonlineardampeddynamicalsystems. Betch[8][9][10]attemptedtoapplydirectlysomeimplicit algorithmstodamp- ing systems. The implicit symplectic algorithms utilized b y Betch[8] possess a few good characteristic, e.g. energy-conservation, momen tum-consistence, etc... In terms of energy-conservation, implicit symplectic algo rithms might be better than explicit symplectic ones. But explicit symplectic sch emes might be more suitablefornonlinearproblems. If one needs to apply symplecticalgorithmsto a dissipative system, one must convertthedissipativesystemintoaHamiltoniansystemor findsomerelationship between thedissipativesystemand aconservativeone. In theliterature[11], wehavestatedapropositiondescrib ingarelationamong adampingdynamicalsystemand conservativeones: Proposition1.1. For any non-conservative classical mechanical system and a r- bitraryinitial condition, there exists a conservative sys tem; both systems sharing one and only one common phase curve; and the value of the Hamil tonian of the conservativesystemisequaltothesumofthetotalenergyof thenon-conservative systemontheaforementionedphasecurveandaconstantdepe ndingontheinitial condition. In other words, a dissipative ordinary equation and a conser vative equation may possess a common particular solution. In the next section, a n analytical exam- ples are given to explain this proposition. Readers can find t he detailed proof of Proposition1.1 inthereference[11] In the Literature [12] a basic explicit canonical integrato r is proposed. Based on this basic scheme, Neri[6] constructed 4-order explicit canonical integrator, andthenYoshida[7]proposedageneralmethodtoconstructh igherorderexplicit symplecticintegrator. UtilizingtheProposition1.1,wea pplythisclassofexplicit canonical integrators to damping dynamical systems. This p oint will be in detail statedin sec. 3. 32. One-dimensional Analytical Example Consideraspecialone-dimensionalsimplemechanical syst em: ¨x+c˙x=0, (1) wherecis aconstant. Theexact solutionoftheequationaboveis x=A1+A2e−ct, (2) whereA1,A2are constants. Differentiationgivesthevelocity: ˙x=−cA2e−ct. (3) From the initial condition x0,˙x0, we find A1=x0+˙x0/c,A2=−˙x0/c. Inverting Eq. (2)yields t=−1 clnx−A1 A2(4) and bysubstitutingintoEq. (3), such wehave ˙x=−c(x−A1) (5) Thedissipativeforce Fin thedissipativesystem(1)is F=c˙x. (6) SubstitutingEq. (5)intoEq. (6), theconservativeforce Fisexpressed as F=−c2(x−A1); (7) Clearly, the conservative force Fdepends on the initial condition of the dissi- pative system (1), in other words, an initial condition dete rmines a conservative force. Consequently,a newconservativesystemyields ¨x+F=0→¨x−c2(x−A1)=0. (8) Thestiffnesscoefficient in thisequationmustbenegative. Onecan readily verify that the particular solution (2) of the dissipativesystem c an satisfy the conserva- tiveone(8). Thispointagrees withProposition(1.1). Thepotentialoftheconservativesystem(8)is V=/integraldisplayx 0/bracketleftbig −c2(x−A1)/bracketrightbig dx=−c2 2x2+c2A1x 4Figure1: Relationshipbetweennonconservativesystem (1) andconservativeone(8) ThereforetheHamiltonianis ˆH=T+V=1 2p2−c2 2x2+c2A1x, wherep=˙x. Furthermore, Proposition (1.1) can be depicted by Fig. 2. Th e phase flow of conservativesystem(2) transforms thered area in phasespa ce to the purplearea; thephaseflowofconservativesystem(8)transformsthereda reatothegreenarea. The blue curve in Fig. 2 illustrates the common phase curve. I f one draws more common phase curves and phase flows, the picture will like a flo wer, the phase flow of the nonconservative system likes a pistil and phase flo ws conservative systemslikepetals. 3. Modification SymplecticNumerical Schemes 3.1. BasicExplicitSymplecticNumericalSchemes In the paper[12][6][7] a symplectic algorithm based second kind generation functionwas stated pppi+1=pppi−τHq(pppi+1,qqqi) qqqi+1=qqqi+τHp(pppi+1,qqqi),(9) wherethesuperscript idenotesthe i-thtimenode, qqqdenotescoordinatesand pppde- notescanonicalmomenta,and HdenotesHamiltonianquantity, Hq=∂H/∂qqq,Hp= 5∂H/∂ppp. IftheHamiltonianisseperable,i.e. H=U(ppp)+V(qqq),Vq=Hq,Up=Hp, thenthesymplecticscheme(9)abovebecomes an explicitsym plecticscheme: pppi+1=pppi−τVq(qqqi) qqqi+1=qqqi+τUp(pppi+1).(10) Forsomenonlinearvibrationmechanicalsystem, Vq=K(qqq)qqq. Let usconsideran n−dimensionalnonlinearoscillator: ¨qqq+C˙qqq+Kqqq=0, (11) whereCdenotes a non-linear damping coefficient matrix which depen ds onqqq, andKdenotes a non-linear stiffness matrix which depends on qqqand consists of twoparts K=ˇK+ˆK(ˇKis adiagonalmatrix). In accordance with Proposition 1.1, a conservative mechani cal system was found associated with the dissipative system (11) in additi on to its initial condi- tions. Subject to these initial conditions, the dissipativ e system (11) possesses a common phase curve γwith the conservativesystem. As in Eq. (7), we can con- sider that the components of the damping force C˙qqqdetermine the components of aconservativeforceon thephasecurve γ c11˙q1=ρ11(q1)...c1n˙qn=ρ1n(q1) ......... cn1˙q1=ρ21(qn)...cnn˙qn=ρnn(qn).(12) For convenience, this conservative force is assumed to be an elastic restoring force: ρ11(q1)=κ11(q1)q1...ρ1n(q1)=κ1n(q1)q1 ......... ρn1(q1)=κn1(qn)qn...ρnn(qn)=κnn(qn)qn.(13) In a similar manner, the components of the non-conservative forceˆKqqqare equalto thecomponentsofaconservativeforceon thephasec urveγ ˆK11q1=χ11(q1)...ˆK1nqn=χ1n(q1) ......... ˆKn1q1=χ21(qn)...ˆKnnqn=χnn(qn).(14) Theconservativeforcecan likewisebeassumedtoan elastic restoringforce: χ11(q1)=λ11(q1)q1...χ1n(q1)=λ1n(q1)q1 ......... χn1(q1)=λn1(qn)qn...χnn(qn)=λnn(qn)qn.(15) 6Byanappropriatetransformation,anequivalentstiffness matrix˜Kthatisdiagonal inform can beobtained ˜Kii=n ∑ l=1κil(ql)+λil(ql). (16) Consequently,an n-dimensionalconservativesystemisobtained ¨qqq+(ˇK+˜K)qqq=0 (17) which shares the commonphase curve γwith then-dimensionaldamping system described by (11). In this paper, the conservativesystem is called the ’substitute’ conservativesystem. TheLagrangian ofEqs.(17)is ˆL=1 2˙qqqT˙qqq−/integraldisplayqqq 000(ˇKqqq)Tdqqq−/integraldisplayqqq 000(˜Kqqq)Tdqqq, (18) withtheHamiltonian ˆH=1 2pppTppp+/integraldisplayqqq 000(ˇKqqq)Tdqqq+/integraldisplayqqq 000(˜Kqqq)Tdqqq, (19) where000isthezerovector,and ppp=˙qqq. HereˆHinEq. (19)isthemechanicalenergy of the conservativesystem (17), because/integraltextqqq 000(˜Kqqq)Tdqqqis a potential function such thatˆHisindependentofthepathtaken inphasespace. Subjecttoacertaininitialcondition,oneneedmerelytoso lvetheconservative system(17). But one must in advance obtain the numerical app roximation of the matrix˜Kfor a time step, such that one can utilize the algorithm (10) t o integrate the conservative system (17) for a time step. One can repeat t his process above up to the end. In this way one obtains the numerical particula r solution of the conservativesystem(17),whichisexactlythenumericalpa rticularsolutionofthe damping one. The he numerical approximation of the matrix ˇKcan be assumed as: ˜K= ˜K11...0 ......... 0...˜Knn (20) ˜Kj(qji)=cjl˙qi l/qji+ˆKjlqi l/qji Hencetheexplicitcanonical scheme(10)can bemodified into ˜Ki j(qji)=cjl˙qi l/qji+ˆKjlqi l/qji pji+1=pji−τ[Kj+˜Ki j(qji)]qi) qji+1=qi+τpji+1(21) 7The scheme above is a one order scheme. Furthermore one can co nstruct higher orderexplicitcanonicalschemesutilizingthemethodrepo rtedintheliteratures[6][7]. Nowconsideramap from zzz=zzz(0)tozzz′=zzz(τ): zzz′′′≈(h ∏ i=1eritEesiτF+O(τn+1))z, (22) where zzz=/bracketleftbigg ppp, qqq/bracketrightbigg ,zzz′=/bracketleftbigg ppp′, qqq′/bracketrightbigg , E=/bracketleftbigg 0 0 1 0/bracketrightbigg F=/bracketleftbigg 0−(K+˜K) 0 0/bracketrightbigg . In fact Eq.(22)is thesuccessionofthefollowingmappings, pj+1=pj−siτVq(qj) qj+1=qj+riτUp(pj+1). (23) Inrealitythedifferencebetweentheequationsaboveand Eq .(21)isthatthecoef- ficientssi,ribefore the time step τ. In the literature [7] a generalized method to determine si,riwere given. Therefore, thehigherorderexplicitcanonical scheme can berepresented as: ˜K(qj)= ˜K1(qj 1) 0 ... 0 ˜Kn(qj n) ˜Kα(qj α)=n ∑ l=1cαl˙qj l/qj α+ˆKαlqi l/qi α E=/bracketleftbigg0 0 1 0/bracketrightbigg F=/bracketleftbigg 0−(K+˜K) 0 0/bracketrightbigg zzzj+1=(h ∏ i=1esiτFeriτE)zzzj(24) 4. Numerical Examples Two exampleswillbegiventoshownthisnumerical method24. 4.1. TheFirstExample To begin,weconsideraVan DerPol’soscillator ¨x+µ˙x(x2−1)+x=0, (25) 8-2.5-2-1.5-1-0.5 0 0.5 1 1.5 2 2.5 0 10 20 30 40 50 60 70 80 90 100x tExplicit Symplectic Runge-Kutta Figure2: TheresonanceoftheVanDerPol’soscillator whereµ=10. The initial conditions are given by x0=1,˙x=0. We employee the4−order explicitsymplecticmethod(24)withcoefficients s1=s4=[2+(3√ 2+1/3√ 2)]/6,s2=s3=[1−(3√ 2+1/3√ 2)]/6, r1=r3=[2+(3√ 2+1/3√ 2)]/3,r2=−[2+(3√ 2+1/3√ 2)]/3,r4=0, and classical explicit 4 −order Runge-Kutta method to compute the resonance of the Van Der Pol’s oscillator (26) respectively, then employ a same method to in- tegrate the results to the total energy, which is the sum of th e mechanical energy and the work done by damping forces in the system (25). The bot h methods are run with the same step size τ=0.01. The resonance is shown in Fig. 2, and the totalenergy isshowninFig. 3. It is aparent from Fig. 3 that the explicit symplectic method (24) has qual- itatively different behavior to the Runge-Kutta method. Th e energy divergence between the explicit symplectic method and the exact soluti on is smaller than that between Runge-Kuttamethod and the exact solution. The energy divergence between the explicit symplectic method and Runge-Kutta met hod increases with the time evolution. Due to the increasement of the energy, th e phase difference between boththeresultsin Fig. 2 increases also withthetim eevolution. 9-2.5-2-1.5-1-0.5 0 0.5 1 0 10 20 30 40 50 60 70 80 90 100Total Energy tExplicit Symplectic Runge-Kutta Exact solution Figure3: Thetotalenergyofthe VanDer Pol’soscillator 4.2. TheSecond Example Inthesecondexample,weconsidera2 −dimensionaldampednonlinearDuff- ingoscillator 2¨q1+0.1˙q1+(2+0.1q2 1)q1+q2=0 3¨q2+0.2˙q2+q1+(2+0.2q2 2)q2=0,(26) with the initial conditions q1=0,q2=0,˙q1=0,˙q2=1. The program of the bothmethodswiththestepsize τ=0.01arecarriedouttosimulateEq. (26). The resonance is shown in Fig. 2, the numerical solution of the to tal energy is shown inFig.5. There is only tiny difference between resonance results of t he two methods, correspondingly,thedifferenceamongthetotalenergy obt ainedbythenumerical methods and anlytical methods is very tiny. As numerical exa mples in the other literatures[13], that explicit Runge-Kutta method must ca use numerical pseudo dissipation which might be positive or negative. The differ ence between our nu- merical examples and the examples in the literature[13] is t he total energy in our examplesand themechanicalenergy intheirexamples1. 1Fig.6.1in theliterature[13] 10-1.5-1-0.5 0 0.5 1 1.5 0 20 40 60 80 100 120 140 160 180 200 220q1 tExplicit Symplectic Runge-Kutta Figure4: The1-thdisplacementofthedampedDuffingoscilla tor 0.4995 0.4996 0.4997 0.4998 0.4999 0.5 0.5001 0 20 40 60 80 100 120 140 160 180 200Total Energy tExplicit Symplectic Runge-Kutta Exact solution Figure5: Total energyofthedampeddissipativeoscillator 115. Conclusions Wehaveintroducedaclassofexplicitsymplecticalgorithm stodissipativeme- chanicalsystemssuccessfully,bychangingthesealgorith msintothescheme.(24). Becausethealgorithms(24)areexplicitandpossessgooden ergypreservingchar- acteristics, the explicit symplectic algorithms (24) is qu ite suitable for long term integrationofarbitrary dimensionalnonlineardissipati vemechanical systems. References [1] K. Feng, On difference schemes and symplectic geometry, in: Ed. Feng Kang Proceeding of the 1984 Beijing Symposium on differenti al geometry and differential equations-computationof partial differ ential equations, Sci- encePress, Beijing,1985,pp. 42–58. [2] H. Wu, M. Qin, K. Feng, Construction of canonical differe nce schemes for hamiltonianformalismviageneratingfunctions, JCM7 (198 9)71–96. [3] H.Wu,M.Qin,K.Feng, Symplecticdifferenceschemesfor thelinearhamil- toniancanonicalsystems, JCM 8 (1990)371–380. [4] K.Feng, Thehamiltonianway forcomputinghamiltoniand ynamics, Math. Appl.56(1991)17–35. [5] J. E. Marsden, G. W. Patrick, S. Shkoller, Multisymplect ic geometry, vari- ational integrators, and nonlinear pdes, Communications i n Mathematical Physics199 (1998)351–395.Cited By (since1996): 129. [6] F. Neri, Lie algebras and canonical integration, Techni cal Report, Depart- mentofPhysics,UniversityofMaryland,1988. [7] H. Yoshida, Construction of higher order symplectic int egrators, Physics LettersA 150(1990)262–268. [8] S. Uhlar, P. Betsch, On the derivation of energy consiste nt time stepping schemes for friction afflicted multibody systems, Computer s & Structures 88(2010)737– 754. [9] S. Leyendecker, P. Betsch, P. Steinmann, Energy-conser ving integration of constrained hamiltonian systems a comparison of approache s, Computa- tionalMechanics 33(2004)174–185.10.1007/s00466-003-0 516-2. 12[10] P. Betsch, Energy-consistent numerical integration o f mechanical systems with mixed holonomic and nonholonomic constraints, Comput er Methods inApplied Mechanics and Engineering195 (2006)7020– 7035. Multibody DynamicsAnalysis. [11] T. Luo, Y. Guo, Infinite-dimensional Hamiltonian descr iption of a class of dissipativemechanicalsystems, ArXive-prints (2009). [12] K. Feng, M. Qin, The symplectic methods for the computat ion of hamilto- nian equations, in: Numerical Methods for Partial Differen tial Equations, Springer,Berlin, 1987,pp. 17–35. [13] C. Kane, J. E. Marsden, M. Ortiz, M. West, Variational in tegrators and the newmarkalgorithmforconservativeanddissipativemechan icalsystems, In- ternational Journal for Numerical Methods in Engineering 4 9 (2000) 1295– 1325. 13

2011-03-08

In this paper an approach is outlined. With this approach some explicit algorithms can be applied to solve the initial value problem of $n-$dimensional damped oscillators. This approach is based upon following structure: for any non-conservative classical mechanical system and arbitrary initial conditions, there exists a conservative system; both systems share one and only one common phase curve; and, the value of the Hamiltonian of the conservative system is, up to an additive constant, equal to the total energy of the non-conservative system on the aforementioned phase curve, the constant depending on the initial conditions. A key way applying explicit symplectic algorithms to damping oscillators is that by the Newton-Laplace principle the nonconservative force can be reasonably assumed to be equal to a function of a component of generalized coordinates $q_i$ along a phase curve, such that the damping force can be represented as a function analogous to an elastic restoring force numerically in advance. Two numerical examples are given to demonstrate the good characteristics of the algorithms.

Application of Explicit Symplectic Algorithms to Integration of Damping Oscillators

1103.1455v1

Spin motive forces due to magnetic vortices and domain walls M.E. Lucassen,1,G.C.F.L. Kruis,2R. Lavrijsen,2H.J.M. Swagten,2B. Koopmans,2and R.A. Duine1 1Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 2Department of Applied Physics, Center for NanoMaterials and COBRA Research Institute, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Dated: November 9, 2021) We study spin motive forces, i.e.,spin-dependent forces, and voltages induced by time-dependent magnetization textures, for moving magnetic vortices and domain walls. First, we consider the voltage generated by a one-dimensional eld-driven domain wall. Next, we perform detailed calcu- lations on eld-driven vortex domain walls. We nd that the results for the voltage as a function of magnetic eld dier between the one-dimensional and vortex domain wall. For the experimentally relevant case of a vortex domain wall, the dependence of voltage on eld around Walker breakdown depends qualitatively on the ratio of the so-called -parameter to the Gilbert damping constant, and thus provides a way to determine this ratio experimentally. We also consider vortices on a magnetic disk in the presence of an AC magnetic eld. In this case, the phase dierence between eld and voltage on the edge is determined by the parameter, providing another experimental method to determine this quantity. PACS numbers: 75.60.Ch, 72.15.Gd, 72.25.Ba I. INTRODUCTION One of the recent developments in spintronics is the study of spin motive forces1and spin pumping2. These eects lead to the generation of charge and spin currents due to time-dependent magnetization textures. The idea of spin motive forces due to domain walls is easily un- derstood on an intuitive level: if an applied current in- duces domain-wall motion,3{8Onsager's reciprocity the- orem tells us that a moving domain wall will induce a current. This idea was already put forward in the eight- ies by Berger9. In the case of a domain wall driven by large magnetic elds ( i.e.,well above the so-called Walker breakdown eld), a fairly simple approach to the prob- lem is justied where one goes to a frame of reference in which the spin quantization axis follows the magnetiza- tion texture.10This transformation gives rise to a vec- tor potential from which eective electric and magnetic elds are derived. Experimentally, the domain-wall in- duced voltage has recently been measured above Walker breakdown11, and the results are consistent with this ap- proach. It has also been shown that the induced voltage well above Walker breakdown is determined from a topo- logical argument that follows from the properties of the above-mentioned vector potential.12 The above approach only captures the reactive con- tribution to the spin-motive forces. When the velocity of the domain wall is below or just above Walker break- down, a theory is needed that includes more contribu- tions to the spin motive forces. Renewed interest has shed light on the non-adiabatic and dissipative contribu- tions to the spin motive forces13{15that are important in this regime. In this paper, we study this regime. The article is organized as follows. In Section II we summarize earlier results that give a general framework to compute electrochemical potentials for given time-dependent magnetization textures. In Section III we con- sider an analytical model for a one-dimensional domain wall and numerically determine the form of the spin accu- mulation and the electrochemical potential. The results agree with the known results for the potential dierence induced by a moving one-dimensional domain wall.13In Section IV we turn to two-dimensional systems and study a vortex domain wall in a permalloy strip. We use a micro-magnetic simulator to obtain the magnetization dynamics, and numerically evaluate the reactive and dis- sipative contributions to the voltage below and just above Walker breakdown and compare with experiment.11An- other example of a two-dimensional system is a vortex on a disk which we treat in Section V. For small enough disks, the magnetic conguration is a vortex. Both ex- perimentally and theoretically, it has been shown that a vortex driven by an oscillating magnetic eld will ro- tate around its equilibrium position.16{20This gives rise to a voltage dierence between the disk edge and cen- ter as was recently discussed by Ohe et al. [21]. Here, we extend this study by including both the reactive and the dissipative contributions to the voltage, that turn out to have a relative phase dierence. This gives rise to a phase dierence between the drive eld and voltage that is determined by the so-called -parameter. II. MODEL The spin-motive force eld F(~ x) induced by a time- dependent magnetization texture that is characterized at position ~ xby a unit-vector magnetization direction m(~ x;t) is given by13,14 Fi=~ 2[m
(@tmrim) +(@tm
rim)]:(1)arXiv:1103.5858v3 [cond-mat.other] 8 Apr 20112 This force eld acts in this form on the majority spins, and with opposite sign on minority spins. In this ex- pression, the rst term is the well-known reactive term.1 The second term describes dissipative eects due to spin relaxation13,14and is proportional to the phenomeno- logical-parameter, which plays an important role in current-induced domain-wall motion.3{8The spin accu- mulationsin the system follows from14 1 2 sdsr2s=r
F; (2) wheresd=p Dis the spin-diusion length, with  a characteristic spin- ip time and Dthe eective spin- diusion constant. Here, we assume that the spin- relaxation time is much smaller than the timescale for magnetization dynamics. The total electrochemical po- tentialthat is generated by the spin accumulation due to a non-zero current polarization in the system is com- puted from14 r2=P(r2sr
F); (3) where the current polarization is given by P= (" #)=("+#). Note that there is no charge accumulation for"=#, with"(#) the conductivity of the majority (minority) spin electrons. The magnetization dynamics is found from the Landau-Lifschitz-Gilbert (LLG) equation given by @m @t=m @Emm[m] ~@m m@m @t: (4) Here,Emm[m] is the micromagnetic energy functional that includes exchange interaction, anisotropy, and ex- ternal eld, and is the Gilbert damping constant. III. ONE-DIMENSIONAL DOMAIN WALL For one-dimensional problems the voltage dierence can be easily found. For example, an analytic expression for the electric current (which is the open-circuit equiva- lent of the chemical-potential dierence) was obtained by one of us for an analytical model for a one-dimensional driven domain wall.13In this section, we solve the po- tential problem for a one-dimensional domain wall and obtain the explicit position dependence of the spin accu- mulation and the chemical potential. A one-dimensional domain wall ( @ym=@zm= 0) is described by23 (x;t) = 2arctann eQ[xX(t)]=o ;  (x) = 0;(5) withm= (sincos;sinsin;cos). Here,Q=1 is called the topological charge of the domain wall since it indicates the way in which an external eld aects the domain-wall motion, i.e., a eld in the direction +^ zwillmove a domain wall in the direction Q^x. Here, we choose Q= 1. The domain wall width is indicated by . To study the time-evolution of a domain-wall, we let (x)!0(t) so that the wall is described by time- dependent collective coordinates fX(t);0(t)g, called position and chirality, respectively. For external elds smaller than the Walker-breakdown eld, there is no domain-wall precession ( i.e., the chirality is constant), and the domain wall velocity vis constant so that @tm=v@xm. Since@ym=@zm= 0, we immediately see that the rst term on the right-hand side of Eq. (1) vanishes, and that the force is pointing along the x-axis. We then nd that Fx= (v~=2)=(2cosh[x=]2). Due to symmetry we have that @y=@z=@ys=@zs= 0. In Figs. 1 and 2 we plot the spin accumulation and the electrochemical potential as a function of x. -10 -5 5 10x/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΛ -1-0.50.512Μs Λ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExtΒv/HBar Λsd/Slash1Λ/Equal100Λsd/Slash1Λ/Equal10Λsd/Slash1Λ/Equal12 Λ3 /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExtΒvs /HBar/Gradient.F FIG. 1: (color online) Spin accumulation as a function of position for several values of the spin-diusion length. The black dotted line gives the value of the source term. The spin accumulation tends to zero for x!1 . -10 -5 5 10xΛsd/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΛ20.511.522ΜΛ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΒv/HBarP Λsd/Slash1Λ/Equal100Λsd/Slash1Λ/Equal10Λsd/Slash1Λ/Equal1 FIG. 2: (color online) Electrochemical potential as a function of position. Note that the potential is proportional to the polarization and that on the horizontal axis the position xis multiplied by the spin-diusion length. From Fig. 2 we see that the total potential dier- ence
=(x!1 )(x!1 ) is independent of3 the spin-diusion length and linear in the parameter :
=~Pv= . Note that this result is only valid below Walker Breakdown. To nd the voltage for all elds Bwe generalize the results for the voltage dierence in Ref. [13] for general domain-wall charge Q. A general expression for the volt- age in one dimension is given by13
=~P 2jejZ dx[m
(@tm@xm) +@tm
@xm]: (6) We insert the ansatz [Eq. (5) with (x) =0(t)] into Eq. (6) and nd
=~P 2jej" Q_0+_X # : (7) To nd a time-averaged value for the voltage we consider the equations of motion for a domain wall that is driven by a transverse magnetic eld B22{24, contributing to the energygB
m, withg >0. The equations of motion forX(t) and0(t) are ultimately derived from the LLG equation in Eq. (4), and given by (1 +2)_0=gB ~K? 2~sin(20); (1 +2)_X =QgB ~QK? 2~sin(20): (8) Here,K?is the out-of-plane anisotropy constant. These equations are solved by h_0i=Sign(B) 1 +2Re2 4sgB ~2 K? 2~23 5; h_Xi=Q 1 +2 gB ~+h_0i ! (9) whereh::idenotes a time average. It follows that the voltage dierence for general topological charge is
=Sign(B)Q 1 +2~P 2jej(  gjBj ~  1 +  Re2 4sgB ~2 K? 2~23 5) :(10) Note that the overall prefactor Sign( B)Qmakes sense: inversion of the magnetic eld should have the same re- sult as inversion of the topological charge. In the above, we used a domain-wall ansatz with mag- netization perpendicular to the wire direction. Using the topological argument by Yang et al.12one can show that the result is more general and holds also for head- to-head and tail-to-tail domain walls. Therefore, for a one-dimensional domain wall, the reactive and dissipa- tive contributions, i.e.,the contributions with and with- outin the above expression, to the voltage always have opposite sign.IV. VORTEX DOMAIN WALL For more complicated two-dimensional structures the spin-motive force eld can have rotation and the simpli- ed expression in Eq. (6) is no longer valid so that we need to treat the full potential problem in Eqs. (1-3). Motivated by recent experimental results11we consider in this section the voltage induced by a moving vortex domain wall. We study the magnetization dynamics using a micro- magnetic simulator25from which we obtain the magneti- zation m(~ x;t). This simulator solves the LLG equation in Eq. (4). For comparison with the experiment by Yang et al.,11we simulate a permalloy sample that has the same dimensions as this experiment, i.e. 20nm 500nm 32m, which is divided in 1 1288192 lattice points. On this sample, we drive a head-to-head vortex domain wall by means of a magnetic eld that is pointing from right to left, such that the vortex moves from right to left. For several eld strengths, we obtain the magnetization m, and its time-derivative which allows us to compute the force eld Fat each lattice point. Next, we solve the matrix problem that is the discrete equivalent to the potential problem in Eqs. (2) and (3). For details on this calculation, see App. B. We rst investigate the velocity of the vortex domain wall as a function of the applied eld. We use the value = 0:02 for the Gilbert-damping parameter to obtain the curve in Fig. 3. 0 0.4 0.8 1.2 1.6 Field/LParen1mT/RParen1050100150200250300Velocity/LParen1m/Slash1s/RParen1 FIG. 3: Velocity of the vortex domain wall as a function of the magnetic eld strength for = 0:02. Above Walker break- down, the velocity is time-averaged. The line is a guide to the eye. The decrease in velocity for B= 1:5 mT signals Walker breakdown. Indeed, up to elds B= 1:4 mT, the vortex moves parallel to the long direction of the sample. ForB= 1:5 mT, the vortex domain wall motion is more complicate and has a perpendicular component.26,27We therefore expect that below Walker Breakdown, just like for the one-dimensional domain wall, the vortex domain wall only has a dissipative contribution to the voltage.4 Comparison with the experimental results of Ref. [11] shows that our velocity is roughly a factor 2 higher. This might be partly caused by a dierence in damping and partly by the presence of defects in the experiment which causes pinning and therefore a decrease of veloc- ity. The exact value of the Walker breakdown eld is hard to compare, since this depends also on the exact value of the anisotropy. Nonetheless our value for the Walker breakdown eld is of the same order as Ref. [11]. Moreover, what is more important is the dependence of wall velocity and wall-induced voltage on the magnetic eld normalized to the Walker-breakdown eld, as these results depend less on system details. 500 1000 1500 2000length50100 width-505 Μ/LParen1ΜV/RParen1 500 1000 1500 length FIG. 4: Electrochemical potential as a function of position for a moving vortex domain wall on the sample. The num- bers on the horizontal axes correspond to lattice points with separation a= 3:9nm. This specic gure is for = 0:02, H= 0:8mT (i.e. below Walker breakdown), P= 1 and sd=a. Note that the peak signals the position of the vortex core. An example of a specic form of the electrochemical potential on the sample due to a eld-driven vortex do- main wall is depicted in Fig. 4. We see that there is a clear voltage drop along the sample, like in the one- dimensional model. Additionally, the potential shows large gradients around the vortex core and varies along the transverse direction of the sample. For each eld strength, we compute the voltage dierence as a func- tion of time. For eld strengths below Walker Breakdown we nd that, as expected, only the dissipative term con- tributes and the voltage dierence rapidly approaches a constant value in time. This is understood from the fact that in this regime, the wall velocity is constant after a short time. The dissipative contribution to the voltage is closely related to the velocity along the sample, as can be seen in Fig. 5. Above Walker breakdown the reactive term con- tributes. We nd that for =, the oscillations in the reactive component compensate for the oscillations 20406080100120140 Time-2000200400/CapDeltaΜ/Slash1P/LParen1nV/RParen1 velocity/CapDeltaΜreac/CapDeltaΜdissFIG. 5: (color online) reactive (blue squares) and dissipative (red triangles) contributions to the voltage as a function of time. The numbers on the horizontal axis correspond to time steps of 0.565 ns. The green line gives the velocity along the sample, it is scaled to show the correlation with the voltage. These curves are taken for = 0:02,=and eld strength B= 1:6mT. in the dissipative component. If we look closely to Fig. 5 we see that the length of the periods is not exactly equal. The periods correspond to a vortex moving to the upper edge of the sample, or to the lower edge. The dierence is due to the initial conditions of our simulation. We average the voltage dierence over time to arrive at the result in Fig. 6. We see that the dissipative contribu- 0 0.4 0.8 1.2 1.6 Field/LParen1mT/RParen10100200300400/CapDeltaΜ/Slash1P/LParen1nV/RParen1/CapDeltaΜdiss/Plus/CapDeltaΜreac/CapDeltaΜreac/CapDeltaΜdiss FIG. 6: (color online) Voltage drop along the sample for = 0:02 and=. tion becomes smaller for elds larger than the Walker breakdown eld, whereas the reactive contribution has the same sign and increases. In fact, for =, the re- duction of the dissipative contribution is exactly compen- sated by the reactive contribution. The dependence is illustrated in Fig. 7. The behavior is fundamentally dif- ferent from the one-dimensional domain-wall situation: for the vortex domain wall, the dissipative contribution has the same sign as the reactive contribution. In order to understand the relative sign, we now dis- cuss general vortex domain walls. A single vortex (i.e.5 0 0.4 0.8 1.2 1.6 Field/LParen1mT/RParen10100200300400/CapDeltaΜ/Slash1P/LParen1nV/RParen1Β/Equal3Α/Slash12Β/EqualΑΒ/EqualΑ/Slash12Β/Equal0 FIG. 7: (color online) Voltage drop along the sample for = 0:02 and several values for . with vorticity q= +1) is described by two parameters: the chargep=1 indicates wether the central magnetic moment points in the positive or negative zdirection and the chirality Cindicates wether the magnetic mo- ments align in a clockwise ( C=1) or anti-clockwise (C= +1) fashion. We have a vortex that is oriented clockwiseC=1. The relative sign is explained from a naive computation of the voltage above Walker break- down that does not take into account the rotation of the spin-motive force eld
/Z dx[@tm
@xm+m
(@tm@xm)] =vxZ dx(@xm)2+vyZ dxm
(@ym@xm) /(+ 1)vx: (11) whereis a positive number and we used that above Walker breakdown m'm(xvxt;yvyt) withvx6= 0 andvy6= 0. Note that if vy= 0 (below Walker breakdown) the reactive term, i.e., the second term in the above expression, indeed vanishes. We used in the last line that above Walker breakdown vx/pvyandR dxm
(@ym@xm)/ p. The former equality is understood from a geometric consideration: consider a sample with a vortex characterized by C= 1,p= 1 and vxvy<0. By symmetry, this is equivalent to C=1, p=1 andvxvy>0. It is therefore clear that the sign ofvxvydepends on either the polarization, or the hand- edness of the vortex. Since we know from the vortex domain wall dynamics that reversal of the polarization reverses the perpendicular velocity,11we conclude that vxvydoes not depend on the handedness of the vortex. The latter equality is understood from a similar argu- ment:RR dxdym
(@ym@xm) changes sign under the transformation m!m. During this transformation, bothp!pandC!C, and therefore their prod- uct cannot account for the total sign reversal. Therefore, the integral depends on the polarization12but not on the handedness of the vortex. The positive number is obtained from our numerical simulation, which suggeststhat the magnetic-eld dependence of the voltage is
=Bconstant + (1=)j
reactivej:(12) Note that the sign of the relative contributions can also be obtained using the topological argument by Yang et al.12, which gives the same result. We compare our results in Fig. 7 with the experiment by Yang et al.11. If we assume that the voltage below Walker breakdown lies roughly on the same line as the voltages above Walker breakdown, their results suggest a slope of 10nV/Oe. For P 0:8, our results suggest a slope of (=)14nV/Oe. Taking into account our higher velocity, we nd that in the experiment is somewhat larger than . The decrease in slope of the voltage in Ref. [11] as Walker breakdown is approached from above also suggests  > . In conclusion, the behavior of the voltage around Walker breakdown allows us to determine the ratio =. In experiment, the potential dierence as a function of the applied magnetic eld would show an upturn or downturn around Walker breakdown as in Fig. 7, which corresponds to  < and > , respectively. V. MAGNETIC VORTEX ON A DISK On small disks (of size m and smaller) of ferromag- netic material the lowest energy conguration is a vor- tex. It has been shown that one can let the vortex rotate around its equilibrium position by applying an AC mag- netic eld16{20. This motion gives rise via Eq. (1) to a spin motive force on the spins, which induces a voltage on the edge of the disk relative to a xed reference voltage, e.g. the disk center. Ohe et al.21have shown that the reactive contribution to the spin motive force eld can be seen as a dipole that is pointing in the radial direction, i.e., the divergence of the force eld consists of a posi- tive and a negative peak along the radial direction (note that the divergence of the force eld can be seen as an eective charge). Rotation of this dipole gives rise to an oscillating voltage on the edge of the sample. Here, we consider also the dissipative contribution to the voltage. We consider a vortex on a disk with radius Rthat moves around its equilibrium position ( i.e., the center of the disk) at a distance r0from the center of the disk with frequency !. We use as a boundary condition that the magnetization on the edge of the disk is pointing perpendicular to the radial direction. In equilibrium, the micro-magnetic energy density of the form Jm
r2m K?m2 zis minimized by mx(x;y) =yp x2+y2cosh 2 arctan eCp x2+y2=i my(x;y) =xp x2+y2cosh 2 arctan eCp x2+y2=i mz(x;y) =psinh 2 arctan ep x2+y2=i ; (13)6 where the center of the vortex is chosen at x=y= 0. Here=p K?=Jis the typical width of the vortex core. For permalloy this length scale is of the order 10nm. The parameters pandCare dened as before, for de- niteness we choose p= 1,C=1. To describe clockwise circular motion of the vortex around its equilibrium po- sition at xed radius r0we substitute x!xr0sin(!t) andy!yr0cos(!t). Note that we assume that the form of the vortex is not changed by the motion, which is a good approximation for r0R. From the magnetization in Eq. (13), we compute the force eld using Eq. (1). The reactive and dissipative contributions to the divergence of the force eld are shown in Fig. 8. The direction of the dipoles follows Π 0Π 0Π Angle00.050.1r/Slash1R FIG. 8: The reactive (left) and dissipative (right) contribu- tions to the divergence of the force eld. White means posi- tive values, black is negative values. The reactive contribution can be seen as a dipole in the radial direction. The dissipative contribution is a dipole perpendicular to the radial direction. directly from Eq. (1) if we realize that for our system @tm
~rm=~ v(@^vm)2is always pointing in the direc- tion of the velocity which shows that the dissipative con- tribution points along the velocity. Likewise the reactive contribution is always pointing perpendicular to the ve- locity. From the relative orientations of the eective dipoles, we expect that the peaks in the reactive and dissipative contributions to the voltage on the edge will dier by a phase of approximately =2 (forr0=R!0 this is exact). We divide our sample in 1000 rings and 100 angles and use the general method in App. B to nd the voltage on the edge shown in Fig. 9. To compare with Ref. [21], we take a frequency !=(2) = 300 MHz and P= 0:8, which yields amplitudes for the reactive contribution of V on the edge. However, Ohe et al. suggest that voltage probes that are being placed closer to the vortex core measure a higher voltage. This indeed increases the voltage up to order 10V atr= 2r0. Placing the leads much closer to the vortex core does not seem to be realistic because of the size of the vortex. Since the voltage scales with velocity, it can also be increased by a larger radius of rotation, i.e. by applying larger magnetic elds. However, for disks larger than 1 m, the vortex 0 Π 2Π 3Π 4Π Ωt-15-10-50510152 Π /CapDeltaΜ/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtΩP/LParen1ΜV/Slash1GHz/RParen1 /CapDeltaΜtotal/CapDeltaΜdiss/Slash1Β/CapDeltaΜreacFIG. 9: (color online) The reactive (red dashed curve) and dissipative (blue dashed curve) contributions to the voltage dierence between opposite points on the edge of the disk. The green line gives the total voltage dierence
total =
reac+
diss, in this example for = 0:4. We used r0= 10sd,R= 100sdand=sd. For realistic spin-diusion lengthsd'5 nm, these parameter values agree with the system of Ohe et al.21. structure is lost. The dissipative contribution becomes important for large values of . In principle, it is possible to determine by looking at the shift of the peak in the total volt- age with respect to the peak in the reactive contribution, which is in turn determined by the phase of the applied magnetic eld. The phase dierence between applied eld and measured voltege then behaves as tan(
)/. VI. DISCUSSION AND CONCLUSION We have investigated the voltage that is induced by a eld-driven vortex domain wall in detail. In contrast to a one-dimensional model of a domain wall, the reactive and dissipative contribution to the voltage have the same sign. The qualitative dierences for dierent values of  provide a way to determine the ratio =experimentally by measuring the wall-induced voltage as a function of magnetic eld. To this end the experimental results in Ref. [11] are in the near future hopefully extended to elds below Walker breakdown, which is challenging as the voltages become smaller with smaller eld. We also studied a magnetic vortex on a disk. When the vortex undergoes a circular motion, a voltage is in- duced in the sample. Earlier work computed the reactive voltage on the edge of the disk,21here we include also the dissipative contribution to the voltage. We nd that the phase dierence between voltage and AC driving eld is determined by the -parameter.7 Acknowledgments This work was supported by the Netherlands Organiza- tion for Scientic Research (NWO) and by the European Research Council (ERC) under the Seventh Framework Program (FP7). Appendix A: Boundary conditions As a boundary condition for the potential problems, we demand that the total spin current and charge current perpendicular to the upper and lower boundaries is zero: j? s=j? "j? #= 0 andj? s=j? "+j? #= 0. Therefore, the majority-and minority spin currents are necessarily zero. They are given by j"="(Fr") andj#= #(Fr#). From this, the boundary conditions on the derivatives of the potentials follow as @?s=@?(" #)=2 =Fand@?=@?("+#)=2 = 0. We consider a two-dimensional sample that is innitely long in the x-direction, and of nite size 2 in the y-direction the boundary conditions are @ys(x;y=) =Fy(x;y=): (A1) To measure the induced voltage, we also put the deriva- tives of the potential at innity to zero so that the bound- ary conditions for the electrochemical potential are @y(x;y=) = 0; @x(x!1;y) = 0: (A2) Appendix B: Potential problem on a Lattice We consider a two-dimensional lattice, where we have spin accumulation i;j sand an electrochemical potential i;jat sitei;j. Between sites ( i;j) and (i;j+ 1), there can be a particle current density of majority spins ji;j+1=2 ";^ =" Fi;j+1=2 ^ +i;j "i;j+1 " ai;j+1=2 ^ ! =" Fi;j+1=2 ^ ^ i;j+1=2 " ; (B1) withai;j+1=2 ^ the lattice spacing in the ^  direction between sites (i;j) and (i;j+ 1), and a particle current density of minority spins ji;j+1=2 #;^ =# Fi;j+1=2 ^ +i;j #i;j+1 # ai;j+1=2 ^ ! =# Fi;j+1=2 ^ ^ i;j+1=2 " ; (B2) and equivalently for currents in the ^  direction. The derivative ^ is dened as ^ Oi;j= (Oi;j+1=2 Oi;j1=2)=ai;j ^ , and likewise for ^ . Note that upper in- dices (i;j) denote a position on the lattices and lower in- dices ^  or ^  denote a direction. We can write "=+sand#=s. The continuity-like equations for the density of majority- and minority spins are (note that spins move in the direction of the current) Ai;jni;j "# =
(`i;jji;j "#) jej; (B3) with characteristic spin- ip time and with the dimen- sionless operator
given by
Oi;j=Oi+1=2;j ^ Oi1=2;j ^ +Oi;j+1=2 ^ Oi;j1=2 ^ : (B4) These denitions allows for non-square lattices with sides at position ( i1=2;j) or (i;j1=2) that have length `i1=2;j ^ or`i;j1=2 ^ (lower index denotes the normal di- rection), respectively, and the area of the site itself given byAi;j. The equation for the electrochemical potential is ob- tained from the continuity equation 0 =jejAi;jni;j "+ni;j # =
[`i;j(ji;j "+ji;j #)] = "
[`i;j(Fi;ji;j ")] +#
[`i;j(Fi;ji;j #)] = ("+#)
f`i;j[i;j+P(Fi;ji;j s)]g: !
(`i;ji;j) =P
[`i;j(Fi;ji;j s)]; (B5) where the current polarization is given by P= (" #)=("+#). This result was already obtained for a continuous system in Ref. [14]. To nd an equation for the spin accumulation, we write jejAi;jni;j "ni;j # =
[`i;j(ji;j "ji;j #)] = "
[`i;j(Fi;ji;j ")]#
[`i;j(Fi;ji;j #)] = ("+#)
f`i;j[Fi;ji;j sPi;j]g= ("+#)(1P2)
[`i;j(Fi;ji;j s)]: (B6) If we compare this in the case of a square lattice to the expression in Ref.14 1 2 sdsr2s=r
F; (B7) we nd that the density of spins that pile up can be expressed in terms of the spin accumulation as ( ni;j " ni;j #)== ("+#)(1P2)i;j s=(jej2 sd). We insert this expression to nd that the spin accumulation on a lattice is determined by 1 2 sdi;j s=1 Ai;j
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2011-03-30

We study spin motive forces, i.e, spin-dependent forces, and voltages induced by time-dependent magnetization textures, for moving magnetic vortices and domain walls. First, we consider the voltage generated by a one-dimensional field-driven domain wall. Next, we perform detailed calculations on field-driven vortex domain walls. We find that the results for the voltage as a function of magnetic field differ between the one-dimensional and vortex domain wall. For the experimentally relevant case of a vortex domain wall, the dependence of voltage on field around Walker breakdown depends qualitatively on the ratio of the so-called $\beta$-parameter to the Gilbert damping constant, and thus provides a way to determine this ratio experimentally. We also consider vortices on a magnetic disk in the presence of an AC magnetic field. In this case, the phase difference between field and voltage on the edge is determined by the $\beta$ parameter, providing another experimental method to determine this quantity.

Spin motive forces due to magnetic vortices and domain walls

1103.5858v3

arXiv:1106.4861v1 [nlin.PS] 23 Jun 2011Ratchet effect on a relativistic particle driven by external forces Niurka R. Quintero1, Renato Alvarez-Nodarse2and Jos´ e A. Cuesta3 1Departamento de F´ ısica Aplicada I, E. S. P., Universidad de Sevilla, C/ Virgen de ´Africa 7, E-41011, Sevilla, Spain 2Departamento de An´ alisis Matem´ atico, Universidad de Sevilla, apdo . 1160, E-41080, Sevilla, Spain 3Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamen to de Matem´ aticas, Universidad Carlos III de Madrid, avda. de la Univers idad 30, E-28911 Legan´ es, Madrid, Spain E-mail:niurka@us.es ,ran@us.es ,cuesta@math.uc3m.es Abstract. We study the ratchet effect of a damped relativistic particle driven b y both asymmetric temporal bi-harmonic and time-periodic piecewise c onstant forces. This system can be formally solved for any external force, providin g the ratchet velocity as a non-linear functional of the driving force. This allows us to explicitly illustrate the functional Taylor expansion formalism recently propo sed for this kind of systems. The Taylor expansion reveals particularly useful to obta in the shape of the current when the force is periodic, piecewise constant. We also illust rate the somewhat counterintuitiveeffectthatintroducingdampingmayinducearatch eteffect. When the force is symmetric under time-reversal and the system is undampe d, under symmetry principles no ratchet effect is possible. In this situation increasing da mping generates a ratchet current which, upon increasing the damping coefficient ev entually reaches a maximum and decreases toward zero. We argue that this effect is no t specific of this example and should appear in any ratchet system with tunable dampin g driven by a time-reversible external force. PACS numbers: Submitted to: J. Phys. A: Math. Gen.Ratchet effect on a relativistic particle 2 1. Introduction The ratchet effect is identified with the motion of particles or solitons induced by zero-average periodic forces [1, 2], sometimes in the presence of t hermal fluctuations. The effect arises as a subtle interplay between nonlinearities in the sy stem and broken symmetries. Ratchets appear in many fields of physics, where net m otion is generated either by an asymmetric, periodic, spatial potential [3, 4, 5, 6, 7, 8 , 9, 10], or by an asymmetric temporal forcing [10, 11, 12, 13, 14, 15, 16, 17, 18]. I n bothcases the ratchet effect can be regarded as an application of Curie’s symmetry principle , which states that a symmetry transformation of the cause (forces) is also a symmet ry transformation of the effect (ratchet velocity) [19, 20]. Most studies of ratchets driven by temporal forces employ a bi-ha rmonic forcing f(t) =ǫ1cos(qωt+φ1)+ǫ2cos(pωt+φ2), (1) wherepandqare positive integers which, without loss of generality, can be taken co- prime (otherwise common factors can be absorbed in the frequenc yω) andǫ1,ǫ2are small non-zero parameters. If both pandqare odd, the force (1) exhibits the shift symmetry (Sf)(t) =f(t+T/2) =−f(t), where T= 2π/ω. In systems invariant under time translations this implies that both, f(t) and−f(t) generate the same ratchet current (or velocity) defined as [21, 22] v= lim t→+∞1 t/integraldisplayt 0˙x(τ)dτ= lim t→+∞x(t) t, (2) wherex(t) is the position of the particle, soliton, or localized structure. If re versing the force changes the sign of the current, this current must be z ero. So shift-symmetric bi-harmonic forces cannot induce a ratchet effect. In contrast, ifpandqhave different parity, shift symmetry is broken by f(t) so the force can induce a nonzero net current [23]. Many attempts have been made to determine quantitatively the dep endence of the ratchetvelocity, v,ontheparametersofthebi-harmonicforce(1)[11,24,25]. Inva riably, the analysis performed in these works rests on the so-called method of moments, where it is assumed that the average ratchet velocity can be expanded as a series of the odd moments of f(t), i.e./summationtext∞ k=1/angbracketleft[f(t)]2k+1/angbracketrightwith/angbracketlefth(t)/angbracketright=/integraltextT 0dth(t). This method seemed to work for some systems but not for others without a clear reaso n and with no known criterion to tell ones from the others. We have recently shown tha t the moment method relies on an assumption that almost never holds, and have provided a n alternative procedure that yields the correct result regardless of the syste m [23]. The aimofthispaper istoprovide explicit examples which illustratesthis otherwise abstractmethod—thefunctionalexpansionof vintermsof f—usingaworkingexample for which an analytic solution can be found. The system represents the motion of a damped, relativistic particle under the effect of two different force s: a bi-harmonic forceRatchet effect on a relativistic particle 3 like (1), and a time-periodic piecewise constant force like f(t) = ǫ1if 0< t < T l, 0 if Tl< t < T−Tl, −ǫ1ifT−Tl< t < T.(3) To this purpose we introduce the model as well as its analytic solution in section 2. In section 3 we discuss the phenomenon of damping-induced ratche ts. The formalism developed in [23] is fully illustrated for this problem in section 4. For the se two specific driving forces it is also shown that the method of moments is valid only w hen the dynamics of the relativistic particle is overdamped, and fails otherwis e. Conclusions are summarized in section 5. 2. Motion of a relativistic particle driven by a bi-harmonic force The equation of motion of a relativistic particle with mass M >0, whose position and velocity at time tare denoted x(t) andu(t), respectively, is dx dt=u(t), x (0) =x0, (4a) Mdu dt=−f(t)(1−u2)3/2−γu(1−u2), u(0) =u0, (4b) wherex0andu0are the initial conditions, γ >0 represents the damping coefficient andf(t) is aT-periodic driving force. Notice that if the force f(t) satisfied ( Sf)(t) = f(t+T/2) =−f(t), then (4 b) would be invariant under a combination of shift symmetry (S:t/mapsto→t+T/2) and the sign change x/mapsto→ −x. Changing the variable u(t) by the momentum P(t) =Mu(t)/radicalbig 1−u2(t)(5) transforms (4 b) into the linear equation dP dt=−βP−f(t), P(0) =P0=Mu0/radicalbig 1−u2 0, (6) whereβ=γ/M. Equation (6) is easily solved to give P(t) =P0e−βt−/integraldisplayt 0dzf(z)e−β(t−z). (7) From (5) one obtains u(t) =∞/summationdisplay k=0/parenleftbigg −1 2/parenrightbiggk(2k−1)!! k!/parenleftbiggP(t) M/parenrightbigg2k+1 . (8) Let usnowfocusourattentiononthe T-periodicdriving force f(t)givenby (1)with p= 2 and q= 1 (the most common choice of parameters [26, 22, 14, 27]). Subst ituting (1) into (7) leads to P(t) =/tildewideP0e−βt−˜ǫ1cos(ωt+φ1−χ1)−˜ǫ2cos(2ωt+φ2−χ2),Ratchet effect on a relativistic particle 4 with /tildewideP0=P0+˜ǫ1cos(φ1−χ1)+˜ǫ2cos(φ2−χ2), ˜ǫ1=ǫ1(β2+ω2)−1/2, χ 1= tan−1(ω/β), ˜ǫ2=ǫ2(β2+4ω2)−1/2, χ 2= tan−1(2ω/β). Ast→ ∞the momentum P(t) behaves, for any β >0, as P(t)∼ −˜ǫ1cos(ωt+φ1−χ1)−˜ǫ2cos(2ωt+φ2−χ2), thus the term P(t)2k+1in (8) isO(ǫr 1ǫs 2) withr+s= 2k+1. Since the time average of P(t) is zero, the leading term of (2) in powers of ǫ1andǫ2will be −1 2M3lim t→∞1 t/integraldisplayt 0P(τ)3dτ=3 2M3T˜ǫ2 1˜ǫ2/integraldisplayT 0cos(ωτ+φ1−χ1)2cos(2ωτ+φ2−χ2)dτ =3 8M3˜ǫ2 1˜ǫ2cos(2φ1−φ2+χ2−2χ1). Therefore, the rachet velocity (2), for small amplitudes ǫ1andǫ2, is given by v=Bǫ2 1ǫ2cos(2φ1−φ2+θ0), (9) with B=3 8M3(β2+ω2)/radicalbig β2+4ω2, θ 0=χ2−2χ1=−tan−1/parenleftbigg2ω3 β(β2+3ω2)/parenrightbigg ,(10) in agreement with the result reported in [23]. Notice that in the undamped limit γ→0 (equivalently β→0) the parameters (10) become B=3 16M3ω3, θ 0=−π 2, whereas in the overdamped limit M→0 (and therefore β→ ∞withMβ=γ) we get B=3 8γ3, θ 0= 0, both limits agree with the predictions of [23]. 3. Ratchet induced by damping The depence of von parameters of the system like the damping coefficient (through β=γ/M) shown in (9) and (10) reveals an interesting effect. If we take φ1=φ2= 0 in f(t) and do some algebra, the ratchet velocity (for small amplitudes of the force) turns out to be v=ǫ2 1ǫ2 ω3M3V(β/ω), V(x) =3x(x2+3) 8(x2+1)2(x2+4). (11) Function V(x) is depicted in Figure 1. The most remarkable observation is that the currentincreases up to a maximum with increasing damping before it begins to show the expected decay. Intuition dictates that the current should d ecrease with dampingRatchet effect on a relativistic particle 5 0 1 2 3 4 5 β/ω00.020.040.060.080.1v (Mω)3/ε12ε2 Figure 1. Plot of the current velocity v, in units of ǫ2 1ǫ2/(Mω)3, vs. the damping coefficient β, in units of the frequency, ω, induced by a biharmonic force like (1) with φ1=φ2= 0. Notice that this force is time reversible, i.e., f(−t) =f(t). because friction opposes movement, so the fast increase it revea ls for small damping is counterintuitive. The cause of this effect is the interplay between the breaking of the time-reversal symmetry R:t/mapsto→ −tthat generates the ratchet current, and the damping that hinde rs it [27]. In the limit β→0 the system (2) is invariant under Rand a sign change of u, because for φ1=φ2= 0 the force (1) satisfies f(−t) =f(t). Accordingly v= 0 in this limit. But introducing damping breaks the symmetry of the equat ion and induces a net movement of the particle. For small damping, the higher the da mping coefficient βthe larger v. If we keep on increasing βeventually the friction it introduces in the movement of the particle causes the decay of vasβ−3. This argument makes it clear that in any ratchet system with a tunab le damping and undergoing the action of a time-reversible bi-harmonic force, t he ratchet effect can be generated upon increasing damping above zero. 4. Ratchet velocity as a functional of the force The starting point to obtain formula (9) for a ratchet system is to r ealize that vis a functional of f(t) and that, under certain regularity assumptions, one such funct ional can be expanded as a functional Taylor series [28, 29, 30] as v[f] =/summationdisplay nodd/integraldisplayT 0dt1 T···/integraldisplayT 0dtn Tcn(t1,...,t n)f(t1)···f(tn), (12) where the kernels cn(t1,...,t n) are proportional to the nth functional derivatives of the functional v[f]. These kernels can be taken T-periodic in each variable and totally symmetric under any exchange of variables. Only odd terms appear in this expansion as a cosequence of the symmetry v[−f] =−v[f] that these systems have.Ratchet effect on a relativistic particle 6 Thatvis indeed a functional of f(t) in this example is obvious from equations (2)– (7). The aim of this section is to determine explicitly the expansion (12 ) for this exactly solvable example. Let us start off by rewriting the integral in (7) as /integraldisplayt 0dzf(z)e−β(t−z)=I1(t)+I2(t), (13) I1(t) =n(t)/summationdisplay k=1/integraldisplayT 0dzf(z)e−β(t−z−(k−1)T), I2(t) =/integraldisplayα(t) 0dzf(z)e−β(α(t)−z), whereα(t) =t−n(t)Tandn(t) = [t/T] ([X] denoting the integer part of X). Notice thatα(t+T) =α(t). Now, since S(t)≡n(t)/summationdisplay k=1eβ(k−1)T=eβnT−1 eβT−1, (14) thenI1(t) = e−βtCS(t), with C=/integraldisplayT 0dzf(z)[eβz−1]. (15) Using this form in (7) we can write P(t) =Ae−βt+/tildewideP(t), (16) whereA=P(0)+C(eβT−1)−1and/tildewideP(t) is theT-periodic function /tildewideP(t) =−1 eβT−1/integraldisplayT 0dyf(y)e−βα(t)[eβy−1]−/integraldisplayα(t) 0dyf(y)e−β(α(t)−y). It is thus enough to obtain /tildewideP(t) in the interval 0 ≤t < T, where it adopts the compact form /tildewideP(t) =−/integraldisplayT 0dyf(y)e−β(t−y)χ(y,t), (17) defining χ(y,t) =1−e−βy eβT−1+Θ(t−y) (18) (as it is customary, Θ( x) denotes the Heaviside function, which is 1 if x >0 and 0 otherwise). Equations (17)–(18) have a well defined β→0+limit, namely /tildewideP(t) =−/integraldisplayT 0dyf(y)χ1(y,t), χ 1(y,t) =y T+Θ(t−y). (19) Ontheotherhand, forzero-averageforces f(t)thekernel χ(t,z)canbefurthersimplified to /tildewideP(t) =−/integraldisplayT 0dyf(y)e−β(t−y)χ2(y,t), χ 2(y,t) =1 eβT−1+Θ(t−y). (20)Ratchet effect on a relativistic particle 7 Whatever the form, it should be periodically extended beyond the int erval [0,T). It is then clear that (2) and (8) boil down to v=∞/summationdisplay k=0/parenleftbigg −1 2/parenrightbiggk(2k−1)!! k!M2k+1/integraldisplayT 0dτ T/tildewideP(τ)2k+1. (21) A direct comparison of this equation with the functional Taylor serie s (12) yields c2k(t1,...,t 2k) = 0, (22a) c2k+1(t1,...,t 2k+1) =/parenleftbigg −1 2/parenrightbiggk(2k−1)!! k!M2k+1T2ka2k+1(t1,...,t 2k+1),(22b) where am(t1,...,t m) =/integraldisplayT 0dτe−βm(τ−¯t)m/productdisplay k=1χ(tk,τ),¯t=1 mm/summationdisplay k=1tk.(23) As expected [23], functions am(t1,...,t m) are, by construction, T-periodic in each variable and symmetric under any exchange of their arguments. The integral in (23) can be performed integrating by parts and tak ing into account thatd dτχ(y,τ) =δ(τ−y) (a Dirac delta). The result is am(t1,...,t m) =eβm¯t βm/braceleftBiggm/productdisplay k=1χ(tk,0)−m/productdisplay k=1χ(tk+T,0)+m/summationdisplay j=1e−βmtjm/productdisplay k=1, k/negationslash=jχ(tk,tj)/bracerightBigg ,(24) where we have used the fact that χ(tk,T)e−βT=χ(tk+T,0). As usual, empty products are assumed to be 1 (the case of the last term for m= 1). The limit β→0 of this expression is better obtained by replacing χ(y,t) byχ1(y,t) in (23) and integrating by parts again. This results in am(t1,...,t m) =Tm/productdisplay k=1χ1(tk,T)−m/summationdisplay j=1tjm/productdisplay k=1, k/negationslash=jχ1(tk,tj). (25) Finally, in the overdamped case ( M→0,β→ ∞), instead of (6) the evolution of Pis given by P(t) =−(1/β)f(t), sovcan be expressed simply as v=−∞/summationdisplay k=0/parenleftbigg −1 2/parenrightbiggk(2k−1)!! k!γ2k+11 T/integraldisplayT 0dtf(t)2k+1. (26) From (12) and (26) it follows that c2k(t1,...,t 2k) = 0 and c2k+1(t1,...,t 2k+1) =−/parenleftbigg −T2 2/parenrightbiggk(2k−1)!! k!γ2k+1δ(t1−t2)···δ(t2k−t2k+1). (27) 4.1. Forcing with a time-periodic piecewise constant force Theexpansion(12)withkernels (22 b)and(23)turnsouttobeuseful toanalyzedifferent types of forcing. For instance, another standard choice in the lite rature (see [1] and references therein), alongside with the bi-harmonic force (1), ha s been the time-periodic piecewise constant force defined in (3). This force is shift-symmet ric only for Tl=T/2, so any other value Tl< T/2 breaks this symmetry and induces a ratchet current.Ratchet effect on a relativistic particle 8 In order to ascertain the effect of this force in system (4 b) for small amplitudes ǫ1≪1, we will compute the first nonzero term in the expansion (12). To t hat purpose we need to evaluate (c.f. equation (23)) Km≡ /angbracketleftam(t1,...,t m)f(t1)···f(tm)/angbracketright=/integraldisplayT 0/bracketleftbig e−βτI(τ)/bracketrightbigmdτ, (28a) I(τ)≡1 T/integraldisplayT 0eβtχ2(t,τ)f(t)dt, (28b) where the choice χ2(t,τ) instead of χ(t,τ) is made because f(t) in (3) has zero average. According to (20) χ2(t,τ) = (1−e−βT)−1¯χ2(t,τ), where ¯χ2(t,τ) =/braceleftBigg 1 if t < τ, e−βTift > τ.(29) Substitution into (28 b) yields I(τ) =ǫ1 βT/bracketleftbigg4 1−e−βTsinh2/parenleftbiggβTl 2/parenrightbigg +Q(τ)/bracketrightbigg , (30a) Q(τ) = eβτ−eβTl if 0< τ < T l, 0 if Tl< τ < T −Tl, eβ(T−Tl)−eβτifT−Tl< τ < T.(30b) It is straightforward to check that K1in (28a) vanishes, so the first term that may not be zero is K3. Lengthy calculations lead to K3=−32ǫ3 1 β4T3eβT (eβT−1)2sinh2/parenleftbiggβ(T−2Tl) 2/parenrightbigg sinh4/parenleftbiggβTl 2/parenrightbigg , (31) that is to say v=4ǫ3 1 (βM)3βTsinh2/parenleftbiggβ(T−2Tl) 2/parenrightbiggsinh4(βTl/2) sinh2(βT/2)+o(ǫ3 1). (32) It is interesting to noticing that K3= 0 ifTl=T/2 because in that case the time- periodic piecewise constant force (3) is shift-symmetric. On the ot her hand, we can determine the value of Tlfor which the ratchet effect is maximum by differetiating (32). This leads to sinh/parenleftbiggβ(T−3Tl) 2/parenrightbigg sinh/parenleftbiggβ(T−2Tl) 2/parenrightbigg sinh3/parenleftbiggβTl 2/parenrightbigg = 0. (33) The only three solutions to this equation are Tl= 0,Tl=T/2 andTl=T/3. The first two do not produce any ratchet current (with Tl= 0f= 0 whereas for Tl=T/2 the force is shift-symmetric), therefore the last one provides its max imum value. As a final remark, expression (32) has well defined overdamped ( M→0,β→ ∞, with finite γ=βM) and undamped ( β→0) limits. In fact, the undamped limit of (32) yields v=ǫ3 1 4(MT)3T4 l(T−2Tl)2+o(ǫ3 1), (34)Ratchet effect on a relativistic particle 9 whereas the overdamped produces v=o(ǫ3 1). Indeed, since f(t)2k+1=ǫ2k 1f(t), in the overdamped case Eq. (26) immediately implies v= 0. This is in marked contrast with the overdamped deterministic dynamic of a particle in a sinusoidal pot ential driven by a bi-harmonic force [31]. In this case, the zero ratchet velocity ca n be explained as a symmetryeffect. Indeed, noticethat f(t) =−f(−t)whenf(t)isgivenby(3)(something that only happens for the bi-harmonic force (1) for specific choice s of the phases), and that the overdamped limit of Eq. (4 b) remains invariant under a simultaneously action of time-reversal and a sign change of xandu(see [23] for further details). 5. Discussion We have studied the dynamics of a damped relativistic particle under t wo zero averageT-periodic forces which breaks the shift-symmetry f(t+T/2) =−f(t). This nonlinearsystem canbeexplicitly solvedthroughatransformationt hatrenders itlinear. Therefore, theratchetaveragevelocity, v, isexactlyobtainedforanyarbitraryforce f(t). This result allows us to show, first of all, that the ratchet velocity ca nnot be obtained in general by using the method of moments (according to which vis obtained as a series of the odd moments of f(t)). And secondly, that vis a functional of f(t), i.e.v[f]. Indeed, for anyT-periodic force we have explicitly found the coefficients of the funct ional Taylor expansion (12). In particular, this expansion shows that the meth od of moments is only justified in the strict overdamped limit (see Eqs. (26) and (27)). Du e to the symmetry v[f] =−v[−f] only odd terms contribute to the Taylor expansion. Besides, since the ratchet velocity is translationally invariant, the kernel c1(t1) must be a constant. So the first order term vanishes because the force has zero average. T herefore, the first term in the expansion (12) that is not necessarily zero is the third one, irre spective of the kind of nonlinearity of the system. We have chosen to illustrate this functional representation the bi- harmonicforce(1) (withp= 2 andq= 1) as well as a time-periodic piecewise constant force (3). We have obtained the leading term of the average velocity for both these fo rces. They are given by equations (9)–(10) and (32), respectively. It is worth emphas izing that the method of moments always predicts a zero ratchet velocity when the syste m is driven by a time- periodic piecewise constant. This is to be compared with the result (3 2) obtained here. We have discussed the two limiting dynamics: undamped and overdamp ed. In these two limits the system remains invariant if the driving force has the symmet riesf(t) =f(−t) andf(t) =−f(−t), respectively. If the relativistic particle is driven by a bi-harmonic forcev∼ǫ2 1ǫ2cos(2φ1−φ2)intheoverdampedlimit, whereas v∼ǫ2 1ǫ2sin(2φ1−φ2)inthe undamped limit. In the latter case this means no ratchet current if w e setφ1=φ2= 0. The unexpected consequence of this is that introducing damping generates a ratchet current, whose intensity grows up to a maximum before it drops to z ero upon a further increase of the damping. This effect is a result of a trade off between symmetry effects and friction and our prediction is that should be observed in any syst em with damping and forced with a time-reversible external force.Ratchet effect on a relativistic particle 10 On its side, if an overdamped relativistic particle is driven by a time-per iodic piecewise constant force like (3), the ratchet velocity is always zer o as a consequence of the symmetry f(t) =−f(−t) exhibited by this force. Summarizing, we hope to have illustrated the predictive power of the Taylor functional expansion method introduced in [23]. This working examp le also shows that this is the only reliable method to analyze the ratchet current a s a function of the parameters of the external force. The most widely used alter native so far, the method of moments, is shown to work only in the overdamped limit of th e dynamics of a relativistic particle driven by a periodic force. When damping is finite a nd the forcing of the system is bi-harmonic, the rathet current predicted by the method of moments still retains some relevant features of the exact one (9). Howeve r, it dramatically fails if the system is driven by a piecewise constant force, because it alwa ys predicts a zero ratchet current, in marked constrast with the result (32) predic ted by the functional Taylor expansion. Acknowledgments We acknowledge financial support through grants MTM2009-1274 0-C03-02 (R.A.N.), FIS2008-02380/FIS (N.R.Q.), and MOSAICO (J.A.C.) (from Ministerio d e Educaci´ on y Ciencia, Spain), grants FQM262 (R.A.N.), FQM207 (N.R.Q.), and P09-F QM-4643 (N.R.Q., R.A.N.) (from Junta de Andaluc´ ıa, Spain), and project MODEL ICO-CM (J.A.C.) (from Comunidad de Madrid, Spain). References [1] P. Reimann. Phys. Rep. , 361:57, 2002. [2] M. Salerno and N. R. Quintero. Soliton ratchets. Phys. Rev. E , 65:025602(R), 2002. [3] M. O. Magnasco. Phys. Rev. Lett. , 71:1477, 1993. [4] F. Falo, P. J. Mart´ ınez, J. J. Mazo, T. P. Orlando, K. Segall, and E. Tr´ ıas. Appl. Phys. A , 75:263, 2002. [5] P. Reimann and P. H¨ anggi. Appl. Phys. A , 75:169, 2002. [6] R. D. Astumian and P. H¨ anggi. Phys. Today , 55:32, 2002. [7] H. Linke, editor. Ratchets and Brownian motors: basics, experiments and appl ications, volume 75 ofAppl. Phys. A Special Issues . 2002. [8] J. E. Villegas, S. Savel’ev, F. Nori, E. M. Gonz´ alez, J. V. Anguita, R. Garc´ ıa, and J. L. Vicent. Science, 302:1188, 2003. [9] M. Beck, E. Goldobin, M. Neuhaus, M. Siegel, R. Kleiner, and D. Koe lle.Phys. Rev. Lett. , 95:090603, 2005. [10] P. H¨ anggi and F. Marchesoni. Rev. Mod. Phys. , 81:387, 2009. [11] W. Schneider and K. Seeger. Appl. Phys. Lett. , 8:133, 1966. [12] A. Ajdari, D. Mukamel, L. Peliti, and J. Prost. J. Phys. I France , 4:1551, 1994. [13] A. Engel, H. W. M¨ uller, P. Reimann, and A. Jung. Phys. Rev. Lett. , 91:060602, 2003. [14] L. Morales-Molina, N. R. Quintero, F. G. Mertens, and A. S´ anc hez.Phys. Rev. Lett. , 91:234102, 2003. [15] M. Schiavoni, L. S´ anchez-Palencia, F. Renzoni, and G. Grynbe rg.Phys. Rev. Lett. , 90:094101, 2003.Ratchet effect on a relativistic particle 11 [16] A. V. Ustinov, C. Coqui, A. Kemp, Y. Zolotaryuk, and M. Salerno .Phys. Rev. Lett. , 93:087001, 2004. [17] D. Cole, S. Bending, S. Savel’ev, A. Grigorenko, T. Tamegai, and F. Nori. Nat. Mater. , 5:305, 2006. [18] S. Ooi, S. Savel’ev, M. B. Gaifullin, T. Mochiku, K. Hirata, and F. No ri.Phys. Rev. Lett. , 99:207003, 2007. [19] P. Curie. J. Phys. (Paris) S´ er. 3 , III:393, 1894. [20] J. Ismael. Curie’s principle. Synthese , 110:167–190, 1997. [21] S. Flach, O. Yevtushenko, and Y. Zolotaryuk. Phys. Rev. Lett. , 84:2358, 2000. [22] M. Salerno and Y. Zolotaryuk. Phys. Rev. E , 65:056603, 2002. [23] N. R. Quintero, J. A. Cuesta, and R. Alvarez-Nodarse. Symme tries shape the current in ratchets induced by a bi-harmonic force. Phys. Rev. E , 81:030102R, 2010. [24] C. E. Skov and E. Pearlstein. Rev. Sci. Inst. , 35:962, 1964. [25] S. Denisov, S. Flach, A. A. Ovchinnikov, O. Yevtushenko, and Y . Zolotaryuk. Phys. Rev. E , 66:041104, 2002. [26] O. H. Olsen and M. R. Samuelsen. Phys. Rev. B , 28:210, 1983. [27] L. Morales-Molina, N. R. Quintero, A. S´ anchez, and F. G. Mert ens.Chaos, 16:013117, 2006. [28] R.F.CurtainandA.J.Pritchard. Functional Analysis in Modern Applied Mathematics . Academic Press, London, 1977. [29] J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman. The Theory of Critical Phenomena: An Introduction to the Renormalization Group . Oxford University Press, Oxford, 1992. [30] J.P.HansenandI. R.McDonald. Theory of Simple Liquids . AcademicPress,London, 3rdedition, 2006. [31] V. Lebedev D. Cubero and F. Renzoni. Current reversals in a ro cking ratchet: Dynamical versus symmetry-breaking mechanisms. Phys. Rev. E , 82:041116, 2010.

2011-06-23

We study the ratchet effect of a damped relativistic particle driven by both asymmetric temporal bi-harmonic and time-periodic piecewise constant forces. This system can be formally solved for any external force, providing the ratchet velocity as a non-linear functional of the driving force. This allows us to explicitly illustrate the functional Taylor expansion formalism recently proposed for this kind of systems. The Taylor expansion reveals particularly useful to obtain the shape of the current when the force is periodic, piecewise constant. We also illustrate the somewhat counterintuitive effect that introducing damping may induce a ratchet effect. When the force is symmetric under time-reversal and the system is undamped, under symmetry principles no ratchet effect is possible. In this situation increasing damping generates a ratchet current which, upon increasing the damping coefficient eventually reaches a maximum and decreases toward zero. We argue that this effect is not specific of this example and should appear in any ratchet system with tunable damping driven by a time-reversible external force.

Ratchet effect on a relativistic particle driven by external forces

1106.4861v1

arXiv:1106.5808v2 [cond-mat.mtrl-sci] 6 Oct 2011Stability of precessing domain walls in ferromagnetic nano wires Yan Gou1, Arseni Goussev1,2, JM Robbins1, Valeriy Slastikov1 1School of Mathematics, University of Bristol, University W alk, Bristol BS8 1TW, United Kingdom 2Max Planck Institute for the Physics of Complex Systems, N¨ othnitzer Straße 38, D-01187 Dresden, Germany (Dated: November 21, 2018) We show that recently reported precessing solution of Landa u-Lifshitz-Gilbert equations in ferro- magnetic nanowires is stable under small perturbations of i nitial data, applied field and anisotropy constant. Linear stability is established analytically, w hile nonlinear stability is verified numerically. PACS numbers: 75.75.-c, 75.78.Fg I. INTRODUCTION The manipulation and control of magnetic domain walls (DWs) in ferromagnetic nanowires has recently be- come a subject of intense experimental and theoretical research. The rapidly growing interest in the physics of the DW motion can be mainly explained by a promising possibility of using DWs as the basis for next-generation memoryandlogicdevices1–5. However,inordertorealize such devices in practice it is essential to be able to posi- tion individual DWs precisely along magnetic nanowires. Generally, this can be achieved by either applying ex- ternal magnetic field to the nanowire, or by generating pulses of spin-polarized electric current. The current study is concerned with the former approach. EventhoughthephysicsofmagneticDWmotionunder the influence of external magnetic fields has been studied for more than half a century6–9, currentunderstanding of the problem is far from complete and many new phenom- ena have been discovered only recently10–14. In particu- lar, a new regime has been reported13,14in which rigid profile DWs travel along a thin, cylindrically symmetric nanowire with their magnetization orientation precessing around the propagation axis. In this paper we address the stability of the propagation of such precessing DWs with respect to perturbations ofthe initial magnetization profile, some anisotropy properties of the nanowire, and applied magnetic field. Letm(x) = (cosθ(x),sinθ(x)cosφ(x),sinθ(x)sinφ(x)) denote the magnetization along a one-dimensional wire. With easy magnetization axis along ˆ xand hard axis alongˆ y, the micromagnetic energy is given by15 E(m) =1 2/integraldisplay/parenleftBig Am′2+K1(1−m2 1)+K2m2 2/parenrightBig dx =1 2/integraldisplay/parenleftBig Aθ′2+sin2θ(Aφ′2+K1+K2cos2φ)/parenrightBig dx(1) whereAis the exchange constant and K1,K2the anistropy constants. Here and in what follows, integrals are taken between −∞and∞(for the sake of brevity, limits of integration will be omitted). We consider here the case of uniaxial anisotropy, K2=0. Minimizers of Esubject to the boundary conditions lim x→±∞m(x) =±ˆ x, (2) describe optimal profiles for a domain wall separating two magnetic domains with opposite orientation. The optimal profiles satisfy the Euler-Lagrange equation m×H= 0, (3) where H=−δE δm=Am′′+K1(m·ˆ x)ˆ x=−e0m+e1n+e2p. (4) Herem,n=∂m/∂θandp=m×nformanorthonormal frame, and the components of Hin this frame are given by e0=Aθ′2+sin2θ(K1+Aφ′2) e1=Aθ′′−1 2sin2θ(K1+Aφ′2), e2=Asinθφ′′+2Acosθθ′φ′. (5) In terms of these components, the energy Eq. (1) (with K2= 0) is given by E(m) =1 2/integraldisplay e0dx, (6) and the Euler-Lagrange equation becomes e1=e2= 0. While the energy Eis invariant under translations along and rotations about the x-axis, the optimal pro- files cannot be so invariant (because of the boundary conditions). Instead, the optimal profiles form a two- parameter family obtained by applying translations, de- notedT(s), and rotations, denoted R(σ), to a given op- timal profile m∗. We denote the family by T(s)R(σ)m∗. In polar coordinates, T(s)R(σ)m∗is given by φ(x) =σ (the optimal profile lies in a fixed half-plane), and θ(x) = θ∗((x−s)/d0), where d0=/radicalbig A/K1and θ∗(ξ) = 2tan−1(e−ξ). (7) It is clear that θ∗(ξ) satisfies θ′ ∗=−sinθ∗,sinθ∗(ξ) = sechξ. (8)2 The dynamics of the magnetization in the presence of an applied magnetic field is described by the Landau- Lifschitz-Gilbert equation16, which for convenience we write in the equivalent Landau-Lifschitz (LL) form, ˙m=m×(H+Ha)−αm×(m×(H+Ha)).(9) Hereα >0 is the damping parameter, and we take the applied field to lie along ˆ x, Ha=H1(t)ˆ x. (10) In polar coordinates, the LL equation is given by ˙θ=αe1−e2−αH1sinθ, (11) sinθ˙φ=e1+αe2−H1sinθ. (12) Theprecessingsolutionisatime-dependenttranslation and rotation of an optimal profile, which we write as T(x0(t))R(φ0(t))m∗. The centre x0(t) and orientation φ0(t)ofthedomainwallfortheprecessingsolutionevolve according to ˙x0=−αd0H1,˙φ0=−H1. (13) It was shown13,14thatT(x0)R(φ0)m∗satisfies the LL equation. It is important to note that the precessing so- lution is fundamentally different from the so-called Walker solution8. Indeed, the latter is defined only for K2>0 (the fully anisotropic case) and time-independent H1less than the breakdown field HW=αK2/2. The Walkersolutionisgivenby m(x,t) =/parenleftbig cosθW(x,t),sinθW(x,t)cosφW,sinθW(x,t)sinφW/parenrightbig with θW(x,t) =θ∗/parenleftbig γ−1(x−VWt)/parenrightbig , (14) sin2φW=H1/HW, (15) and VW=γ(α+α−1)d0H1, (16) γ=/parenleftbiggK1 K1+K2cos2φW/parenrightbigg1 2 . (17) Equations (14)-(17) describe a DW traveling with a constant velocity VWwhose magnitude cannot exceed γ(α+α−1)d0HW; note that VWdoes not depend lin- early on the applied field H1. In contrast, the velocity ˙ x0 of the precessing solution is proportional to H1, and can be arbitrarily large. Also, while for the Walker solution the plane of the DW remains fixed, for the precessing solution it rotates about the nanowire at a rate propor- tional to H1. Finally, for the Walker solution, the DW profile contracts ( γ <1) in response to the applied field, whereas for the precessing solution the DW profile prop- agates without distortion. In this paper we consider the stability of the precess- ing solution. We establish linear stability with respectto perturbations of the initial optimal profile (Sec. II), small hard-axis anisotropy (Sec. III), and small trans- verse applied magnetic field (Sec. IV); specifically, we show, to leading order in the perturbation parameter, that up to translation and rotation, the perturbed solu- tion converges to the precessing solution (in the case of perturbed initial conditions) or stays close to it for all times (for small hard-axis anisotropy and small trans- verse magnetic field). The argument is based on consid- erations of energy, and depends on the fact that for all t, the precessing solution belongs to the family of global minimizers. The analytic argument establishes only lin- ear stability. Nonlinear stability is verified numerically for all three cases in Sec. V. For convenience we choose units so that A=K1= 1. II. PERTURBED INITIAL PROFILE Letmǫ(x,t) denote the solution of the LL equation with initial condition m∗+ǫµ, a perturbation of an opti- mal profile. Let T(xǫ(t))R(φǫ(t))m∗denote the optimal profile which, at time t, is closest to mǫ; that is, the quantity ||mǫ−T(s)R(σ)m∗||2=/integraldisplay/parenleftbig mǫ(x,t)−R(σ)m∗(x−s)/parenrightbig2dx (18) is minimized for s=xǫ(t) andσ=φǫ(t). Then the following conditions must hold: /integraldisplay mǫ·/parenleftbigg T(xǫ(t))R(φǫ(t))∂m∗ ∂x/parenrightbigg dx= 0, /integraldisplay mǫ·(ˆ x×T(xǫ(t))R(φǫ(t))m∗)dx= 0.(19) It is clear that xǫ(t) =x0(t) +O(ǫ) andφǫ(t) = φ0(t) +O(ǫ), but we shall not explicitly calculate the O(ǫ) corrections produced by the perturbation. Rather, our approach is to show that to leading order O(ǫ2), ||mǫ−T(xǫ)R(φǫ)m∗||2decays to zero with t. This will imply that the precessingsolution is linearlystable under perturbations of initial conditions up to translations and rotations. Letθǫ(x,t) andφǫ(x,t) denote the spherical coordi- nates of mǫ(x,t). We expand these in an asymptotic series, θǫ(x,t) =θ∗(x−xǫ(t)) +ǫθ1(x−xǫ(t),t)+···, φǫ(x,t) =φ∗(t) + ǫφ1(x−xǫ(t),t)+···(20) where the correction terms θ1(ξ,t),φ1(ξ,t), etc are ex- pressedin areferenceframe movingwith the domain wall Then to leading order O(ǫ2), ||mǫ−T(xǫ)R(φǫ)m∗||2=ǫ2/integraldisplay (θ2 1+sin2θ∗φ2 1)dξ =ǫ2/angbracketleftθ1|θ1/angbracketright+ǫ2/angbracketleftsinθ∗φ1|sinθ∗φ1/angbracketright,(21)3 where for later convenience we have introduced Dirac no- tation, expressing the integral in Eq. (21) in terms of inner products. It is straightforward to show that the conditions Eq. (19) imply (using θ′ ∗=−sinθ∗)) that /angbracketleftsinθ∗|θ1/angbracketright=/angbracketleftsinθ∗|sinθ∗φ1/angbracketright= 0,(22) which expressesthe fact that the perturbations described byθ1andφ1are orthogonal to infinitesimal translations (described by sin θ∗) along and rotations about ˆ x. Since the difference between mǫandT(xǫ)R(φǫ)m∗ isO(ǫ), the difference in their energies is O(ǫ2) (as T(xǫ)R(φǫ)m∗satisfies the Euler-Lagrange equation Eq. (3)), and is given to leading order by the second variation of Eaboutm∗, ∆Eǫ=E(mǫ)−E(T(xǫ)R(φǫ)m∗) = E(mǫ)−E(m∗) =ǫ2 2/integraldisplay f0dξ, (23) wheref0=θ′ 12+cos2θ∗θ2 1+sin2θ∗φ′ 12. Using the relations Eq. (8) and performing some integra- tions by parts, we can write /integraldisplay f0dξ=/angbracketleftθ1|H|θ1/angbracketright+/angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright,(24) whereHistheSchr¨ odingeroperator −d2/dξ2+V(ξ) with potential given by V(ξ) = 1−2sech2ξ. (25) V(ξ) is a particular case of the P¨ oschl-Teller potential, forwhichthespectrumof Hisknown17.Hhastwoeigen- states, namely sin θ∗(ξ) = sechξwith eigenvalue λ0= 0, and cosθ∗(ξ) = tanh ξwith eigenvalue λ1= 1, and its continuous spectrum is bounded below by λ= 1. This is consistent with the fact that the optimal profiles are global minimizers of E(subject to the boundary condi- tions Eq. (2)), which implies that the second variation ofEaboutm∗is positive for variations transverse to translations and rotations of m∗. It follows that, for any (smooth) square-integrable function f(ξ) orthogonal to sinθ∗, we have that /angbracketleftf|Hj+1|f/angbracketright ≥ /angbracketleftf|Hj|f/angbracketright (26) forj≥0(wewillmakeuseofthisfor j= 0andj= 1). In particular, since θ1and sinθ∗φ1are orthogonal to sin θ∗ (cf Eq. (22)), it follows that /angbracketleftθ1|H|θ1/angbracketright ≥ /angbracketleftθ1|θ1/angbracketright, (27) /angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright ≥ /angbracketleftsinθ∗φ1|sinθ∗φ1/angbracketright.(28) Therefore, from the preceding Eqs. (27)–(28) and Eqs. (21) and (23)–(24), we get, to leading order O(ǫ2), that ||mǫ−T(xǫ)R(φǫ)m∗||2≤2∆Eǫ.(29)Below we show that, to leading order O(ǫ2), for small enoughH1(it turns out that |H1|<1/2 is sufficient), we have the inequality d dt∆Eǫ≤ −γ∆Eǫ (30) for some γ >0. Taking Eq. (30) as given, it follows from the Gronwall inequality that ∆Eǫ≤1 2Cǫ2e−γt(31) for some C >0 (which depends only on the form of the initial perturbation). From Eq. (29), it follows that ||mǫ−T(xǫ)R(φǫ)m∗||2≤Cǫ2e−γt.(32) The result Eq. (32) shows that, to O(ǫ2),mǫconverges toanoptimalprofilewithrespecttothe L2-norm. Infact, with a small extension of the argument, we can also show that, to O(ǫ2),mǫconverges to an optimal profile uni- formly (that is, with respect to the L∞-norm). Indeed, making use of the preceding estimates, one can obtain a bound on ||m′ ǫ−T(xǫ)R(φǫ)m′ ∗||, theL2-norm of the difference in the spatial derivatives of the perturbed so- lution and the optimal profile. To O(ǫ2), ||m′ ǫ−T(xǫ)R(φǫ)m′ ∗||2 =ǫ2(/angbracketleftθ′ 1|θ′ 1/angbracketright+/angbracketleftsinθ∗φ′ 1|sinθ∗φ′ 1/angbracketright+/angbracketleftsinθ∗θ1|sinθ∗θ1/angbracketright) ≤ǫ2(3(/angbracketleftθ1|H|θ1/angbracketright+/angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright) ≤6ǫ2∆Eǫ.(33) Arguing as in Eqs. (29)–(32), we may conclude that ||m′ ǫ−T(xǫ)R(φǫ)m′ ∗||decays exponentially with t. Thus,mǫconverges to an optimal profile with respect to the Sobolev H1-norm (where ||f||2 H1=||f||2+||f′||2). It is a standard result that this implies that the conver- gence is also uniform (again, to O(ǫ2)). It remainsto establish Eq. (30). From Eq. (9), we have that for any solution m(x,t) of the LL equation, d dtE(m) =−/integraldisplay H·˙ mdx =/integraldisplay (m×H)·Hadx− −α/integraldisplay/parenleftbig m×H)2+(m×H)·(m×Ha/parenrightbig dx =−α/integraldisplay/parenleftbig e2 1+e2 2+H1sinθe1/parenrightbig dx,(34) wheree1ande2are given by Eq. (5), and we have used the fact that the term ( m×H)·Havanishes on inte- gration. Substituting the perturbed solution mǫinto Eq.(34)andnotingthatthe E(T(xǫ)R(φǫ)m∗) =E(m∗) does not vary in time, we obtain after some straightfor-4 ward manipulation that d dt∆Eǫ= −αǫ2/parenleftbig /angbracketleftθ1|H2|θ1/angbracketright+/angbracketleftsinθ∗φ1|H2|sinθ∗φ1/angbracketright+H1F/parenrightbig (35) to leading O(ǫ2), where F=/integraldisplay/parenleftbig cosθ∗f0+cosθ∗sin2θ∗θ2 1/parenrightbig dξ.(36) For the first two terms on the rhs of Eq. (35), we have, from Eq. (26) and Eqs. (23)–(24), that /angbracketleftθ1|H2|θ1/angbracketright+/angbracketleftsinθ∗φ1|H2|sinθ∗φ1/angbracketright ≥ /angbracketleftθ1|H|θ1/angbracketright+/angbracketleftsinθ∗φ1|H|sinθ∗φ1/angbracketright =2 ǫ2∆Eǫ.(37) The term H1Fin Eq. (35) is not necessarily positive, asH1can have arbitrary sign. But for sufficiently small|H1|, it is smaller in magnitude than the preceding two terms. Indeed, we have, again using Eq. (26) and Eqs. (23)–(24), that |F| ≤/integraldisplay/parenleftbig |f0|+θ12/parenrightbig dξ≤2 ǫ2∆Eǫ+/angbracketleftθ1|θ1/angbracketright ≤2 ǫ2∆Eǫ+/angbracketleftθ1|H|θ1/angbracketright ≤4 ǫ2∆Eǫ.(38) Substituting Eqs. (37) and (38) into Eq. (35), we get that d dt∆Eǫ≤ −2α(1−2|H1|)∆Eǫ, (39) from which the required estimate (30) follows for |H1|< 1/2. It is to be expected that the stability of the pre- cessing solution depends on the applied field not being too large. Indeed, it is easily shown that, for H1>1 (resp.H1<−1), the static, uniform solution m=−ˆ x (resp. m= +ˆ x) becomes linearly unstable. As the precessing solution is nearly uniform away from the do- main wall, one would expect it to be similarly unstable for|H1|>1. The numerical results of Sec. VA bear this out. Finally, we remark that the stability criterion ob- tained here, namely |H1|<1/2, is certainly not optimal. III. SMALL HARD-AXIS ANISOTROPY Next we suppose the hard-axis anisotropy is small but nonvanishing, taking K2=ǫ >0. Letmǫ(x,t) denote the solution of the LL equation with initial condition mǫ(x,0) =m∗(x). As above, let T(xǫ(t))R(φǫ(t))m∗ denote the translated and rotated optimal profile closest tomǫat timet. Adapting the argument of the preceding section, we show below that, to leading order O(ǫ2), ||mǫ−T(xǫ)R(φǫ)m∗||2≤C2ǫ2for allt >0 (40)for some constant C2>0. In contrast to the preceding result Eq. (32) for perturbed initial conditions, here we do not expect mǫto converge to T(xǫ)R(φǫ)m∗. Indeed, while an explicit analytic solution of the LL equation is not available for small K2(the Walker solution is valid only forK2>2|H1|/α), it is easily verified that there are no exact solutions of the form T(xǫ(t))R(φǫ(t))m∗. The resultEq.(40)demonstratesthat, throughlinearorderin ǫ, the solution for K2=ǫremains close to the precessing solution, up to translation and rotation. To proceed, let ∆ Eǫdenote, as above, the difference in theuniaxialmicromagnetic energy, i.e. the energy given by Eq. (1) with K2= 0, between mǫandT(xǫ)R(φǫ)m∗. Then, as in Eq. (29), we have that ||mǫ−T(xǫ)R(φǫ)m∗||2≤2∆Eǫ.(41) AsE(T(xǫ)R(φǫ)m∗) =E(m∗) is constant in time, we have that d dt∆Eǫ=d dtE(mǫ). (42) The hard-axis anisotropy affects the rate of change of the uniaxial energy through additional terms in ˙ m. In- deed, foranysolution m(x,t)oftheLLequation, wehave that d dtE(m) =d dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle K2=0E(m)+G(m),(43) whered/dt|K2=0E(m) denotes the rate of change when K2= 0, as given by Eq. (34), and G(m) =−ǫ/integraldisplay R(m·ˆy)(m×H(m))·ˆydx +ǫα/integraldisplay (m×H(m))·(m׈y)(m·ˆy)dx.(44) Takingm=mǫ, we recall from the preceding section (c.f. Eq. (30)) that, for |H1|<1/2, d dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle K2=0E(mǫ)≤ −γ∆Eǫ (45) for some γ >0. Below we show that there exists con- stantsC1,γ1withγ1< γsuch that |G(mǫ)| ≤γ1∆Eǫ+C1ǫ2. (46) Taking Eq. (46) as given and substituting it along with Eq. (45) into Eqs. (42)–(43), we get that d dt∆Eǫ≤ −(γ−γ1)∆Eǫ+C1ǫ2.(47) From Gronwall’s equality it follows that ∆Eǫ≤C1 γ−γ1ǫ2, (48)5 which together with Eq. (41) yields the required result Eq. (40). It remains to show Eq. (46). Substituting the asymp- totic expansion Eq. (20), we obtain after straightforward calculations that, to leading order O(ǫ2), G(mǫ) =−ǫ2cos2φ∗(t) ×/integraldisplay/parenleftbig sin4θ∗φ′ 1+4/3αsin3θ∗θ′ 1/parenrightbig dξ.(49) This can be estimated using the elementary inequality 2|ab| ≤βa2+b2 β, (50) which holds for any β >0. Indeed, recalling Eqs. (8), (23), (27), and using integration by parts where neces- sary, we have that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sin4θ∗φ′ 1dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β 2/integraldisplay sin2θ∗φ′ 12dξ+1 2β/integraldisplay sin6θ∗dξ ≤β ǫ2∆Eǫ+8 15β, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sin3θ∗θ′ 1dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β 2/integraldisplay θ′ 12dξ+1 2β/integraldisplay sin6θ∗dξ ≤β ǫ2∆Eǫ+8 15β. (51) From Eqs. (49)–(51), it is clear that β,γ1andC1can be chosen so that Eq. (46) is satisfied. IV. SMALL TRANSVERSE APPLIED FIELD Suppose the applied magnetic field has a small trans- verse component, so that Ha=H1ˆ x+H2ˆ y, where H2=ǫh2(x) (52) (h2depends on xbut not t). For simplicity, let K2= 0. Letmǫ(x,t) denote the solution of the LL equation with initial condition mǫ(x,0) =m∗(x). As above, let T(xǫ(t))R(φǫ(t))m∗denote the translated and rotated optimal profile closest to mǫat timet. We first note that, unless h2vanishes as x→ ±∞, mǫwillnotremain close to T(xǫ(t))R(φǫ(t))m∗. For example, if h2is constant, then away from the domain wall,mǫwill relax to one of the local minimizers of the homogeneous energy K1(1−m2 1)−Ha·m, and these do not lie along ±ˆ xforH2/negationslash= 0. It follows that ||mǫ− T(xǫ(t))R(φǫ(t))m∗||will diverge with time. Physically, this divergence is spurious. It stems from the fact that we are taking the wire to be of infinite ex- tent. One way to resolve the issue, of course, would be to take the wire to be of finite length. However, one would then no longer have an explicit analytic solution of the LL equation.Here we shall take a simpler approach, and assume that the transverse field h2(x) approaches zero as xap- proaches ±∞. In fact, for technical reasons, it will be convenienttoassume that the integralof h2 2+h′ 22, i.e. the squared Sobolev norm ||h2||H1, is finite. Then without loss of generality, we may assume ||h2||2 H1=/integraldisplay (h2 2+h′ 22)dξ= 1. (53) Under this assumption, the main result of this section is thatmǫstaysclosetoanoptimalprofileuptotranslation and rotation. That is, for some C1>0, ||mǫ−T(xǫ)R(φǫ)m∗||2≤C1ǫ2.(54) The demonstration proceeds as in the preceding sec- tion, so we will discuss only the points at which the present case is different. The main difference is that, in place of Eq. (49), we get (by considering the LL equa- tion with H2/negationslash= 0 rather than K2/negationslash= 0) the following expression for G(mǫ) to leading order O(ǫ2): G(mǫ) =ǫ2/parenleftBig αcosφ∗(t)/integraldisplay cosθ∗(θ′′ 1−cos2θ∗θ1)h2dξ −αsinφ∗(t)/integraldisplay sinθ∗(φ′′ 1−2cosθ∗φ′ 1)h2dξ −sinφ∗(t)/integraldisplay (θ′′ 1−cos2θ∗θ1)h2dξ −cosφ∗(t)/integraldisplay sinθ∗cosθ∗(φ′′ 1−2cosθ∗φ′ 1)h2dξ/parenrightBig . (55) After some straightforward manipulations including in- tegration by parts, and making use of the inequality Eq. (50), one can show that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay cosθ∗(θ′′ 1−cos2θ∗θ1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β 2/bardblθ1/bardbl2 H1+1 2β, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sinθ∗(φ′′ 1−2cosθ∗φ′ 1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β 2||sinθ∗φ′ 1||2+1 2β, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay (θ′′ 1−cos2θ∗θ1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤β 2/bardblθ1/bardbl2 H1+1 2β, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay sinθ∗cosθ∗(φ′′ 1−2cosθ∗φ′ 1)h2dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤β 2||sinθ∗φ′ 1||2+1 2β.(56) From Eqs. (23), (24) and (27) it follows that /integraldisplay/parenleftBig θ′ 12+sin2θ∗φ′ 12/parenrightBig dξ≤4 ǫ2∆Eǫ,(57) and /integraldisplay θ2 1dξ≤2 ǫ2∆Eǫ. (58)6 Substituting Eqs. (56)–(58) into Eq. (55), we get that |G(mǫ)| ≤(1+α)/parenleftbigg 3β∆Eǫ+1 βǫ2/parenrightbigg .(59) This estimate is of the same form as (46), and the ar- gument given there, with βchosen appropriately, estab- lishes Eq. (54). V. NUMERICAL STUDIES In the preceding Sections II–IV we have shown that the precessing solution is linearly stable; to leading or- derO(ǫ), a perturbed solution either approaches or stays close to the precessing solution up to a translation and rotation, according to whether the perturbation is to the initial conditions or to the anistropy and transverse ap- plied magnetic field in the LL equation. Here we present numerical results which verify nonlinear stability for the precessing solution under small perturbations. To this end, weinvestigatetheenergy,∆ Eǫ=E(mǫ)−E(m∗), of the numerically computed perturbed DW, mǫ(x,t), rela- tive to the minimum energy E(m∗) of an optimal profile, as a function of time t. Throughout, Eis taken to be theuniaxial micromagnetic energy given by Eq. (1) with K2= 0. As in the preceding sections, we choose units so thatA=K1= 1. In these units, E(m∗) = 2. In typical ferromagnetic microstructures, the value of the Gilbert damping parameter αis known to lie between 0.04 and 0.22 (see e.g. Ref.18and references within), so we take α= 0.1 throughout our numerical study. A. Perturbed initial profile We first investigate the evolution of a DW, mǫ(x,t), from an initial perturbation of an optimal profile. We take the initial condition in polar coordinates to be given by θǫ(x,0) =θ∗/parenleftbiggx 1+ǫ1/parenrightbigg , φǫ(x) =φ0+ǫ2x,(60) which corresponds to stretching the unperturbed profile along and twisting it around the axis of the nanowire. The applied field is directed along the nanowire, Ha= H1ˆx, and we take K2= 0. Figure 1 shows the dependence of the relative energy ∆Eǫon timetfor different values of the applied field H1. The figure presents 13 curves corresponding, from top to bottom, to H1varying from −1.2 to 0 at the increment of 0.1. In the initial condition given by Eq. (60), we take ǫ1= 0.1 andǫ2=π/50. Figure 1 clearly indicates that ∆ Eǫ(t) decays exponen- tially for weak applied fields, |H1| ≤1/2, in accord with the analytic result Eq. (31). However, for |H1| ∼1, devi- ations from exponential decay are evident, and the pre- cessing solutionappears to become unstable for |H1|/greaterorsimilar1.0 50 100 150 20010−810−610−410−2100 t∆Eǫ FIG. 1: (Color online) Relative energy, ∆ Eǫ(t), of the per- turbed DW for 13 different values of the applied field H1. See text for discussion. B. Small hard-axis anisotropy We consider next the evolution of a DW from an opti- mal profile at t= 0 when the hard-axis anisotropy K2is nonvanishing. We fix H1=−0.5. 0102030405060708000.0010.0020.0030.0040.0050.0060.0070.0080.0090.01 t∆Eǫ FIG. 2: (Color online) Relative energy, ∆ Eǫ(t), of the per- turbed DW for 5 different values of the hard-axis anisotropy constant K2. See text for discussion. Figure 2 shows the dependence of the relative energy ∆Eǫon time tfor different values of K2. The figure presents 5 curves corresponding, from top to bottom, to K2varying from 0 .1 to 0.02 at the decrement of 0 .02. (The blue and red colorings alternate to make adjacent curves more easily distinguishable.) It is evident that the relative energy remains small, verifying the linear analy- sis of Sec. III. Figure 3 shows the maximum value of the relative en- ergy ∆Eǫ(over the interval 0 ≤t≤80) as a func- tion ofK2. Red squares represent numerically computed values. The black solid curve is the parabola CKK2 2, withCK= 1.3207 fitted by the method of least squares7 0 0.02 0.04 0.06 0.08 0.100.0010.0020.0030.0040.0050.0060.0070.0080.0090.01 K2max(∆ Eǫ) FIG. 3: (Color online) Maximum value of the relative en- ergy ∆Eǫof the perturbed DW as a function of the hard- axis anisotropy K2. Numerically computed values are repre- sented by (red) squares. The (black) solid curve is a parabol a, max(∆Eǫ) =CKK2 2withCK= 1.3207, fitted by the method of least squares through the data points with K2≤0.04. through the data points with K2≤0.04. We obtain con- vincing confirmation of the leading-order analytical re- sult Eq. (48). For larger values of K2, we see departures from quadratic dependence; for sufficiently large values ofK2(not shown), the Walker solution was recovered. C. Small transverse applied field Finally, we address the stability of the precessing solu- tion under an applied magnetic field, Ha=H1ˆx+H2ˆy, with a small transverse component, H2(x). As discussed in Sec. IV, we want H2(x) to vanish as x→ ±∞. Here we take H2(x) =¯H2w(x), (61) wherew(x) is equal to one inside the window 0 ≤x≤20 andvanishesoutside(the argumentofSectionIViseasily modified to establish the linear stability result Eq. (48) in this case). We consider the evolution of a DW given att= 0 by the optimal profile m∗centred at x= 0. We takeH1=−0.5, so that in the absence of the transverse field, the DW velocity is positive (cf. Eq. (13)) and the DW crosses the window. We take K2= 0. Figure 4 shows the dependence of the relative energy ∆Eǫon timetfor different values of the transverse field amplitude ¯H2. The figure presents 5 curves correspond- ing, from top to bottom, to ¯H2varying from 0 .1 to 0.02 at the decrement of 0 .02. (The blue and red colorings al- ternate to make adjacent curves more easily distinguish- able.) The relative energy ∆ Eǫ(t) is presented over the time interval 0 ≤t≤400, which, for small values of ¯H2, is sufficient for the DW to traverse the spatial win- dow 0≤x≤20 (cf. Eq. (13)). The results confirm that05010015020025030035040010−310−210−1100 t∆Eǫ FIG. 4: (Color online) Relative energy, ∆ Eǫ(t), of the per- turbed DW for 5 different values of the transverse field am- plitude¯H2. See text for discussion. the relative energy of the perturbed magnetization pro- file remains small for small values of ¯H2, in accord with the leading-order results of Section IV. 0 0.02 0.04 0.06 0.08 0.100.20.40.60.81 ¯H2max(∆ Eǫ) FIG. 5: (Color online) Maximum value of the relative energy ∆Eǫof the perturbed DW as a function of the amplitude of the transverse applied field, ¯H2. Numerically computed values are represented by (red) squares. The (black) solid curve is a parabola, max(∆ Eǫ) =CH¯H2 2withCH= 99.6586, fitted by the method of least squares through the data points with¯H2≤0.04. Figure 5 shows the maximum value of the relative en- ergy ∆Eǫ(over the interval 0 ≤t≤400) as a function of ¯H2. Red squares represent numerically computed values. The black solid curve corresponds to the parabola CH¯H2 2 withCH= 99.6586 fitted by the method of least squares through the data points with ¯H2≤0.04. The figure provides a confirmation of the leading-order analytical result of Sec. IV that the maximum relative energy de- pends quadratically on ¯H2for small ¯H2. Deviations from the parabolic dependence can be seen for ¯H2/greaterorsimilar0.08.8 VI. CONCLUSIONS The precessing solution is a new, recently reported ex- act solution of the Landau-Lifschitz-Gilbert equation. It describes the evolution of a magnetic domain wall in a one-dimensional wire with uniaxial anisotropy subject to a spatially uniform but time-varying applied magnetic field alongthe wire. We haveanalysedthe stabilityofthe precessing solution. We have proved linear stability with respect to small perturbations of the initial conditions aswellastosmallhard-axisanisotropyandsmalltransverse applied fields, provided the applied magnetic field along the wire is not too large. We have also carried out nu- merical calculations that confirm full nonlinear stability under these perturbations. Numerical calculations suggest that, for sufficiently large perturbations and applied longitudinal fields, the precessing solution becomes unstable, and new stable so- lutions appear. It would be interesting to analyse these bifurcationsandstudythesenewregimesforDWmotion. 1D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, R. P. Cowburn, Science 309, 1688 (2005). 2R. P. Cowburn, Nature (London) 448, 544 (2007). 3S. S. P. Parkin, M. Hayashi, L. Thomas, Science 320, 190 (2008). 4M. Hayashi, L. Thomas, R. Moriya, C. Rettner, S. S. P. Parkin, Science 320, 209 (2008). 5L. Thomas, R. Moriya, C. Rettner, S. S. P. Parkin, Science 330, 1810 (2010). 6L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. Sowietu- nion8, 153 (1935). 7T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans. Mag.40, 3443 (2004). 8N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). 9A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep.194, 117 (1990). 10M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008).11X. R. Wang, P. Yan, J. Lu, Europhys. Lett. 86, 67001 (2009). 12X. R. Wang, P. Yan, J. Lu, C. He, Ann. Phys. (N.Y.) 324, 1815 (2009). 13Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104, 037206 (2010). 14A. Goussev, J. M. Robbins, V. Slastikov, Phys. Rev. Lett. 104, 147202 (2010). 15V. Slastikov and C. Sonnenberg, IMA J. Appl. Math. XXX, XXX (2011), doi:10.1093/imamat/hxr019 16A. Hubert and R. Sch¨ afer, Magnetic Domains: The Anal- ysis of Magnetic Microstructures (Springer, Berlin, 1998). 17P.M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I McGraw-Hill, New York, 1953 18Y. Tserkovnyak and A. Brataas, Phys. Rev. Lett. 88, 117601 (2002).

2011-06-28

We show that recently reported precessing solution of Landau-Lifshitz-Gilbert equations in ferromagnetic nanowires is stable under small perturbations of initial data, applied field and anisotropy constant. Linear stability is established analytically, while nonlinear stability is verified numerically.

Stability of precessing domain walls in ferromagnetic nanowires

1106.5808v2

In uence of randomness and retardation on the FMR-linewidth Thomas Bose and Steen Trimper Institute of Physics, Martin-Luther-University, D-06099 Halle, Germany (Dated: October 25, 2018) Abstract The theory predicts that the spin-wave lifetime Land the linewidth of ferromagnetic resonance
Bcan be governed by random elds and spatial memory. To that aim the eective eld around which the magnetic moments perform a precession is superimposed by a stochastic time dependent magnetic eld with nite correlation time. The magnetization dynamics is altered by inclusion of a spatial memory eect monitoring a non-local interaction of size . The underlying Landau- Lifshitz-Gilbert equation (LLG) is modied accordingly. The stochastic LLG is equivalent to a Fokker-Planck equation which enables to calculate the mean values of the magnetization vector. Within the spin-wave approximation we present an analytical solution for the excitation energy and its damping. The lifetime and the linewidth are analyzed depending on the strength of the random eld Dand its correlation time cas well as the retardation strength 0and the size . WhereasLdecreases with increasing D, retardation strength 0andc, the lifetime is enhanced for growing width of the spatial retardation kernel. In the same manner we calculate the exper- imentally measurable linewidth
Bis increased strongly when the correlation time cranges in the nanosecond interval. thomas.bose@physik.uni-halle.de; steen.trimper@physik.uni-halle.de 1arXiv:1107.0638v1 [cond-mat.mes-hall] 4 Jul 2011I. INTRODUCTION Ferromagnetic resonance (FMR) is a powerful technique to study magnetic materials, in particular the inherent magnetization dynamics [1, 2]. So the observable FMR-linewidth is very sensitive to the underlying dynamical processes as well as the real structure of the ma- terial like anisotropy. From a theoretical point of view the Landau-Lifshitz-Gilbert equation (LLG) [3, 4], see Eq. (1) in the present paper, is an appropriate tool to investigate magnetic excitations and dissipative processes as the damping of the excitations. Although, the LLG is known since a few decades it is still a standard model to analyze magnetodynamics. Recently the Gilbert damping parameter was determined experimentally for ferromagnetic thin lms in [5] and by rst-principle calculations for itinerant ferromagnets in [6]. Obviously, the ap- plicability of the LLG depends on the physical situation in mind. In case the magnetization is not conserved the Landau-Lifshitz-Bloch (LLB) equations are more appropriate, in partic- ular in the vicinity of the phase transition as demonstrated in [7]. The LLB equations were used to investigate magnetization switching near the Curie temperature in [8, 9]. Moreover, the geometrical conguration of the sample are able to play an important role in measuring the FMR-linewidth. Related to this fact the contribution of the Gilbert damping to the linewidth can be superimposed by extrinsic eects as magnon-magnon scattering processes [10] which become of the same order of magnitude or even exceed the Gilbert damping. Es- pecially for an in-plane conguration where the external eld as well as the magnetization lie in the lm plane the in uence of two magnon processes to the FMR-linewidth cannot be ne- glected [11, 12]. Those theoretical results predicting a nonlinear dependence of the linewidth on the frequency were extended to the case when the magnetization is tipped out of plane [13]. Dierent experimental ndings emphasize the importance of extrinsic contributions for in-plane setups, see [14{17]. A quantitative separation of Gilbert damping and two magnon scattering contributions was carried out [14, 16, 17]. Contrary to these observations there are other investigations [18], which oer no qualitative dierence between in-plane and normal- to-plane measurements. In both realizations the linewidth depends linearly on the frequency even for frequencies smaller than 10 GHz . Such theoretical and experimental works suggest, among others, that the FMR-linewidth is exclusively controlled by the Gilbert damping and exhibits a pure linear frequency dependence in a perpendicular conguration with respect to thin lms measurements. Furthermore, the two magnon scattering is supposed to be of 2less importance in bulk ferromagnets [16]. Thus the LLG equation seems still applicable to describe magnetization dynamics provided the physical situation is carefully analyzed as pointed out in [2]. A more realistic magnetization dynamics requires a modication of the LLG. Recently, the anisotropic damping and its manifestation in the FMR-linewidth has been discussed by several authors [19{21]. An alternative formulation of Gilbert damping by means of scattering theory was discussed in [22]. In addition, ferromagnetic resonance measurements were used as well to investigate spin transport in magnetic single and double layer structures [23]. Moreover, very recently it was shown that the transfer of spin angular momentum can induce ferromagnetic resonance dynamics in a ferromagnetic lm due to the spin Hall eect in an adjacent lm with strong spin-orbit scattering [24]. Related to this phenomena it was reported on the direct time-resolved measurement of spin torque in magnetic tunnel junctions to detect resonant magnetic precession due to an oscillating spin torque [25]. A theory of ferromagnetic resonance in perpendicular magnetized nanodisks is suggested in [26]. To push forward the theory stochastic forces and non-local interactions should be included into the model to gain a more realistic description of magnetic materials and to reveal unex- pected behavior as for example the noise suppression by noise behavior argued in [27]. The eects of noise in magnetic nanosystems obeying spin torque dynamics are investigated in [28, 29]. Experimentally, the role of noise in magnetic systems was prospected in [30, 31]. The present work is addressed to the in uence of randomness on the magnetization dynam- ics. As the two new aspects the system considered is simultaneously subjected to feedback coupling and to a stochastic eld with colored noise. The starting point is the LLG equation which is generalized in a manner that both spatial memory eects and a temporal stochastic eld with a nite correlation time is incorporated into the model. Previously the in uence of colored noise [32] and retardation eects [33] within the LLG were analyzed separately. Otherwise, both eects can occur simultaneously. Consequently we study a combined model concerning both kind of impacts, feedback and randomness. As demonstrated in former papers there exits the possibility that the total damping, originated by the Gilbert damp- ing and that one induced by memory eects are able to cancel by the distinct damping mechanisms. In this paper we are interested in the FMR-linewidth. The corresponding parameters range in such reasonable intervals where dierent dissipation sources are not observable. The main goal is to calculate the FMR-linewidth and to discuss its dependence 3on the parameters characterizing randomness and retardation. Let us give a brief outline of the paper. In Sec. 2 we present the mathematical model and its underlying basic assumptions. The stochastic LLG equation is equivalent to a Fokker- Planck equation which is derived approximately in Sec. 3. This equation enables to compute the mean values of the magnetization. The results are discussed in detail in Sec. 4. Finally, we conclude by summarizing the results and by an outlook in Sec. 5. II. MODEL As already indicated in the introduction we are interested in micro- and nanosized mag- nets. Therefore a coarse-grained description is an appropriate tool to investigate magnetic material. In this mesoscopic description the discrete magnetic moments are replaced by a spatiotemporal vector eld m(r;t). The interaction and the dynamics of the moments are formulated in a continuous approximation. The situation is schematically illustrated in Fig. 1. Here, the magnetization m(r1) represents the magnetic properties within the mesoscopic microscopic νm(r1)∝/summationtext {iln}∈d3r1siln d3r1 sh11 sh1kshj1 shjk /Bullet x yzr1 FIG. 1. Illustration of the of the coarse-grained mesoscopic model. The sirepresent microscopic magnetic moments which are related to the magnetization m. Further explanation can be found in the text. volume-element d3r1which is build around the position r1. The eld m(r1) stands for the total set of microscopic spins which will be visible if one zooms into the microscopic struc- 4ture. The huge number of microscopic degrees of freedom within d3r1are substituted by a single degree of freedom, namely the mesoscopic quantity mwhich can be considered as the sum over the microscopic spins located at equivalent crystal positions. Moreover, the magnetization vector eld m(r1) is assumed to be oriented continuously in space. The basics of our model consists in this mesoscopic description discussed before. Further, the system is supposed to oer an uniaxial anisotropy where the direction of the anisotropy axis is denoted by. Our calculations refer to weak excitations which evolve as spin waves and possess a nite life time. Both quantities are found in the long wave-length limit qa1, whereqis the amount of the wave vector and ais the lattice constant. This assumption re ects the mesoscopic level of description. Experimentally the dynamic behavior of the magnetization mcan be detected for instance by means of ferromagnetic resonance (FMR). Because the main goal of the paper is to put forward the modeling towards more realistic systems we develop a dynamic model for the magnetization eld m(r) in which retardation eects as well as stochastic elds are included. In particular, the aim is to relate our ndings for the magnetic excitations and their damping to an experimentally accessible quantity, namely the FMR-line width
B, cf. Eq. (23). As underlying model we start from is the Landau-Lifshitz-Gilbert equation @m @t= 1 +2mh Be+[mBe]i ; (1) which will be generalized accordingly. In Eq. (1) the quantities andare the gyromagnetic ratio and the dimensionless Gilbert damping parameter, respectively. In this description m(r;t) is the unit vector m=M=Mswith the magnetization Mthe saturation magneti- zation. The local eective eld Be(r;t) causes the precession of the magnetization. In general, the eective eld Beis composed of dierent contributions, an internal eld due to the interaction of the spins, the magnetic anisotropy and an external eld. This eective eld can be derived from the Hamiltonian of the system by functional variation with respect tom Be=M1 sH m: (2) 5The Hamiltonian Hcan be expressed as [32, 34] H=Z d3rfwex+wan+wextg wex=1 2Ms~J(rm)2; wan=1 2MsKsin2 ; w ext=Bext
M:(3) The quantities ~J=Ja2andKdesignate the exchange energy density and the magneto- crystalline anisotropy energy density. Here, Jis the coupling strength between nearest neighbors referring to the isotropic Heisenberg model [35] and ais the lattice constant. Further, Bextis the static external magnetic eld. The quantity is the angle between the the local magnetization mand the anisotropy axis = (0;0;1). We assume that  points in the direction of the easy axis in the ground state. Therefore, K > 0 characterizes the strength of the anisotropy. In deriving Eq. (3) we have used m2= 1. Let us stress again that this assumption seems to be correct if the temperature is well below the Curie temperature [7]. Our calculations based on the LLG suggest that other damping mechanism such as an extrinsic magnon-magnon scattering due to magnetic inhomogeneities should be inactive and hence they are irrelevant. In thin lms this situation can be achieved when both the magnetization and the static external eld are perpendicular to the lm plane. In our model this situation is realized when both the easy axis of the anisotropy as well as the external eld Bextpoint into the z-direction. Hence the equilibrium magnetization is likewise oriented parallel to the z-axis. This situation corresponds to a normal-to-plane conguration. From here we conclude that the application of the LLG leads to reasonable results. For a dierent realization an alternative dynamical approach seems to be more accurate, see also the conclusions. To proceed further, the vector mis decomposed into a static and a dynamic part termed asand = ( 1; 2; 3), respectively. In the frame of spin wave approximation we make the ansatz m(r;t) =+ (r;t) =+ ;  = const:; (4) Combining Eqs. (2) and (3) yields the eective eld Be=~Jr2 K 0+Bext; 0= ( 1; 2;0): (5) 6It is appropriate to introduce dimensionless quantities: l2 0=~J K=Ja2 K;  = (l0q)2+ 1; = K; t= t;jBextj K=":(6) The quantity l0is called the characteristic magnetic length [36] whereas the parameter "re ects the ration between the strengths of the external and the anisotropy eld. For convenience later we will substitute t!tagain. So far we have introduced the LLG in Eq.(1) in its conventional form and incorporated our special basic model assumptions for a ferromagnetic material below its Curie temperature. To proceed toward a more realistic description of magnets the LLG will be extended by the inclusion of retardation eects and random magnetic elds. Whereas retardation is implemented by a memory kernel (r;r0;t;t0) a stochastic eld (r;t) contributes additionally to the eective eld, i.e. Be(r;t)!be(r;t) =Be(r;t) +(r;t): (7) Taking both eects into account we propose the following generalized LLG @m(r;t) @t=Zt 0dt0Z ddr0(rr0;tt0)  1 1 +2m(r0;t0) be(r0;t0)+ +[m(r0;t0)be(r0;t0)] ;(8) where the stochastic eld is included in the dimensionless eective eld as be=l2 0r2 0+"b0+(r;t): (9) The unit vector b0indicates the direction of the external magnetic eld. In general, the kernel should respect the retardation concerning temporal and spatial processes. More precise, a change of the magnetic moment at position rshould in uence another moment at position r0and vice versa. This in uence is thought to be an additional contribution which should not be confused with parts of the exchange interaction in the eective eld, i.g. the lengthon which spatial retardation eects are relevant could be of a dierent order of magnitude in comparison with the lattice constant a. Insofar, a purely coordinate dependent part of the kernel re ects a kind of non-local interaction. All moments within a radius  7contribute to the interaction. Likewise a temporal feedback mechanism can be taken into account due to the fact that the transport of information from one magnetic moment to its neighbors needs at least a nite albeit small time. Such an in-time retardation mechanism is considered already in [33]. Here we concentrate on instantaneous retardation in time whereas the spatial part is realized for simplicity by a Gaussian shape (r;t) =(t)( 0 (p)3exp" r 2#) ; (10) where 0anddetermine the strength and the size of the retardation, respectively. The - function in the last equation signalizes that all contribution to the interaction within a sphere with radius contribute simultaneously to the interaction. As discussed below a typical value foris assumed to be of the order 108m, i.e. the time for the signal propagation within  is about 10151016s. Because this time is much smaller as the lifetime of the spin-waves, see the discussion below, we conclude that delay eects within the region with radius can be neglected. As indicated in Eq. (7) the noise (r;t) can also depend on space and time, i.e. in general random forces can eect the value of the magnetization at dierent positions in a distinct manner while additionally their uctuations are also time dependent. Such a behavior maybe lead back to local innitesimal temperature gradients or defects. The random eld (r;t) is regarded as a colored noise the statistical properties of which obey the following relations h(t)i=0; (t;t0) =h(t)(t0)i =D exp jtt0j  !0! 2D(tt0):(11) The components (t) have a zero mean and a nite correlation time. As an aside in the limit!0 the usual white noise properties are recovered. However, we want to concentrate on the more realistic colored noise case with  > 0. In Eq. (11) we assume (r;t) =(t). In other words the total system is aected by the same random in uences. This may be reasonable if we have a well controllable constant temperature over the whole sample and an ideal sample without defects. Let us brie y summarize the new properties of the model dened by Eqs. (8) and (9). The in uences of retardation and a multiplicative 8noise as well are implemented in the conventional Landau-Lifshitz-Gilbert equation. After a general incorporation into the model we had to limit the properties of both retardation and noise to an idealized situation in order to obtain analytical results in the subsequent section. However, although each of the Eqs. (10) and (11) represents a simplied version of a more general case the linking between both by means of the equation of motion for the magnetization in Eq. (8) models a quite complex behavior which is partly indicated in Fig. 2. While the exchange interaction is a short range coupling over a lattice constant a, si si+1 si+2 si+n−1si+n∝Jfeedback ∝Γ0 a ρ=n a FIG. 2. Schematic depiction of the dierence between the exchange interaction Jand the coupling due to retardation /0. As is visible feedback mechanisms can range over a larger distance '=na, wherenis integer. the interplay due to retardation with strength 0can cover a distance which is a multiple of the lattice constant. If this distance is comparable to the characteristic length scale in Eq. (10) retardation eects should be relevant. This microscopic picture can be transferred to a mesoscopic one and means a kind of non-local interaction. On the one hand at every spatial point the same kind of noise aects the magnetization. Otherwise, the magnetization m(r;t) takes dierent values at distinct positions rand therefore, the impact of the noise might be slightly dierent, too. Although spatial alterations of the noise are not regarded in the correlation function dened in Eq. (11) the memory kernel respects spatial correlations withinas seen in Eq. (10). Insofar the eect of noise at dierent spatial positions is transmitted by the memory kernel ( r;r0;t;t0). Another important hallmark is that the noise-noise correlation function (t;t0) is featured by a nite lifetime, cf. Eq. (11). For the forthcoming calculations we assume that =c. Likewise the matrix of the noise 9correlation strength is supposed to be diagonal, i.e. Dkl=Dkl. Hence, the two important stochastic parameters are the correlation time cand the correlation strength Dwhereas the relevant parameters originated by the retardation are the retardation strength 0and the retardation length , see Eq. (10). The results will be discussed in terms of the set of parameters D;c;0and. III. STATISTICAL TREATMENT Eqs. (8) and (9) represents the stochastic LLG. Due to the coupling to the stochastic eld (r;t) the magnetization eld m(r;t) becomes a stochastic variable. To calculate the mean values of mone needs the probability distribution P(m;t). To that aim the current section is devoted to the derivation of an approximated Fokker-Planck equation which allows to nd the equations of motion for averaged quantities. To that purpose let us reformulate the model presented in Eqs. (4), (8) and (9). After Fourier transformation '(q;t) =FTf (r;t)g we nd in linear spin-wave approximation d dt'(q;t) =A['(q;t)] + B['(q;t)](t): (12) The vector Aand the matrix Bposses the components A=f(q;) 1 +20 BBB@(+) (' 1+'2) (+) ('1' 2) 01 CCCA; (13) and B=f(q;) 1 +20 BBB@' 3'3('2+' 1) '3' 3'1' 2 '2'1 01 CCCA: (14) Here the function f(q;) is the Fourier transform of the memory kernel ( r;t) dened in Eq. (10) and andare introduced in Eqs. (4) and (6), respectively. Notice that fdepends only on the absolute value qof the wave vector and takes the form f(q; ) = 0exp 1 42q2 : (15) To get the probability distribution function of the stochastic process determined by Eqs. (11) and (12)-(14) we dene according to [37, 38] P(';t) =h['(t)']i: (16) 10Here the symbol <:::> means the average over all realizations of the stochastic process. As usual'(t) represents the stochastic process whereas 'are the possible realizations of the process at time t. Due to the colored noise the corresponding Fokker-Planck equation can be obtained only approximatively in lowest order of the correlation time. The time evolution of Eq. (16) can be written in the form @ @tP(';t) =LP(';t): (17) In deriving this expression we have used the time evolution of '(t) according to Eq. (12), the Novikov theorem [39] and the correlation function given by Eq. (11) with =c, D=D. The form of the operator Lis given in a correlation time and cumulant expansion while transient terms have been neglected [40{42] L(';c) =@ @'A(') +@ @'B(')@ @' ( D B (')cM (') +D2c K (')@ @'B(') +1 2B (')@ @'K(')) ;(18) with M =A@B  @'B@A @' K =B@B  @'@B  @'B:(19) Notice that summation over double-indices is understood. The single probability distribution is determined by the operator Lin Eq. (18) which enables us to nd the equation of motion for the expectation values h'i. It follows d dth'(t)i=hAi+D@B @' B cM 
 D2c(@ @'@B @' K  B +1 2@ @'@B @' B  K) :(20) Notice that in the white noise case all terms /cwould vanish. 11IV. RESULTS AND DISCUSSION We nd an analytical solution for the colored noise problem in Eq. (20) by standard Greens' function technique and Laplace transformation. After performing the summation in Eq. (20) while making use of Eqs. (13), (14) and the expressions in Eq. (19) the result reads h'(t)i=0 BBB@etcos( t)etsin( t) 0 etsin( t)etcos( t) 0 0 0 et1 CCCA
h'0i; (21) whereh'0i=h'(t= 0)iare the initial conditions. Physically, the parameters ; and play the roles of the inverse magnon lifetimes and the frequency of the spin wave, respectively. They are determined by =["+]f(q;) 1 +2+D[222]f(q;) 1 +22 + 2Dc["+]f(q;) 1 +23 +D2c[1622]f(q;) 1 +24 ; =["+]f(q;) 1 +2+ 3Df(q;) 1 +22 +Dc[221] ["+]f(q;) 1 +23 +D2c 2[11322]f(q;) 1 +24 ; =2Df(q;) 1 +22 [4Dc("+)]f(q;) 1 +23 +D2c[322+ 1]f(q;) 1 +24 :(22) Note that the parameters of the retardation mechanism, the strength 0and the length scale , are included in the function f(q;) dened in Eq. (15). The two important parameters originated from the noise are the correlation time cand the correlation strength Dof the random force. Both aect the quantities in Eq. (22) as well. We proceed by studying the system under the variation of these four model parameters. To be comparable to FMR experiments we refer to the following quantities L= ( K )1;
B= 1:16! = 1:16K ; (23) 12i.e. the lifetime Lof the spin waves and the FMR-linewidth
B, compare [1, 17], which are related to the dimensionless inverse lifetime and frequency from Eq. (22). Here the frequency!is tantamount to the resonance frequency of the spin waves. The lifetime L and the linewidth
Bare given in SI-units. Notice that the frequency independent part
B0, typically added on the right-hand side of the equation for
Bis already subtracted in Eq. (23). The contribution
B0is supposed to take into account magnetic inhomogeneities. For a quantitative evaluation we need to set the model parameters to reasonable values. In doing so we also refer to Eq. (6). First let us start with xed values. For the Gilbert damping parameter we choose the bulk value for Co which was found to be '0:005 [43, 44]. A similar value ( '0:0044) was measured for a FE 4=V4multilayer sample in perpendicular conguration where only intrinsic Gilbert damping is operative [16]. The anisotropy eld Kis estimated as follows. Since the exchange interaction is typically about 104times larger than relativistic interactions which are responsible for anisotropy [45] and the magnetic exchange eld can adopt large values we estimate the anisotropy asK= 0:1 T. Since we are interested in small excitations transverse to the anisotropy axiswe suppose = 0:9 for the time independent part of the magnetization pointing in the direction of the anisotropy axis , compare Eq. (4). Moreover, the gyromagnetic ratio '1:761011(Ts)1. The characteristic magnetic length dened in Eq. (6) is of the order l0'108m. For the calculations a static magnetic eld of about 0 :5 T is taken into account which corresponds to the scaled external eld "= 5. Notice that the dispersion relation in Eq. (22) is q-dependent. In the following we assume a medial value q= 106m1. The parameters which are altered in the upcoming analysis are the noise correlation strength D and the retardation strength 0. We investigate our model for both values ranging in the interval [0,10]. After this estimation the two parameters, the noise correlation time cand the retardation length are left over. For a comprehensive estimation we suggest that is ranged in 1012m< < 106m. The lower limit is smaller than a typical lattice constant a'1010m where the upper limit is a few orders of magnitude larger than the lattice constant. Likewise the correlation time ccaptures a quite large interval. Remark that we keep the notation c, especially with regard to Fig. 3, although we also designated the dimensionless correlation time as c. In order to cover a wide range the time interval is chosen in between atto- and nanoseconds. The results are depicted in Fig. 3 and Fig. 4. In Fig. 3 the behavior of the FMR-linewidth
Bas well as the lifetime of the spin waves L, introduced 13(a) (c)(b) (d)FIG. 3. The FMR-linewidth and the lifetime depending on: (a) the noise correlation strength Dfor c= 568 as, 0= 1,= 108m; (b) the noise correlation time cforD= 1, 0= 1,= 108m; (c) the retardation strength 0forc= 568 as,D= 1,= 108m; (d) the retardation length for c= 568 as, 0= 1, 0. The other parameters take l0= 108m,q= 106m1,"= 5,= 0:9 and = 0:005. in Eq. (23), are shown in dependence on the dierent model parameters explained above. The in uence of the correlation noise strength on
BandLis shown in Fig. 3(a). Whereas the linewidth decreases only very weak linearly when the noise strength Dis increased, the lifetime of the spin waves Lreveals a strong dependency on D. This is indicated by the fact that Lis monotonic decaying while it covers several orders of magnitude with growing noise strength D. The curve shape for the lifetime Lseems to be comprehensible because the stronger the stochastic forces are correlated and interact mutually the faster the coherent motion of the spin moments is destroyed. This microscopic picture is reasonable under the premise that the evolution of spin waves is based on the phase coherence between adjacent magnetic moments. Apparently the frequency and consequently the linewidth
B show only a quite small eect, compare Eq. (23). Therefore, the variation of Dreveals no signicant in uence on the frequency velocity of the moments. A distinct behavior is depicted in Fig. 3(b) for
BandLas a function of the noise correlation time c. Both the linewidth and the lifetime remain constant for large interval of the correlation time c, 14roughly speaking for cranging from as to ps. If the correlation time is in between ps and ns the linewidth
Bincreases about a factor of 20 and the lifetime Ldecreases to a value about 9-times smaller. Thus caects both
BandLin an opposite manner provided the noise- noise correlations occur on time scales larger than ps. In this regime a growing correlation timecimplicates likewise an enhancement of the resonance frequency of the spin waves !/
B, see Eq. (23). Simultaneously the spin wave lifetime Ldeclines strongly. Such a behavior may be attributed to a 'stochastic acceleration' which on the one hand enhances the frequency but on the other hand drives neighboring magnetic moments out of phase coherence. Remark that for times c>1ns the linewidth
Btends to innity. This eect is not shown in the picture. Concerning the in uence of the retardation parameters we refer to Fig. 3(c), which illustrates the in uence of the retardation strength 0. As recognizable the FMR-linewidth exhibits a seemingly linear dependence as function of 0while
Bgrows with increasing retardation strength. The lifetime Ldecreases in a non-linear manner. The decay covers a range of 3 orders of magnitude. We suggest the following mechanism behind this eect: Let us consider two moments both localized at arbitrary positions within the retardation length as schematically displayed in Fig. 2. The mutual coupling due to retardation between both characterized by 0leads to a phase shift between neighboring spins. Therefore, the phase coherence originated by the self-organized internal magnetic eld is interfered in view of an interplay within the feedback coupling in coordinate space. The stronger this interaction 0is the faster is the damping of the spin waves. Accordingly, spin wave solutions for dierent values of 0are plotted exemplary in Fig. 4. The retardation lengthin uences
BandLas well as is visible in Fig. 3(d). Here the quantities
Band Lremain constant for a retardation strength ranging within the pm regime and a few tenthm. For larger -values the linewidth
Bdecreases while the lifetime Lincreases. In the regime >1m the linewidth
Btends to zero and the lifetime L!1 . This behavior is not depicted in Fig. 3(d). Notice that for reasonable values of which not exceed the sample size the mentioned situation is not realized. The shapes of the curves in Fig. 3(d) may be explained as follows. This graph corresponds to a xed retardation strength 0 while the retardation length is enlarged. Again we refer to the physical picture where the internal eld, originated by the mutual interaction of the moments, and the coupling due to the retardation operate as opposite mechanisms. The interplay happens in such a manner that an increasing retardation strength 0weakens or destroys the phase coherence 15-0.6-0.4-0.20.00.20.40.60.81.0mean value 0 10 20 30 40 50 60 time[ps]0= 2 0= 3 0= 5 /angbracketleftϕ1/angbracketright /angbracketleftϕ0/angbracketrightFIG. 4. Evolution of spin waves for dierent values of the retardation strength 0. The other parameters take l0= 108m,q= 106m1,"= 5,= 0:9 and= 0:005,c= 568 as,D= 0:5 and = 108m. between adjacent spins. Yet it is found that a growing retardation length counteracts the damping of the spin waves. As a consequence we suppose that the more spins are involved into the retardation eect, i.e. the larger the parameter becomes, the more the damping is reduced. In other words it seems that retardation eects can average out if suciently many magnetic moments are involved. V. CONCLUSIONS In the present paper we have studied a model on a mesoscopic scale realized by means of Landau-Lifshitz-Gilbert dynamics. The magnetization is driven by an eective magnetic eld. This eld consists of an internal eld due to the exchange interaction, an anisotropy eld and a static external eld. Additionally, the eective eld is supplemented by a time 16depending random one obeying colored noise statistics. Moreover, the stochastic LLG is generalized by the introduction of a retardation kernel depending on the spatial coordinates only. Such a kernel simulates a kind of non-local interaction of size . After deriving an approximated Fokker-Planck equation we were able to calculate the mean values of the components of the magnetization in the linear spin wave approach. They depend strongly on the parameters characterizing the retardation (strength 0, length) as well as the stochastic (strength D, correlation time c) processes. As a result of the analysis we found that the increase of the retardation strength 0compared with the growth of the retardation lengthcan entail con ictive eects on the lifetime L. The main results are depicted in Fig. 3. There, in addition to the lifetime of the spin waves Lthe FMR-linewidth
Bis displayed. In doing so we want to provide comparability to experimental investigations based on ferromagnetic resonance for cases when the LLG is applicable. Let us remark that also other mechanisms are able to contribute to the damping process. As suggested in [2, 46] the Bloch-Bloembergen equations [47, 48] are more appropriate for in-plane congu- rations in thin lms. These equations are characterized by two relaxation times. Another approach with dierent relaxation processes is based upon the Landau-Lifshitz-Bloch equa- tions [49, 50]. Our method including the inevitable stochastic forces can be likewise applied to the modied set of equations. One of us (T.B.) is grateful to the Research Network 'Nanostructured Materials' , which is supported by the Saxony-Anhalt State, Germany. Furthermore, we are indebted to Dr. Khalil Zakieri (MPI of Microstructure Physics) for valuable discussions. [1]Ultrathin Magnetic Structures II + III , edited by B. Heinrich and J. Bland (Springer, 2005). [2] D. L. Mills and S. M. Rezende, \Spin dynamics in conned magnetic structures ii, edited by b. hillebrands and k. ounadjela," (Springer, Berlin, 2003) Chap. Spin Damping in Ultrathin Magnetic Films, pp. 27{59. [3] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935). [4] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [5] G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Appl. Phys. 17Lett. 94, 102501 (2009). [6] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). [7] O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and D. Garanin, Phys. Rev. B 74, 094436 (2006). [8] D. A. Garanin and O. Chubykalo-Fesenko, Phys. Rev. B 70, 212409 (2004). [9] N. Kazantseva, D. Hinzke, R. W. Chantrell, and U. Nowak, Europhys. Lett. 86, 27006 (2009). [10] M. Sparks, R. Loudon, and C. Kittel, Phys. Rev. 122, 791 (1961). [11] R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999). [12] R. Arias and D. L. Mills, J. 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Steiauf, and M. F ahnle, Phys. Rev. B 81, 174414 (2010). [22] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 101, 037207 (2008). [23] B. Kardasz and B. Heinrich, Phys. Rev. B 81, 094409 (2010). [24] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). [25] C. Wang, Y.-T. Cui, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Nature Physics (published online 27 Feb., doi:10.1038/nphys1928)(2011), doi:\bibinfo doi 10.1038/nphys1928. [26] R. E. Arias and D. L. Mills, Phys. Rev. B 79, 144404 (2009). [27] J. M. G. Vilar and J. M. Rub , Phys. Rev. Lett. 86, 950 (2001). [28] J. Foros, A. Brataas, G. E. W. Bauer, and Y. Tserkovnyak, Phys. Rev. B 79, 214407 (2009). 18[29] J. Swiebodzinski, A. Chudnovskiy, T. Dunn, and A. Kamenev, Phys. Rev. B 82, 144404 (2010). [30] Z. Diao, E. R. Nowak, G. Feng, and J. M. D. Coey, Phys. Rev. Lett. 104, 047202 (2010). [31] F. Hartmann, D. Hartmann, P. Kowalzik, L. Gammaitoni, A. Forchel, and L. 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2011-07-04

The theory predicts that the spin-wave lifetime $\tau_L$ and the linewidth of ferromagnetic resonance $\Delta B$ can be governed by random fields and spatial memory. To that aim the effective field around which the magnetic moments perform a precession is superimposed by a stochastic time dependent magnetic field with finite correlation time. The magnetization dynamics is altered by inclusion of a spatial memory effect monitoring a non-local interaction of size $\xi$. The underlying Landau-Lifshitz-Gilbert equation (LLG) is modified accordingly. The stochastic LLG is equivalent to a Fokker-Planck equation which enables to calculate the mean values of the magnetization vector. Within the spin-wave approximation we present an analytical solution for the excitation energy and its damping. The lifetime and the linewidth are analyzed depending on the strength of the random field $D$ and its correlation time $\tau_c$ as well as the retardation strength $\Gamma_0$ and the size $\xi$. Whereas $\tau_L$ decreases with increasing $D$, retardation strength $\Gamma_0$ and $\tau_c$, the lifetime is enhanced for growing width $\xi$ of the spatial retardation kernel. In the same manner we calculate the experimentally measurable linewidth $\Delta B$ is increased strongly when the correlation time $\tau_c$ ranges in the nanosecond interval.

Influence of randomness and retardation on the FMR-linewidth

1107.0638v1

1Magnetization Dynamics, Throughput and Ener gy Dissipation in a Universal Multiferroic Nanomagnetic Logic Gate with Fan-in and Fan-out Mohammad Salehi Fashami1, Jayasimha Atulasimha1* and Supriyo Bandyopadhyay2 1Department of Mechanical and Nuclear Engineering, 2Department of Electrical and Computer Engineering, Virginia Commonwealth University, Richmond VA 23284, USA. Abstract The switching dynamics of a multiferroic nanomagnetic NAND gate with fan-in/fan-out is simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation while neglecting thermal fluctuation effects. The gate and logic wires are implemented with dipole-coupled 2-phase (magnetostrictive/piezoelectric) multiferroic elements that are clocked with electrosta tic potentials of ~50 mV applied to the piezoelectric layer generating 10 MPa stress in the magnetostrictive layers for switching. We show that a pipeline bit throughput rate of ~ 0.5 GHz is achievable with pr oper magnet layout and sinusoidal four-phase clocking. The gate operation is completed in 2 ns with a late ncy of 4 ns. The total (internal + external) energy dissipated for a single gate operation at this throughput rate is found to be only ~ 1000 kT in the gate and ~3000 kT in the 12-magnet array comprising two input and two output wires for fan-in and fan-out. This makes it respectively 3 and 5 orders of magnitude more energy-efficient than complementary-metal- oxide-semiconductor-transistor (CMOS) based and sp in-transfer-torque-driven nanomagnet based NAND gates. Finally, we show that the dissipation in th e external clocking circuit can always be reduced asymptotically to zero using increasingly slow adiaba tic clocking, such as by designing the RC time constant to be 3 orders of magnit ude smaller than the clocking period. However, the internal dissipation in the device must remain and cannot be eliminated if we want to perform fault-tolerant classical computing. Keywords: Nanomagnetic logic, multiferroics, straintr onics and spintronics, Landau-Lifshitz-Gilbert . ----------------------------------------------------------------------------- * Corresponding author. E-mail: jatulasimha@vcu.edu 2I. Introduction A major challenge in designing digital computi ng machinery is to reduce energy dissipation during the execution of a computational st ep since excessive dissipation is the primary impediment to device downscaling envisioned in Moore’s law. A state-of-t he-art CMOS nanotransistor presently dissipates over 50,000 kT (2×10-16 Joules) of energy at room temperature to switch in isolation , and over 106 kT (4.2×10- 15 Joules) to switch in a circuit at a clock rate of few GHz [1], which makes further downscaling problematic. Therefore, nanomagnet-based computing and signal processing are attracting increasing attention. Single-domain nanomagnets are intrinsica lly more energy-efficient than transistors as logic switches and do not suffer from cu rrent leakage that results in standby power dissipation. Consequently, magnetic architectures hold the promise of outpacing tr ansistor-centric architect ures in energy-efficient computing. A nanomagnetic logic switch is typically implemen ted with a shape-anisotropic nanomagnet that has two stable magnetization orientations along the easy axis. These two magnetizations encode the logic bits 0 and 1. The nanomagnet’s advantage over the transistor accrues from the fact that when the magnetization is flipped, all the spins (information carriers) in a single-domain nanomagnet rotate in unison like a giant classical spin (because of exchange interaction between spins), so that ideally the magnet has but a single degree of freedom [2]. Accordingly, the minimum energy dissipated in switching it non-adiabatically is ~ kTln(1/p) where p is the static error probability associated with random switching due to thermal fluctuations. In contrast, all the di fferent electrons or holes (information carriers) in a transistor act independently, so that the minimum energy dissipated in switching a transistor non- adiabatically is NkTln(1/p) [2], where N is the number of information carri ers (electrons or holes) in the transistor. The advantage of the single-domain nanomagnet thus accrues not from any innate advantage of spin over charge as an information carrier, but from the fact that spins mutually interact in a way which reduces the degrees of freedom, and hence the energy dissipation. 3There are numerous ways of impl ementing nanomagnetic logic (NML) [3], [4]. In one type of architecture, termed ‘magnetic quantum cellular auto mata’ [5], logic gates are configured by placing nanomagnets in specific geometric patterns on a su rface so that dipole interactions between neighbors elicit the desired logic operations on the bits en coded in the magnetization orientations of the nanomagnets[4] ，[5]. This approach is the same as that envisioned in the Single Spin Logic (SSL) paradigm, where exchange interaction between spin s played the role of dipole interaction between magnets, while up- and down-spin pol arizations encoded logic bits [6]. However, SSL required cryogenic operation while NML can operate at room temperature. Curiously, the minimum energy dissipated per bit flip in NML is the same as that in SSL because a si ngle spin and a giant classi cal spin have the same degree of freedom. This is a remarkable feature that makes NML particularly attractive. Of course, the actual energy dissipated per bit flip in NML will always be somewhat higher than kTln(1/p) because of internal dynamics such as sp in-orbit interaction giving rise to Gilb ert damping within the magnet, but that is still a small price to pay for room temperature operation. a. Magnet switching schemes Unfortunately, the nanomagnet’s advantage over th e transistor will be squandered if the method employed to switch the nanomagnet becomes so ener gy-inefficient that the energy dissipated in the switching circuit vastly exceeds the energy dissipated in the nanomagnet. In the end, this can make magnetic architectures less energy-efficient than tran sistor based architectures, thereby defeating the entire purpose of using magnetic switches. Therefore, the switching scheme is vital. Magnets are typically switched with either a magnetic field generated by a current [7], or with spin transfer torque [8], or with domain wall motion indu ced by a spin polarized current [9]. In the first approach, a local magnetic field is generated by a local current based on Ampere’s law: cI Hd l (1) where the integral is taken around the current loop. 4Now, the minimum magnetic field minH required to flip a magnet can be estimated by equating the magnetic energy in the field to the energy barrier bEseparating the two magnetization directions encoding the bits 0 and 1, i.e. 0m i n s b MHE (2). Here0is the permeability of free space, sMis the saturation magnetization which we assume is 105 A/m (typical value for nickel or cobalt), and is the nanomagnet’s volume which we assume is 100 nm 100 nm 10 nm since such a volume results in a single domain ferromagnet at room temperature. The energy barrier Eb is determined by the static bit error probabilitybEk Tethat we can tolerate. For reasonable error probability (< e-30), we will need that 30kTbE . This yields from Equations (1) and (2) that minI= 6 mA if we assume that the loop radius is 100 nm so that it can comfortably encircle the magnet. The resistance of loop will be ~50 ohms if we assume that it is made of silver (lowest resistivity among metals; 2.6 S-cm ) and has a wire radius of 10nm, so that the energy dissipated – assuming that the magnet flips in 1 ns – is at least 1.8 pJ, or ~ 4 108 kT, which makes the magnet switch ~400 times more dissipative than a state-of-the-art transistor switch because of the inefficient switching scheme. There is also a nother disadvantage; the magnetic field cannot be confined to small spaces, which means that indi vidual magnets cannot be addressed unless the magnet density is sparse (magnet separation 0.5 m). That not only reduces device density, but might make dipole interaction between magnets so weak as to ma ke magnetic quantum cellular automata inoperable. Therefore, this method is best adapted to addr essing not individual magnets, but groups of (closely spaced) magnets together, as envisioned in ref. [7]. However, that approach makes magnetic quantum cellular automata architecture non-pipelined and hence very slow [10]. In the end, this is clearly a sub- optimal method of switching magnetic switches. Spin transfer torque (STT) is better adapted to addressing indi vidual magnets since it switches magnets with a spin polarized current passed dir ectly through the magnet. It dissipates about 108 kT of 5energy to switch a single-domain nanomagnet in ~ 1 ns , even when the energy barrier within the magnet is only ~ 30 kT [11]. Thus, it is hardly better than th e first approach in terms of energy efficiency. A more efficient method of switching a magnet is by induci ng domain wall motion by passing a spin polarized current through the magnet. There is at least one report of switching a multi-domain nanomagnet in 2 ns by this approach while dissipating 104 kT – 105 kT of energy [12]. This makes it 1-2 orders of magnitude more energy-efficient than a transistor in a circuit. Recently, we devised a much more efficient magne t switching scheme. We showed that a 2-phase multiferroic nanomagnet, consisting of a piezoelectric layer elastically coupled with a magnetostrictive layer, can be switched by applying a small voltage of few mV to the piezoelectric layer [13] ，[14]. This voltage generates uniaxial strain in the piezoelectric layer that is transferred almost entirely to the magnetostrictive layer by elastic coupling if the latter layer is much thinner than the former. Uniaxiality can be enforced in two ways: either by applying the electric field in the direction of expansion and contraction ( d33 coupling) or by mechanically clamping th e multiferroic in one direction and allowing expansion/contraction in the pe rpendicular direction through d31 coupling when the voltage is applied across the piezoelectric layer. The substrate is assumed to be a soft material (e.g. a polymer) that allows uniaxial expansion/contraction. The uniaxial strain/stress will cause the magnetization of the magnetostrictive layer to rotate by a large angle. Su ch rotations can be used for Bennett clocking of NML gates for logic bit propagation [13]. In ref. [14] ，[15], we showed that the energy dissipated in the magnet and clock together is a few hundreds of kT for a switching delay of 1 ns or less. This makes it one of the most energy-efficient magnet switching schemes. In this paper, we have studied the switching d ynamics of a NAND gate with fan-in/fan-out wires implemented with multiferroic elements and calcu lated the energy dissipation in the entire block assuming low enough temperature when effects of thermal fluctuations can be neglected. At room temperature, thermal fluctuations will act as a random magnetic field that will increase the switching error 6probability and mandate higher stress le vels (along with larger energy dissipation) for reliable gate operation. This study is deferred to a future date. The present paper is organized as follows: In S ection II, the theoretical framework for studying magnetization dynamics in the NAND ga te with fan-in and fan-out wires is discussed. Section III presents and discusses simulation results. Section IV discusses energy dissipation and strategies to reduce the energy overhead in the clocking circuit. Fina lly in section V, we present our conclusions. II. Magnetization Dynamics in a Multiferroic Nano magnetic Logic Gate with Fan-in and Fan- out Consider an all-multiferroic NAND gate with fan- in and fan-out wires as shown in Fig 1. Each multiferroic element has the shape of an ellip tical cylinder with homogeneous magnetization ()Mrin the magnetostrictive layer. We assume that the piezoelectric layer has a thickness of 40 nm and the magnetostrictive layer has a thickness of 10 nm, wh ich will ensure that strain generated in the piezoelectric layer is mostly transferred to the magne tostrictive layer through elastic coupling. There are mechanical clamps along the minor axis of the magne t (not shown) that prevent expansion/contraction along that sideward direction, so that application of a voltage across the piezoelectric layer generates uniaxial stress along the major axis via the 31dcoupling. The magnets are fabricated on a soft substrate that does not hinder expansion and contraction of the elements along any direction by clamping from below. If the planar dimensions of each element are ~ 101. 75 nm × 98.25 nm, the ex change coupling penalty precludes the formation of multi-domain states in the magnetostrictive layer [16], [17] so that we can model it as a single-domain nanomagnet. The shape anis otropy of the element gi ves rise to an energy barrier of 32 kT (at room temperature) between the two stable orientations along the easy axis (major axis) of the ellipse. 7The magnetization dynamics of any nanomagnet under the influence of an effective fieldeffH acting on it is described by the Landau-Lifs hitz -Gilbert (LLG) equation [17]: eff eff sdM tMtHt M t M tHtdt M (3). Here i effH is the effective magnetic field on the ith nanomagnet, which is the partial derivative of its total potential energy ( Ui) with respect to its magnetization ( iM ), is the gyromagnetic ratio, sMis the saturation magnetization of the magnetostrictive layer and is the Gilbert damping factor [18] associated with internal dissipation in the magnet when its magnetization rotates. Accordingly, 0011 i i eff m i s iUtH tU tM Mt , (4). where is the volume of any nanomagnet (only that of th e magnetostrictive layer) in the chain shown in Fig 1. The total potential energy of any element in this chain is given by: 2 222 0 ___ 22sin cos sin sin cos2 3sin sin2dipole dipole shape anisotropy dipoleij is d x x i i d y y i d z z i i ji E E si iUt E t M N t t N t t N t t other terms stress anisotropyi Et (5). Here dipole dipoleijE is the dipole-dipole interaction energy due to in teraction between the i-th and j-th magnets, shape anisotropyE is the shape anisotropy ener gy due to the elliptical shape of the multiferroic element, and stress anisotropyE is the stress anisotropy energy caused by the stress transferred to the magnetostrictive layer of the multiferroic upon application of an elect rostatic potential to the piezoelectric layer. The quantities dk kNare the demagnetization factors in the k-th direction. We assume that the magnetostrictive layer is polycrystalline so that we can neglect magnetocrystalline anisotropy. The quantity sis the magnetostrictive coefficient and the quantities and are the polar and azimuthal 8angles of the magnetization vector iM . We assume that the major axis ( easy axis) of the ellipse is in the y-direction, the minor axis (in-plane hard axis) is in the x-direction and the out-of-p lane hard axis is in the z-direction as shown in Fig. 1(b). The ‘other term s’ in the above equation are time-dependent (they include the chemical potential, the mechanical potential due to stress/strain etc.), but do not depend on iM (or ,ii) and hence do not affect i effH . Consider two adjacent multiferroic elem ents in the chain (labeled as the ith and jth element), whose magnetizations subtend an angle of ,ii and ,j j respectively with the positive z-direction and x- direction in the x-y plane. The dipole-dipole interaction energy is [19]: 22 0 323.. .4| | | |ij s dipole dipole i j i i j j i j ij ijMEt m t m t m t r m t rrr (6). where ijr is the vector distance between the ith and jth magnet and kmis the magnetization of the k-th magnet normalized to Ms. For two neighboring magnets whose in-plane hard axes are collinear with the line joining their centers, the dipole coupling energy is: 22 0 32 sin cos sin cos sin sin sin sin4 cos cosii j j ij s dipole dipole i j j i j ijtt t t MEt t t t tr tt (7). If the line joining the centers subtends an angle γ with their hard axes as shown in Fig 1 (b), the dipole coupling energy is: 22 2222 0 3sin cos )(sin cos 2(cos ) (sin ) sin sin sin sin 2(sin ) (cos ) 4 sin cos sin sin sin cos sin sin 3sin cos cos cosii j j ii j jij s dipole dipole ii jj j j i i ijtt tt tt ttMEtr tt tt tt tt tt j (8). where r is the separation between their centers. 9The total potential energy of a magnet given in Equa tion (5) is used to find the effective field effH acting on it in accordance with Equation (4): 0 0 0() 1() s i n () c o s ()() () 13() s i n () s i n () () s i n () s i n ()() 1()ij dipole dipole i eff x s d xx i i sx ji ij dipole dipole i eff y s d yy i i s i i i bias sy s ji i eff zEtHt M N t tMm t EtHt M N t t t t t HMm t M Ht 0()cos ( )()ij dipole dipole sd z z i sz jiEtMN tMm t (9). where biasH is any external time-invariant magnetic field a pplied in the y-direction (see later discussion as to when and why it might be needed). Further, it appears from equation (9) that a stre ss applied in the y-direct ion, produces only a H eff along the y-direction and could not rotate the magnetization along th e x or z-direction. However, a deeper analysis of its effect on the magnetization shows this is not true. We will explain this by writing the stress anisotropy ener gy term and the associated Heff in Cartesian co-ordinates as: 2 _ 033( ) H ( )2stress anisotropy s i y eff y s i y sE tm t t m tM Now suppose a compressive stress (negative ) is applied to a material with positive magnetostriction ( λ) and the magnetization is initially direct ed close but not parallel to +y-axis so that m y ~ 1. The Heff is along the -yaxis and would make th e magnetization rotate away from +y towards the x-direction (also lifting the magnetizati on vector a little out of plane; the lifting is negligible when N d_ZZ is very large) . However, when the magnetization rotates close to the hard (x-axis), m y ~0, and the Heff due to stress vanishes. If the magnetization had initially pointed along the -y-axis, Heff would have been direct ed along the +y-axis, which would have again rotated the magnetization away from the -y-axi s towards the in plane hard axis (x-axis). The effective field Heff for each nanomagnet described by Equation (9) includes the effect of dipole coupling with neighboring na nomagnets, shape anisotropy and stress anisotropy. For dipole coupling, the summation is performed over nearest, second-nearest and sometimes third-nearest neighbors since the 10interaction with remote neighbors may not be negligib le compared to that with the nearest neighbor. We elucidate this below: In Fig 1 (a), consider a magnet marked "I". The nearest neighbor marked “II” is at a distance r, but the second nearest neighbor marked 'III' is at a distance of 52r which results in a dipole field (proportional to 31r) that is ~25 % of the dipole field due to the magnet marked 'II'. Hence both have to be considered. Similarly, for a magnet marked "III", in teractions with magnets marked 'I", "II" and "IV", whose centers are respectively at distances of 52r, 32rand r, are considered. For magnets marked "IV" , "V", and "VII" both nearest and second nearest neighbor dipole coupling terms were considered. However, for the magnet marked "VI" only the nearest neighbors' dipole coupling was considered since the second nearest neighbor is at a distance 2 r away and contributes only 12.5% of the interaction caused by the nearest neighbor. The effective magnetic field Heff evaluated from Equation (9) is used in the scalar version of equation (3) which is described in ref. [15] to determine th e magnetization state of any magnet at any instant of time. Equation (3) leads to two coupled ordinary differential equations (ODEs) for each nanomagnet. Thus, for 12 nanomagnets, 24 coupled ODEs have to be solved simultaneously. This allows us to compute the temporal evolution of the magnetization orientations ,iitt of all 12 magnets in Fig 1. In order to flip the magnetization of any magnet, it is first subjected to stress with a voltage V applied across the piezoelectric layer. The si gn of the stress (compressive or tensile) depends on the sign of the magnetostrictive coefficient sof the material. For a material with positives, such as Terfenol-D, we apply a compressive (negative) stress to rotate th e magnetization away from the easy axis (y-axis) towards the in-plane hard axis (x-axis). If the str ess is of sufficient magnitude to overcome the shape anisotropy energy barrier, the magnetization will ultima tely align along the in-plane hard axis (x-axis). Once that happens, the stress is reversed to tensile so that the magnetization relaxes back to the easy axis 11(y-axis), but in a direction anti-parallel to the orig inal orientation along the easy axis. What ensures the anti-parallel orientation is that stress not only causes in-plane rotation of the magnetization vector, but also lifts it slightly out of plane in such a way that the resulting y-component ofeffH prefers the anti- parallel orientation over the parallel orientation. This results in a magnetization flip or bit flip. The time evolution of the magnetization during this process is tracked by solving the LLG equation (Equation (3)) starting with the initial condition. If we choose the initial orientation of the magnetization to be exactly along the easy axis 090 , then the effective field on the magnet due to stress vanishes [see Equation (9)] and therefore stress becom es ineffective in causing any rotation. Hence, we always assume that the initial state of any magnetizati on is never exactly along an easy axis but deflected by 0.1 degrees from the easy axis 0090 , 89.9 . Such deflections ~ 5º can easily occur because of thermal fluctuations [20] but we are conserva tive and assume only 0.1º deflection to ensure our simulation results are applicable over a greater confidence interval . The energy dissipated in flipping a bit has two components: i) Energy dissipated while applying, reversing a nd removing a voltage on the piezoelectric layer for generating stress. This is the energy dissipated in the clocking circuit and is given by [21]: 2 21 2 1clockRCEC V RC (10a). where C is the capacitance of the piezoelectric laye r, R is the resistance of the wires and V is the voltage applied across it. We assume that the volta ge waveform is sinusoidal with a period 2. However, the problem with this RC circuit is that the en ergy stored in the capacitor (piezoelectric layer), when it is fully charged, minus the dissipati on in the resistor, will be dissipated in the power source in each cycle. A better scheme is to use an LCR circuit where energy is merely transferred between the capacitive and inductive elements and the onl y energy lost per cycle is the energy dissipated in the resistive element [22]. In such a clocking circuit, the energy dissipated is: 212 ( ) CV 12 (10b). ii) Internal energy dissipated in the magnet during magnetization rotation [14, 15, 17]. This energy dEis calculated as: 0 . d effdE t dMHddt dt (11). By substituting Equation (3) for dM dt in equation (11) and integrating one obtains: 2 0 2 00 || (1 )d de f f sdEE dt H t M t dtdt M (12). This expression clearly shows that this dissipation is associated with damping in the magnet because it disappears when 0. III. Results and Discussions We have used 4th order Runge -Kutta method as described in [15] to solve the system of 24 coupled ordinary differential equations for a chain of twelve dipole coupled mu ltiferroic elements shown in Fig. 1(a). These 12 magnets comprise the NAND-gate and wiri ng for fan-in of 2 and fan-out of 3. The stress applied on the four nanomagnets comprising the actual gate follows a 4-phase si nusoidal clocking scheme shown in Fig 1(c). The magnets are grouped into 8 gr oups I through VIII. The sinusoidal clocks applied to each group and the relative phase lags between the clock signals for different groups is shown in Fig. 1(c). Clearly, a 4-phase clock is required. When the phase for the clock on magnets marked "I" goes past 90º so that the compressive stress on these magnets be gins to decrease, the compressive stress on magnets marked II just begins to increase. Thus, when the stress on magnets "I" has decreased to 12of the 2 2()clockVE RCR 13maximum applied compression, the magnets marked "II" are at a state of 12of the maximum compression and have been sufficiently rotated away from the easy direction. Consequently, as compressive stress decreases to a point where the shap e anisotropy begins to dominate and therefore the magnetizations of magnets marked "I" rotate towards their easy axes, their orientation is influenced strongly and ultimately uniquely determined by the orientations of the "input" magnets ensuring uni- directionality of information propagation [15]. From the time-dependent voltages on any magnet, we derive the time-depende nt stresses and hence the time dependent effective fields i effHt on each magnet. These are used to solve the LLG equation (24 coupled ODEs). The solutions yield the orientation () , ()iitt of each element. The in-plane magnetization orientation ( ()it ) of each of the 12 nanomagnets (Fig 2 a-d) is plotted to demonstrate: (i) successful NAND operation for any arbitrary input combina tion [(1,1), (0,0), (1,0), (0,1)] starting with the initial input state (1, 1), and (ii) the complete ma gnetization dynamics showing that the primitive gate operation is always completed in 2 ns and the latency is 4 ns. In this study, we assumed that the magnetostric tive layers were made of polycrystalline Terfenol-D with material properties and dimensi ons given in Table I. The piezoelectric layer is assumed to be lead- zirconate-titanate (PZT) that has a reasonably large d31 coefficient (10-10m/V[27]), albeit also a large relative dielectric constant of 1000. Terfenol-D was chosen for its high magnetostriction [23] (even in the nanoscale [24]). The geometric parameters for the i ndividual magnets and the array were chosen to ensure: (i) the shape anisotropy energy of the elements was sufficiently high (~0.8 eV or ~32 kT at room temperature) so that the bit error probability due to spontaneous magnetizatio n flipping was very low (~32 1410 e ), (ii) the dipole interaction energy was limited to 0.26 eV which was significantly lower than the shape anisotropy energy to prevent spontan eous flipping of magnetization, but large enough to ensure that the magnetization of the multiferroic elemen ts always flipped to the correct orientation when stress was applied, even under the influence of ra ndom thermal fluctuations, and (iii) the maximum 14applied stress of 10 MPa corresponded to a stress-anisotropy energy 3 2s= 172 kT that was significantly larger than the shape anisotropy energy barrier of 32 kT. The reason why such large stress was required are: (1) some magnets (for example th e magnet marked "III") had to overcome significant amount of dipole coupling from interaction with multiple neighbors to rotate close to the hard axis; (2) the stress anisotropy is least effective close to Φ=0 and hence the stress had to be large to ensure fast magnetization rotation for angles close to the hard axis. In all our simulations (Fig 2 a-d), the initial ma gnetizations of the nanoma gnets always correspond to the ground state of the array corresponding to input bits “1” and “1”. When a new input stream arrives, the input bits are changed to conform to the new inputs. Thus, at time t = 0, the magnetizations of input-1 and input-2 are respectively set to (1, 1) [Fig 2(a)], (0 , 0) [Fig 2 (b)], (1, 0) [Fig 2 (c)], (0, 1) [Fig 2(d)]. We then consider the time evolution of the in-plane magnetization orientations of every multiferroic nanomagnet when a 4-phase stress cycle is a pplied, as shown in Fig 1 (c), to clock the array. In Fig 2(a), the inputs are unchanged as input-1 = 1 a nd input-2 = 1. This is a trivial case as the ground state already corresponds to the correct output. But it is still important to simulate the magnetization dynamics to verify that the gate works correctly. As seen in Fig 2(a), all magnetizations rotate through ±90º to the hard axis under compressive stress and then rotate back to their initial (correct) orientations under the influence of dipole coupling as the stresses are reversed to tensile. This results in a logical NAND output of "0". As expected there is a phase (a nd time) lag between instants when the compressive stress reaches a maximum and the magnetization is closes t to the hard axis. This is because magnetization takes a finite time to respond to the applied st ress, as is evident from the LLG equations. In Fig 2(b), the inputs are both changed so that input-1 = 0 and input-2 = 0. Therefore, all the magnets in the input wire, gate and output wire f lip through 180º, rotating first through ±90º on application of a compressive stress and then furthe r rotating through ±90º under the influence of dipole coupling. The phasing of the clock not only ensures the correct logical NAND output of "1" is reached but that the information is propagated unidirectionally through the input branches as well as the three 15output branches. The 4-phase clock achieves the fo llowing: As the compressive stress on a magnet is lowered to a point where the shape an isotropy barrier is about to be re stored, the compressive stress on its right (subsequent) neighbor has already rotated it toward s its hard axis. Therefore, the state of its left (previous) neighbor determines the easy direction to wards which the stressed magnet will relax as the stress in lowered. This ensures unidirectional logic bi t propagation as in the case of Bennett clocking [28]. Finally, Figs. 2(c) and Fig 2(d) show magnetization dy namics for the cases when one of the inputs is set to "1" while the other is set to "0". Here again the correct logical NAND output of "1" is achieved and propagated to the three fan-out branches. In summary, we have proved through simulation that the NAND gate, fan-in and fan-out work correctly for all four input combinations for a given initial state of the nanomagnets. This is repeated in the supplementary material accompanying this paper for different initial ground states – (0, 0), (0, 1) and (1, 0) – in order to be exhaustive. IV Energy Considerations: Power dissipated in the magnets and the clock There are two important sources of energy dissipation: (i) internal energy dissipated in the magnets due to Gilbert damping and (ii) energy dissipated in the clock while charging the capacitance of the PZT layer that can be modeled as a parallel-plate capacitor. (i) Internal energy dissipation The internal energy dEdissipated in the magnets during magnetization rotation under stress was estimated using Equation (12). The energy dissipated in the 4 nanomagnets (magnets 3, 4, 5 and 8) that comprise the NAND gate over one clock cycle vari ed depending on the operation performed. For example, when both inputs were "1" (Fig 2(a)) the energy dissipated was 767 kT while for both inputs set to "0" (Fig 2(b)), the energy dissipated was 948 kT. On the average, the energy dissipated in these four 16nanomagnets over one clock cycle is ~1000 kT. When all 12 nanomagnets are considered, the average energy dissipated over one clock cycle is ~3000 kT. Ultimately, this energy (~250kT/nanomagnet/bit) is well over the Landauer limit of kTln(2) [29] but consider ably less than that dissipated in a transistor, or a nanomagnet switched with spin transfer torque, or domain wall motion or current-generated magnetic field when the gate operation is completed in 2 ns. The dissipation is governed by the (i) strength of dipole coupling needed to ensure the nanomagnets switc h to the correct state with low dynamic error [30] under thermal noise and (ii) the large stress anisot ropy needed to ensure that the switching is accomplished in ~ 2 ns. (ii) Energy dissipated in the clock The energy dissipated in the clock is governed by th e electrostatic potential that needs to be applied across the PZT layer to generate the stress (10 MP a) that can beat the sh ape anisotropy to flip magnetization and complete the gate op eration in 2 ns. In this paper, we assume that the PZT layer is 40 nm thick (to ensure it is stiff compared to the magneto strictive layer, ensuring most of the strain in it is transferred). On application of an electrostatic poten tial of ~50 mV across the PZT layer, an electric field of 1.25 MV/m is generated. Since the d31 coefficient of PZT is ~ -10-10m/V [27], a strain of ~1250610 is generated and transferred to the Terfenol-D layer resulting in a stress ~10 MPa since the Young’s modulus of Terfenol-D is 1081 0 Pa. Next, we estimate the capacitance of the ~40 nm thick PZT layer of surface area 101.75 nm × 98.25 nm and thickness 40 nm as 1.74 fF (relative dielectric constant ~ 1000 [27]). Thus the energy dissipated in applying 50 mV, then switching it to -50 mV and discharging to zero (to generate the stress cycle shown in Fig 1 c) is ~ 3200 kT per nanomagnet if this is done abruptly (the energy dissipated in charging the capacitor abruptly with a square wave pulse is 2 1 2CV so that charging it up to + V from 0, reversing it to – V, and then discharging it back to 0 dissipates an energy of 3CV2. In contrast, driving an 2 2()clockVER CR 17LCR circuit with a sinusoidal source dissipates , resulting in an energy saving by a factor3RC. Abrupt (non-adiabatic) switching with a square wave pulse will cause a total energy dissipation of ~ 40,000 kT in the clocking circuit (or more than 10 times the internal energy dissipated in the nanomagnets). However, if the LCR circuit is driven with a sinusoidal voltage of low frequency (large time period), then clocking becomes quasi-adiabatic. Th is reduces dissipation considerably because of the large energy saving factor. From Equation 10 (b), we can see that if the time period is much larger than RC, the dissipation is greatly reduced. In our case, we assume that the PZT layer is electrically accessed with a silver wire of resistivity 2.6 μΩ-cm [31] so that an access line of length10 μm and cross section 50 nm×50 nm has resistance ~100 Ω. Hence, the RC time constant is ~0.174 ps. The clock period is 2 ns, so that the reduction factor 3RC= 45.47 10 . This makes the dissipation in the clock only about 22 kT, which is negligible compared to the internal energy dissipation of 3000 kT. V Conclusions We have modeled the nonlinear magnetization dyna mics of an all-multiferroic nanomagnetic NAND gate with fan-in/fan-out and shown that a throughput of ~ 1 bit per 2 ns and latency ~4 ns can be achieved, so that the clock rate can be 0.5 GHz. Su ch a gate circuit is estimated to dissipate ~ 3000 kT/clock cycle internally in the 12 nanomagnets combin ed and much less energy (20 kT/clock cycle) in the external access circuitry for the clock signal, if we use a 4-phase clocking scheme with a sinusoidal voltage source driving an LCR circuit. All this begs the question as to whether it is po ssible to reduce the internal energy dissipation by some appropriate scheme. This was discussed in ref. [2 ]. Imagine a magnet made of a material that has no Gilbert damping 0 . If we remove the shape anisotropy barrier and make the magnet isotropic (circular disk), then a magnetic field applied pe rpendicular to the magnet’s plane will make the 18magnetization vector precess around it without any damping in accordance with the first term in the right hand side of Equation (3). There is now no internal dissipation. The magnetic field however must be removed precisely at the juncture when the magnetization completes 1800 rotation if we wish to flip the bit. This requires exact precision; otherwis e, the magnet will either not have completed 1800 rotation, or overshot, resulting in more than 1800 rotation. This error will continue to build up with time and finally become too large to endure. In other words, there is no fault tolerance. This is a well-known problem that has been discussed by numerous authors starting from the Fredkin billiard ball computer which can compute without dissipating energy [32], but cannot to lerate any error. Clearly, if we require fault tolerance, we must have damp ing, and hence some internal dissipation. In the presence of damping, fluctuations can deviate the magnetization from the de sired orientation (minimum energy state), but the latter will return to the correct orientation (minimum energy state) by dissipatin g energy. Therefore, the dissipation in the clocking circuit can be eliminat ed by adiabatic approaches (increasingly slow switching), but the internal dissipation must remain for the sake of fault tolerance. The internal energy dissipated in the magnet must be provided by the power source driving the clock. This source need not dissipate any energy to raise and lower the barrier separating the logic bits as long as we raise and lower the barrier adiabatically, but it mu st dissipate some energy internally in the logic device to maintain fault tolerance. Acknowledgement: This work is supported by the US National Science Foundation under the “Nanoelectronics Beyond the Year 2020” grant 1124714. 19References : [1] ITRS Report CORE9GPLL_HCMOS9_TEC_4.0 Databook, STMicro (2003). [2] Salahuddin S and Datta S 2007 Appl. Phys. Lett. 90, 093503. [3] Ney A, Pampuch C , Koch R &. Ploog K. H 2003 Nature 425, 485. [4] Cowburn R.P , Welland M.E 2000 Science 287 1466. [5] Casba G, Imre A, Bernstein G.H , Porod W, Metlushko V , 2002 IEEE Transaction on Nanotechnology , 1, 209. [6] Bandyopadhyay S, Das B and Miller A E 1994 Nanotechnology 5 113. [7] Alam M. T, Siddiq M. J,Bernstein G. H, Niemier M, Porod W and Hu X. S 2010 IEEE Trans. Nanotech. 9 348 . [8] Ralph D C and Stiles M D 2008 J. Magn. Magn. Mater. 320 1190. [9] Yamanouchi M, Chiba D, Matsukura F and Ohno H, 2004 Nature (London) 428 539. [10] Bandyopadhyay S and Cahay M 2009 Nanotechnology 20 412001. [11] Roy K, Bandyopadhyay S and Atulasimha J, 2011, Phys. Rev. B, 83, 224412. 20[12] Fukami S, Suzuki T, Nagahara K, Ohshima N, Ozaki Y, Saito S, Nebashi R, Sakimura N, Honjo H, Mori K, Igarashi C, Miura S, Ishiwata N, and Sugibayashi T 2009 Symposium on VLSI Technology , Symposium on VLSI Technology, Digest of Technical Papers , 230-231. [13] Atulasimha J, Bandyopadhyay S 2010 Appl. Phys. Lett. 97 173105 . [14] Roy K, Bandyopadhyay S and Atulasimha J 2011 Appl. Phys. Lett. 99, 063108. [15 ] Salehi Fashami M, Roy K, Atulasimha J, Bandyopadhyay S 2011 Nanotechnology 22 155201. [16] Cowburn R. P, Koltsov D. K, Adeyeye A. O, Welland M. E and Tricker D. M 1999 Phys. Rev. Lett. 83 1042. [17] Bertotti G, Serpico C, Isaak and Mayergoyz I D 2008 Nonlinear Magnetization Dynamics in Nanosystems (Elsevier Series in Electromagnetism ) (Oxford: Elsevier). [18] Gilbert L 2004 IEEE Transaction on magnetics, Vol 40, No 6 . [19] Chikazumi S 1964 Physics of Magnetism (New York: Wiley) [20] Nikonov, D. E., Bourianoff, G. I., Rowlands, G., Krivorotov, I. N 2010 J. of Appl. Phys , 107, 113910. [21] Boechler G. P,. Whitney J.M, Lent C. S, Orlov A. O., Snider G. L., 2010 Appl. Phys. Lett. , 97, 103502. 21 [22] A. J. G. Hey, Feynman and Computing: Exploring the limits of computers, (c) 1999 Perseus Book Publishing, L.L.C, Chapter 15 Computing Machines in th e future (transcript of 1985 Nishira Memorial Lecture by R.P. Feynman). [23] Abbundi R and Clark A E 1977 IEEE Trans. Mag. 13 1547. [24] Ried K, Schnell M, Schatz F, Hirscher M, Ludescher B, Sigle W and Kronmuller H 1998 Phys. Stat. Sol. (a) 167, 195. [25] Kellogg R A and Flatau A B 2008 J. of Intelligent Material Systems and Structures 19 583. [26] Walowski J, Djorjevic Kaufman M, Lenk B, Hamann C, McCord J and Münzenberg M 2008 J. Phys. D: Appl. Phys. 41 164016. [27] http://www.piezo.com/prodmaterialprop.html [28] Bennett, C.H 1982 Int. J. Theoretical Physics 21 905. [29] Landauer R 1982 Inter, J. Theoretical Physics 21 283. [30] Likharev K. K 1982 Int. J. Theor. Phys. 21 311. [31] http://www.itrs.net/Links/2009ITRS/2009Chapters 2009Tables/2009 Interconnect.pdf [32] Fredkin E and Toffoli T 1982 Int. J. Theor. Phys . 21 219 . 22 Table.1 Material parameters a nd geometric design for Terfenol-D 3()2s 491 0 [23]，[24] sM 610.8 10 A m Young’s modulus 1081 0 P a [25] 0.1 [26] Dimension abt 101.75 98.25 10 nm nm nm r 200 nm 23 Figure captions Fig 1(a): Design of all multiferroic NAND gate w ith input "logic wires" and fan-out. The magnetization directions shown depict the corr ect initial (ground) state corresponsing to input- 1 = 1 and input-2 = 1. Fig 1 (b): Two nanomagnets whose hard axes are at an angle γ to the line joining their centers . Fig 1 (c): 4 phase clock showing sinusoidal stress applied to the nanomagnets. Fig 2: LLG simulation of magnetization dynamics of a ll magnets in the chain with initial (ground) states corresponding to the input-1=1 and input-2=1. Thereafter, inputs are changed to: (a) Input-1=1 and input-2=1 followed by applying 4-phase clock, which results in output=0. (b) Input-1=0 and input-2=0 followed by applying 4-phase clock, which results in output=1. (c) Input-1=0 and input-2=1 followed by applying 4-phase clock, which results in output=1. (d) Input-1=1 and input-2=0 followed by applying 4-phase clock, which results in output=1. 24 Fig 1(a) 25 Fig 1 (b) 26 Fig 1 (c): 4 phase clock 27 Fig.2 (a) 28 (b) 29 (c) 30 (d) SUPPLEMENTARY INFORMATION Magnetization Dynamics, Throughput and Energy Dissipation in a Universal Multiferroic Na nomagnetic Logic Gate Mohammad Salehi-Fashamia, Jayasimha Atulasimhaa, Supriyo Bandyopadhyayb Email: {salehifasham, jatulasimha, sbandy}@vcu.edu (a) Department of Mechanical Engineering, (b) Department of Electrical a nd Computer Engineering, Virginia Commonwealth Univer sity, Richmond, VA 23284, USA. In the main paper we theoreti cally demonstrated the switchi ng dynamics for the case when the ground state (initial conditon) of all nanomagnets in th e input logic wire s, NAND gate, and fan-out correspond to the correct states for input-1 =1 and input-2=1. Thereafte r, different inputs combinations as shown in Table-1 were applied and it was theoretically demonstrated that the magnetizations switch (from the above ground state) to give the correct output states. Thus, we proved that this device works infallibly for all combinations of inputs wh en the initial (ground) state of the nanomagnetic logic wires, NAND gate , and fan-out correspond to the correct states for input-1=1 and input-2=1. (NOTE: That Case 1 is trivial as the ground state already corresponds to the correct output, bu t it is still important to verify that the gate works correctly for this state.) Table 1 . Different initial conditions. Different Inputs to NAND gate INPUT-1 INPUT-2 OUTPUT 1. 1 1 0 2. 0 0 1 3. 0 1 1 4. 1 0 1 In the supplement we present 12 more switch ing diagrams to show that the NAND gate works for 3 other initial conditions of the na nomagnetic logic wires, NAND gate, and fan-out: CASE I. Initial (groud) state correspo nding to input-1=0 and input-2=0 Fig S1: shows the correct initial (gr ound) state for input-1=0 and input-2=0. Fig S2 (a)-(d):show the switchi ng diagrams for four different combinations of input-1 and input-2. In each case the co rrect output corresponsing to a NAND gate is acheived. CASE II. Initial (groud) state correspo nding to input-1=0 and input-2=1 Fig S3: shows correct initial (ground) state for input-1=0 and input-2=1. Figures S4 (a)-(d):show the switching diagrams for four different combinations of input-1 and input-2. In each case the correct output corresponsing to a NAND gate is acheived. CASE III. Initial (groud) state correspo nding to input-1=1 and input-2=0 Fig S5: shows correct initial (ground) state for input-1=1 and input-2=0. Figures S6 (a)-(d):show the switching diagrams for four different combinations of input-1 and input-2. In each case th e correct output corresponsing to a NAND gate is acheived. These simulations in the supplemant, taken together with the simulations in the main paper, prove that the NAND gate works for different combin ations of inputs for a ll possible initial conditions. Figure S1: Shows the correct initial (ground) state of th e nanomagnets correspondi ng to input-1=0 and input-2=0. Figure S2: LLG simulation of magnetization dynamics of a ll magnets in the chain with initial (ground) states corresponding to the input-1=0 and inpu t-2=0. Thereafter, inputs are changed to: (a) Input-1=1 and input-2=1 followed by applyi ng 4-phase clock, which results in output=0. (b) Input-1=0 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. (c) Input-1=0 and input-2=1 followed by a pplying 4-phase clock, which results in output=1. (d) Input-1=1 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. Figure S3: Shows the correct initial (ground) state of th e nanomagnets correspondi ng to input-1=0 and input-2=1. Figure S4: LLG simulation of magnetization dynamics of a ll magnets in the chain with correct initial (ground) state corresponding to the input-1=0 and input-2=1. Thereafter, inputs are changed to: (a) Input-1=1 and input-2=1 followed by applyi ng 4-phase clock, which results in output=0. (b) Input-1=0 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. (c) Input-1=0 and input-2=1 followed by a pplying 4-phase clock, which results in output=1. (d) Input-1=1 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. Figure S5: Shows the correct initial (ground) state of th e nanomagnets correspondi ng to input-1=1 and input-2=0. Figure S6: LLG simulation of magnetization dynamics of a ll magnets in the chain with correct initial (ground) state corresponding to the input-1=1 and input-2=0. Thereafter, inputs are changed to: (a) Input-1=1 and input-2=1 followed by applyi ng 4-phase clock, which results in output=0. (b) Input-1=0 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. (c) Input-1=0 and input-2=1 followed by a pplying 4-phase clock, which results in output=1. (d) Input-1=1 and input-2=0 followed by a pplying 4-phase clock, which results in output=1. Supplementary Figures : Case I. Fig.S1 Fig.S2(a) Fig.S2(b) Fig.S2(c). Fig.S2(d) Case II Fig.S3. Fig.S4(a) Fig.S4(b). Fig.S4(c). Fig.S4(d). Case III: Fig.S5. Fig.S6(a) Fig.S6(b) Fig.S6(c) Fig.S6(d)

2011-08-29

The switching dynamics of a multiferroic nanomagnetic NAND gate with fan-in/fan-out is simulated by solving the Landau-Lifshitz-Gilbert (LLG) equation while neglecting thermal fluctuation effects. The gate and logic wires are implemented with dipole-coupled 2-phase (magnetostrictive/piezoelectric) multiferroic elements that are clocked with electrostatic potentials of ~50 mV applied to the piezoelectric layer generating 10 MPa stress in the magnetostrictive layers for switching. We show that a pipeline bit throughput rate of ~ 0.5 GHz is achievable with proper magnet layout and sinusoidal four-phase clocking. The gate operation is completed in 2 ns with a latency of 4 ns. The total (internal + external) energy dissipated for a single gate operation at this throughput rate is found to be only ~ 1000 kT in the gate and ~3000 kT in the 12-magnet array comprising two input and two output wires for fan-in and fan-out. This makes it respectively 3 and 5 orders of magnitude more energy-efficient than complementary-metal-oxide-semiconductor-transistor (CMOS) based and spin-transfer-torque-driven nanomagnet based NAND gates. Finally, we show that the dissipation in the external clocking circuit can always be reduced asymptotically to zero using increasingly slow adiabatic clocking, such as by designing the RC time constant to be 3 orders of magnitude smaller than the clocking period. However, the internal dissipation in the device must remain and cannot be eliminated if we want to perform fault-tolerant classical computing. Keywords: Nanomagnetic logic, multiferroics, straintronics and spintronics, Landau-Lifshitz-Gilbert equation.

Magnetization Dynamics, Throughput and Energy Dissipation in a Universal Multiferroic Nanomagnetic Logic Gate with Fan-in and Fan-out

1108.5758v1

arXiv:1109.2465v1 [cond-mat.mes-hall] 12 Sep 2011Externally-driven transmission and collisions of domain w alls in ferromagnetic wires Andrzej Janutka∗ Institute of Physics, Wroclaw University of Technology, Wybrze˙ ze Wyspia´ nskiego 27, 50-370 Wroc/suppress law, Poland Analytical multi-domain solutions to the dynamical (Landa u-Lifshitz-Gilbert) equation of a one- dimensional ferromagnet including an external magnetic fie ld and spin-polarized electric current are found using the Hirota bilinearization method. A standard a pproach to solve the Landau-Lifshitz equation (without the Gilbert term) is modified in order to tr eat the dissipative dynamics. I estab- lish the relations between the spin interaction parameters (the constants of exchange, anisotropy, dissipation, external-field intensity, and electric-curr ent intensity) and the domain-wall parameters (width and velocity) and compare them to the results of the Wa lker approximation and micromag- netic simulations. The domain-wall motion driven by a longi tudinal external field is analyzed with especial relevance to the field-induced collision of two dom ain walls. I determine the result of such a collision (which is found to be the elastic one) on the domai n-wall parameters below and above the Walker breakdown (in weak- and strong-field regimes). Si ngle-domain-wall dynamics in the presence of an external transverse field is studied with rele vance to the challenge of increasing the domain-wall velocity below the breakdown. PACS numbers: 75.78.Fg, 85.70.Kh Keywords: domain wall, Landau-Lifshitz equation, magneti c dissipation, spin-transfer torque, soliton colli- sion I. INTRODUCTION Description of the magnetic-field- and electric-current- induced motionsof domain walls(DWs) in nanowireshas became a hot topic because of novel methods of storing and switching the (magnetically encoded) binary infor- mation. These proposals offer a progress in the minia- turization of memory and logic elements, utilizing crucial advantages of magnetic information encoding, (when a bit is identified with a single magnetic domain). Such in- formation is insensitive to the voltage fluctuations while its maintenance does not cost any energy, which enables the data processing with the production of a small heat amount. Currentlyinvestigatedrandom-accessmemories are built of metallic nanowires, formed into a parallel- column structure, which store magnetic domains sepa- rated by DWs. Such a three-dimensional (3D) magnetic systemhasthepotentialofstoringmoreinformationthan devices based on 2D systems, like hard-disk drives or electronic memories, in a given volume [1, 2]. Also, an interesting concept of logical operation via transmitting magnetic DWs through nanowires of specific geometries is being developed [3, 4]. I mention that ferroelectric nanosystems offer similar capabilities while their basic propertiesarestudied with thesamedynamical(Landau- Lifshitz-Gilbert) equation even though the effects of elec- troelastic coupling are strong [5, 6]. In orderto write and switchinformation, one can move the DWs via the application of an external magnetic field (parallel to the easy-axis) or via the application of a voltage which induces the spin-polarized electric cur- rent through the DW. The directions of the field-driven ∗Andrzej.Janutka@pwr.wroc.plmotion are different for the tail-to-tail and head-to-head DWs; thus the magnetic field induces the DW collisions, while the direction of the voltage-drivenmotion uniquely corresponds to the current direction. Field-driven mo- tion and current-driven (below the Walker breakdown) motion are possible due to the magnetic dissipation and itsdescriptiondemands inclusionofthe Gilbert terminto the Landau-Lifshitz (LL) equation. However, existing many-domain analytical solutions to the LL equation do not include the dissipation [7, 8], while approximate so- lutions using the Walker ansatz describe a single DW only [9]. Since the parameters of the DW solutions to the Landau-Lifshitz-Gilbert (LLG) equation determine accessible values of technological characteristics (e.g. the minimal domain length, the bit-switching time, etc.), it is of interest to know the analytic DW solution to the LLG equation. Knowledge of the many-domain solution is of importance for preventing unwanted DW collisions which can result in an instability of the record and it en- ables verification of DW-collision simulations regardless of the internal structure or the geometryof the simulated system. In the present paper, I perform an analytical study of the dynamics of multi-domain systems including the dis- sipation. The dynamical LLG system is bilinearized fol- lowing the Hirota method of solving nonlinear equations and it is extended, via doubling the number of freedom degreesand the number of equations, into a time-reversal invariant form. The field solving the extended system contains proper and virtual (unphysical) dynamical vari- ables and the physical components of the solution are shown to satisfy the pimary LLG system in a relevant time regime. In particular, aiming to analytically de- scribe the field-induced collision, I establish asymptotic three-domainmagnetizationprofiles(relevant in the time limitst→ ±∞). With connection to the phenomenon of2 the Walker breakdown, (a cusp in the dependence of the DW velocity on the external field and in its dependence on the current intensity), I modify the LLG model in a way to make it applicable below the breakdown. I re- duce it into a model of plane rotators. In this weak-field regime, the spin alignment in the DW area is saturated to a plane while, above the breakdown (the strong-field regime), the spins rotate about the magnetic-field axis (the easy axis). These considerations supplement micro- magnetic simulations of the DW collisions in terms of the studied DW-parameter regimes [10]. Within the present method, I verify the Walker-ansatz predictions on the current-driven DW motion above and below the break- down,(includingadiabaticandnon-adiabaticpartsofthe spin-transfer torque) [11]-[15], thus, showing the applica- bility of the present formalism to the field-driven motion of multi-domain systems with an additional voltage ap- plied. With relevance to the challenge of controlling the maximum DW velocity below the Walker breakdown, we analyze the longitudinal-field-driven motion and current- drivenmotionoftheDWsinthepresenceofanadditional perpendicular (with respect to the easy axis) field. A complementary study of the DW collision of mag- netic DWs in the subcritical regime is performed in a separate paper [16]. In Sec. II, I extend the dissipative equations of motion of the ferromagnet in such a way as to make equations applicable to the unlimited range of time t∈(−∞,∞). Section III is devoted to the analysis of its single- and double-DW solutions in the presence of a magnetic field andan electriccurrent. Istudy the field-induced collision in detail. The plane-rotatorapproachto the DW dynam- ics is described in Sec. IV. In Sec. V, consequences of the application of an external field perpendicular to the easy axis for the DW statics and dynamics are consid- ered. Conclusions are given in Sec. VI. II. DYNAMICAL EQUATIONS The dynamics of the magnetization vector m(|m|= M) in the 1D ferromagnet is described with the LLG equation ∂m ∂t=J Mm×∂2m ∂x2+γm×H+β1 M(m·ˆi)m׈i −β2 M(m·ˆj)m׈j−δ∂m ∂x−δβ Mm×∂m ∂x −α Mm×∂m ∂t.(1) The first term of the right-hand side (RHS) of (1) re- lates to the exchange spin interaction while the second term depends on the external magnetic field H; thus,γ denotes the giromagnetic factor (up to its sign). The constant β1(2)determines the strength of the easy axis (plane) anisotropy and ˆi≡(1,0,0),ˆj≡(0,1,0). Note that the long axis of the magnetic nanowire is an easy axis for the majority of real systems; however, anotherchoice of the anisotropy axes does not influence the mag- netization dynamics. The constant δis proportional to the intensity of the electric current through the wire and δchanges its sign under time-arrow reversal (the inver- sion ofthe electronflow) [17, 18]. The non-adiabaticpart of the current-induced torque (which depends on β) is of dissipative origin; thus, β→0 with decreasing Gilbert damping constant α→0 (one takes α,β≪1). Its inclu- sionisnecessaryifonedescribesanobservedmonotonous motion of the DW below the Walker breakdown [11]-[14]. Notice that including the magnetic-dissipation term fol- lowing the original LL approach (changing the last term of (1) into −αm×[m×heff], where heffj=−δH/δmj, andHdenotes the Hamiltonian) would lead to changing the constant βintoβ−α, [13]. Although the discussion of the relevance of both approaches to the magnetic dis- sipation remains open [13, 19], I believe, the clinching argument for the Gilbert approach is the expectation for the proper dissipative term to be dependent on the time derivative of the dynamical parameter m. In other case, the dissipative term could influence static solutions to the LL equation while one expects the magnetic friction to be the kinetic. Since (1) is valid only when the constraint |m|=Mis satisfied, I intend to write equations of the unconstrained dynamics equivalent to (1). Introducing the complex dy- namical parameters m±=my±imz, I represent the magnetization components using a pair of complex func- tionsg(x,t),f(x,t). This way I reduce the number of independent degrees of freedom. The relation between the primary and secondary dynamical variables m+=2M f∗/g+g∗/f, m x=Mf∗/g−g∗/f f∗/g+g∗/f(2) [where (·)∗denotes the complex conjugate (c.c.)] ensures that|m|=Mwhile there areno constraintson g,f. The transform (2) enables bilinearization (”trilinearization”) of (1) following the Hirota method of solving nonlinear equations [7, 8]. In the particular case Hy=Hz= 0, from (1) and (2), we arrive at the trilinear equations for f,g f/bracketleftbig −iDt+JD2 x+δ(β−i)Dx+αDt/bracketrightbig f∗·g +Jg∗D2 xg·g−/parenleftbigg γHx+β1+β2 2/parenrightbigg |f|2g −β2 2f∗2g∗= 0, g∗/bracketleftbig −iDt−JD2 x+δ(β−i)Dx+αDt/bracketrightbig f∗·g −JfD2 xf∗·f∗+/parenleftbigg −γHx+β1+β2 2/parenrightbigg |g|2f∗ +β2 2g2f= 0,(3) whereDt,Dxdenote Hirota operators of differentiation which are defined by Dm tDn xb(x,t)·c(x,t)≡(∂/∂t−∂/∂t′)m ×(∂/∂x−∂/∂x′)nb(x,t)c(x′,t′)|x=x′,t=t′.3 The inclusion of the dissipation into the LLG equation is connected to breaking the symmetry with relevance to the time reversal. Therefore, neither (1) nor (3) can describe the magnetization evolution on the whole time axis. In particular, the application of an external mag- netic field or current to the DW system (or creation of a domain in the presence of an external field) initiates a nonequilibrium process of the DW motion. Such a mo- tioncannotbepresentinthedistantpast( t→ −∞)since anonzerovalueofthedissipativefunction[relevanttothe Gilbert term in (1)] would indicate unlimited growth of the energy with t→ −∞. Thus,solitary-wave solutions to (1) are relevant only in the limit of large positive val- ues of time [11, 13]. This fact makes impossible an exact analysis of the DW collisions using (3) and motivates extension of the dynamical system within a formalism applicable to the whole length of the time axis. We write modified equations of motion using a similar trick to the one proposed by Bateman with application to the Lagrangian description of the damped harmonic oscillator [20]. It is connected to the concept by Laksh-manan and Nakamura of removing the dissipative term fromtheevolutionequationofferromagnetsviamultiply- ing the time variable by a complex constant [21], how- ever, it demands an improvement in the spirit of Bate- man’s idea [22]. The concept is to extend the dynami- cal system doubling the number of freedom degrees and adding the equations which differ from the original ones by the sign of the dissipation constants. Resulting ex- tended system is symmetric with relevance to the time- arrow reversal, however, its solution consists of physical and virtual fields. Let us mention that different quantum dissipative formalisms (non-equilibrium Green functions, thermo-field dynamics, rigged Hilbert space) are based on the Bateman’s trick [23]. We extend the system of secondary dynamical equa- tions (3) since it describes unconstrained dynamics un- like the primary LLG equation (1). We replace g,g∗,f, f∗in (3) with novel fields of the corresponding set g1,g∗ 2, f2,f∗ 1and of the set of their c.c.. For α,β= 0, in the absence of dissipation, g1=g2=g,f1=f2=f. For the caseHy=Hz= 0, the secondary dynamical equations transform into f2/bracketleftbig −iDt+JD2 x+δ(β−i)Dx+αDt/bracketrightbig f∗ 1·g1+Jg∗ 2D2 xg1·g1−/parenleftbigg γHx+β1+β2 2/parenrightbigg f2f∗ 1g1−β2 2f∗ 2f∗ 1g∗ 1= 0, g∗ 2/bracketleftbig −iDt−JD2 x+δ(β−i)Dx+αDt/bracketrightbig f∗ 1·g1−Jf2D2 xf∗ 1·f∗ 1+/parenleftbigg −γHx+β1+β2 2/parenrightbigg g∗ 2g1f∗ 1+β2 2g2g1f1= 0.(4) Writing (4), we have replaced the last terms on the lhs of (3) in a way t o be linear in g(∗) 2,f(∗) 2, which ensures that they vanish (diverge) with time in the presence of Hx∝negationslash= 0 with similar damping (exploding) rates as all other terms of these equations, (in particular, their damping does not modify th e anisotropy). The additional equations of the dynamical system differ from (4) by the sign of the dissipation const antsα,β f1/bracketleftbig −iDt+JD2 x+δ(−β−i)Dx−αDt/bracketrightbig f∗ 2·g2+Jg∗ 1D2 xg2·g2−/parenleftbigg γHx+β1+β2 2/parenrightbigg f1f∗ 2g2−β2 2f∗ 1f∗ 2g∗ 2= 0, g∗ 1/bracketleftbig −iDt−JD2 x+δ(−β−i)Dx−αDt/bracketrightbig f∗ 2·g2−Jf1D2 xf∗ 2·f∗ 2+/parenleftbigg −γHx+β1+β2 2/parenrightbigg g∗ 1g2f∗ 2+β2 2g1g2f2= 0.(5) Though the previous dynamical variables g(f),g∗(f∗) were mutually independent, they had to be c.c. to each other in order that the system of equations (3) and their c.c. was closed. In the system of eight equations; (4)- (5) and their c.c., g1(f1) is not a c.c. to g2(f2), while comparing (4) and (5), one sees that g2(x,t) (f2(x,t)) can be obtained from g1(x,t) (f1(x,t)) via changing the sign of its parameters α,β. Under the time-arrow inversion, the system of the novel equations transforms into itself if one accompanies this operation by the transform of the novel dynamical variables g1(2)→f2(1),f1(2)→ −g2(1). The equations (4) and their c.c., which determine the magnetization dynamics for large positive values of time (in particular, fort→ ∞), contain the differentials of the functions g1, g∗ 1,f1,f∗ 1. Therefore, the magnetization vector shouldbe expressed with these functions in the relevant time regime. Writing the magnetization in the form m+=2M f∗ 1/g1+g∗ 1/f1, mx=Mf∗ 1/g1−g∗ 1/f1 f∗ 1/g1+g∗ 1/f1(6) ensures that their components satisfy |m|=M,mx= m∗ x, and they reproduce (2) for α=β= 0. In the regime of large negative values of time, in particular, for t→ −∞, we can analyze the evolution of the magnetization with the inversed time arrow. It is described with the reversed magnetization vector ˜m+=−2M f∗ 2/g2+g∗ 2/f2,˜mx=−Mf∗ 2/g2−g∗ 2/f2 f∗ 2/g2+g∗ 2/f2.(7)4 III. DOMAIN-WALL MOTION Let us analyze the multi-domain solutions to (4)-(5) in the absence of any external magnetic field, H= 0. We search for the solutions which describe a single DW and two DWs in the forms f∗ 1= 1,g1=w1ek1x−l1t, andf∗ 1= 1+v∗ek1x−l1tek2x−l2t,g1=w1ek1x−l1t+w2ek2x−l2t, re- spectively, where kj= Re(kj), and sign( k1) =−sign(k2). Inserting these ansatz into (4)-(5), one finds lj=1 1+iα/braceleftbigg −/radicalBig −/parenleftbig Jk2 j−β1/parenrightbig/bracketleftbig Jk2 j−(β1+β2)/bracketrightbig +δ(1+iβ)kj/bracerightbigg , kj=/radicalBigg β1+β2(wj+w∗ j)2/(4|wj|2) J.(8) The single-wall (two-domain) solutions with |k1| ∈ (/radicalbig β1/J,/radicalbig (β1+β2)/J) describe moving solitary waves (topological solitons), [7, 24]. One sees the correspon- dence between the wall width and the spin deviation from the easy plain since the dependence of kjonwj. Whenδ= 0, static two-domain solutions represent the Bloch DW, ( |k1|=/radicalbig β1/J,w1=−w∗ 1), or Neel DW, (|k1|=/radicalbig (β1+β2)/J,w1=w∗ 1), respectively (the nomenclature of [24] which differs from Neel and Bloch wall definition of e.g. [26]). These two solutions corre- spond to the ones found in [25, 27] within the XY model. The DW profiles can be described with the functions m+(x,t) =Mw1e−iIml1t |w1|sech[k1x−Rel1t+log|w1|], mx(x,t) =−Mtanh[k1x−Rel1t+log|w1|],(9) (k1>0relatestothehead-to-headstructurewhile k1<0 to the tail-to-tail structure). At the time points of the discrete set t=πn/Iml1,n= 0,±1,±2,..., one finds g1=g2=g,f1=f2=fand (4) coincide with (3). Therefore, (9) is a solitary-wave solution to (1), (in par- ticular, it coincides with the one representing a Bloch DWoraNeelDw for δ∝negationslash= 0, [12]). Throughoutthe paper, we focus our attention on the externally-driven dynam- ics of the Bloch and Neel DWs, since these initially-static structures are the most important with relevance to the magnetic data storage. Weestablishthatstaticdouble-wall(three-domain)so- lutions to the LLG equation cannot be written with the above Hirota expansion when k1=−k2. In this case the coefficient v; v=−β2Jk2 1w∗ 1w∗ 2 (Jk2 1−β1)(Jk2 1−β1−β2)(10) diverges with |k1| →/radicalbig β1/Jor|k1| →/radicalbig (β1+β2)/J. Analogously to the XY model, the Hirota expansion is inapplicable to static three-domain configurations of the Bloch walls or Neel walls, while there exists staticsolution to (4)-(5) which describes a pair of different- type (Neel and Bloch) walls [28]. In particular, for k1=/radicalbig β1/J,k2=−/radicalbig (β1+β2)/J, andw1=−w∗ 1, w2=w∗ 2, one finds v=−β2w1w2 2β1+β2−2/radicalbig β1(β1+β2)(11) Let us emphasize that we have not excluded the coex- istence of a pair of Neel walls or Bloch walls in a mag- netic wire. However, the overlap of both the topological solitons induces their interaction which leads to an insta- bility of their parameters and, unlike for nontopological solitons, is not a temporal one [29]. Solving (4)-(5) in the presence of a longitudinal mag- netic field Hx∝negationslash= 0, we apply the ansatz f∗ 1=/parenleftbig 1+v∗ek1x−l1tek2x−l2t/parenrightbig eγHxt/(−2i+2α), g1=/parenleftbig w1ek1x−l1t+w2ek2x−l2t/parenrightbig e−γHxt/(−2i+2α)(12) at the discrete time points t=tn≡4πn(1+α2)/(γHx), wheren= 0,±1,±2..., (letδ= 0forsimplicity). Behind thesetimepoints,inthepresenceofthelongitudinalfield, the last terms on the lhs of (4)-(5) change faster (they oscillate with the three-times higher frequency) than the other ones. Therefore, taking the above ansatz, we apply an approach similar to the ’rotating wave approxima- tion’ in the quantum optics. Since this ansatz describes the spin structure rotation about the x-axis, it is appli- cable when the external field exceeds Walker-breakdown critical value, |Hx|> HW. From the single-wall solu- tion, (the case of w2= 0 orw1= 0), for k1=/radicalbig β1/J, k2=−/radicalbig (β1+β2)/J, we establish that applying the magnetic field in the easy-axis direction drives the DW motion with the velocity c1(2)=γ|Hx|α/[|k1(2)|(1+α2)]. (13) Correspondingly, for Hx= 0, applying the electric cur- rent through the initially static wall drives it to move with the velocity c=δ(1+αβ) 1+α2(14) which is independent of the DW width. The essential difference between both kinds of the driven motion emerges from the analysis of the three- domain solutions. Under the external field, the two con- secutive DWs move in the opposite directions. The walls which are closing up to each other collide and eventually they can annihilate or wander off each other. The ap- plication of the electric current along the magnetic wire drives both the DWs to move in the same direction with the samevelocity. Analyzinglong-timelimits ofthe mag- netization vector in different regions of the coordinate x, we establish the consequences of the field-induced colli- sion of the complex of a Bloch DW interacting with a Neel DW. We use the ansatz (12) and assume δ= 0. Letηj≡kj(x−x0j)−γHxαt/(1 +α2), ˜ηj≡kj(x− x0j) +γHxαt/(1 +α2). ForHx>0, att=tn(within5 the above ’rotating wave approximation’), we find the distant-future limit of the magnetization (6) m+≈/braceleftBigg m(1) +η2≪η1∼0 m(2) +η1≪η2∼0= lim t→∞m+, mx≈/braceleftBigg m(1) xη2≪η1∼0 m(2) xη1≪η2∼0= lim t→∞mx,(15) where m(j) += 2Mv/w∗ je˜ηke−iγHxt/(1+α2) 1+|v|2/|wj|2e2˜ηk, m(j) x=−M1−|v|2/|wj|2e2˜ηk 1+|v|2/|wj|2e2˜ηk,(16) andj∝negationslash=k. Identifying the parameters x0jwith the DW-center positions, we introduce the restriction on wj, |v|/|wj|= 1. We notice that m(1) +,m(1) xas well as m(2) +, m(2) xare the Walker single-DW solutions to the primary LLG equation which describe the motion of well sepa- rated DWs [9, 30]. Thus, our three-domain profiles of the fields (6) tend to satisfy (1) in the limit t→ ∞ac- cording to the requirement formulated in the previous section. In the distant-past limit, we describe the magnetiza- tion evolution with the reversed time arrow. Following (7), ˜m+≈/braceleftBigg ˜m(1) +˜η1≪˜η2∼0 ˜m(2) +˜η2≪˜η1∼0= lim t→−∞˜m+, ˜mx≈/braceleftBigg ˜m(1) x˜η1≪˜η2∼0 ˜m(2) x˜η2≪˜η1∼0= lim t→−∞˜mx,(17) where ˜m(j) +=−2Mv/w∗ keηje−iγHxt/(1+α2) 1+|v|2/|wk|2e2ηj, ˜m(j) x=M1−|v|2/|wk|2e2ηj 1+|v|2/|wk|2e2ηj,(18) andj∝negationslash=k. In orderto consider the collision of the pair of DWs which are infinitely distant from each other at the beginning of their evolution, we determine the magneti- zation dynamics in the limit t→ −∞. For this aim, one has to invert the propagation direction of the kinks of ˜m andtoreversethearrow’sheadofthefield vector ˜m. Uti- lizing the properties ˜ m(j) +(x+x0k,0) = ˜m(j) +(−x+x0k,0), ˜m(j) x(x+x0k,0) =−˜m(j) x(−x+x0k,0), we arrive at m+(x,t) =/braceleftBigg −˜m(1) +(−x+2x01,t)η1≫η2∼0 −˜m(2) +(−x+2x02,t)η2≫η1∼0, mx(x,t) =/braceleftBigg −˜m(1) x(−x+2x01,t)η1≫η2∼0 −˜m(2) x(−x+2x02,t)η2≫η1∼0.(19)The applicability of the above procedure to study the asymptotic evolution of a single DW is easy to verify since any single-DW solution satisfies m+(x,t) =−˜m+(−x+2x01,t), mx(x,t) = ˜mx(−x+2x01,t). (20) Typically, one should consider the formulas (15), (19) with relevancetothe case β1≫β2, thusk1≈ −k2, which corresponds to commonly studied crystalline magnetic nanowires, e.g. for Fe, FePt, β2/β1∼10−1, [31]. For noncrystalline(permalloy)nanowiresFe 1−xNixdeposited on a crystalline substrate, the easy-axis anisotropy con- stant determined from uniform-resonance measurements wasfound to be, unexpectedly, asbig asin the crystalline nanowires [32]. Therefore, even when neglect structural effects in real systems which lead to the saturation of the spin alignment in the DW area to the easy plain or hard plain (the Walker breakdown) which suppresses theirspontaneousmotion, the spectrumofspontaneously propagatingDWsin nanowireswouldbeverynarrowand their velocities would be very small. According to (15), (19), two initially closing up DWs have to diverge after the collision. If one of the colliding DWs that was initially, for t→ −∞, described with the field ingredient ˜ m(j) +, ˜m(j) x, it is finally, for t→ ∞, de- scribed with the field ingredient m(j) +,m(j) x. Therefore, reflecting DWs exchange their parameters x01↔x02, w1↔w2,k1↔ −k2. It is connected to exchanging the directions of the spin orientation in the yz-plane in the wall areas, (after the collision the Neel wall changes into the Bloch wall and vice versa as shown in Fig. 1). Our prediction corresponds to the result of the collision anal- ysis performed for spontaneously propagating DWs (in absence of external field, electric current, and dissipa- tion). According to findings of [24, 33], the DWs reflect during the collision in a way that one can say they pass through each other without changing their widths and velocities, however, with changing their character from the head-to-head one into the tail-to-tail one and vice versa. Up till now, we have considered systems of infinite do- mains whose energy cannot be defined. However, the smaller a domain is the bigger percentage of the Zeeman part ofits energy is lost per time unit due to the DW mo- tion. The condition of the domain-energy minimization determines the direction of this motion. The domains aligned parallelly to the external field grow while the do- mains aligned antiparallelly to the field diminish. Any DW reflection induces a motion which contradicts this rule. Such a motion has to be decelerated and, eventu- ally, it has to be suppressedwhen the decreaseofthe DW interaction energy equals the increase of the Zeeman en- ergy. The outcome of a many-collisionprocess in a finite- size system is the appearance of a 1D magnetic-bubble structure similar to widely known 2D bubble structures [34]. The bubble size and concentration depend on the magnetic field intensity. Each bubble is ended with a6 mmz yx t=0 t= t/c68 t=2 t/c68 t=3 t/c68v2 v1mx x t=0 t= t/c68 t=2 t/c68 t=3 t/c68v1v2 v2 v2v1 v1v1 v1 v1 v1v2 v2 v2 v2 FIG. 1. The magnetization dynamics of a system of one Neel DW and one Bloch DW in a longitudinal field above the Walker breakdown. Their reflection takes place in the time region (∆ t,2∆t) and it is accompanied by a change of the Bloch wall (of the velocity v2) into the Neel wall (of the velocity v1) and vice versa. Since |Hx|> HW, the spin struc- ture monotonously rotates about the x-axis. Neel DW at one of its sides and with a Bloch DW at the other side. Some analogy with a complex boundary of hard (quasi-2D) magnetic bubbles can be noticed since such a border contains alternating Neel and Bloch points in its structure[35]. Let us mention, that interestingcon- cepts of storing and transforming the binary information have had been developed several decades ago with rele- vance to 2D magnetic bubble systems, though, they have been abandoned because of technologicalproblems of the time [36]. Numerical analyses of the field-induced DW collision (micromagnetic studies) have been performed with rele- vance to flattened nanowires (quasi 1D nanostripes) be- low and just above the Walker breakdown using the dis- sipative LL equation [10]. The systems below the thresh- old correspond to a plane-rotator model studied in sec- tion 4, while the systems above the threshold are qual- itatively described with the present model. The men- tioned simulations focus on the collisions of similar-type (Neel or Bloch) DWs neglecting the anisotropy. They have predicted mutual annihilation or reflection of the walls depending on (parallel or anti-parallel) spin align- ments in both the DW centers. This result is partially supported by a perturbation analysis of Bloch-wall in- teractions within the XY model which has shown such DWs to repel or attract each other depending on their chiralities [29]. The method of present study cannot be appliedtothesecollisionssinceoneisunabletodetermine neither the double-Neel nor double-Bloch wall analytical solutionsto the dynamical equations. However, except in the case of a periodically distributed DWs, multi-Bloch or multi-Neel structures are unstable because of unbal- anced DW interactions, thus, they seem to be less suit- able for the information-storing purposes than the Neel- Bloch DW structures. To the best of the author knowl- edge, the field-induced collision of the Neel DW with the Bloch DW has not been simulated.IV. PLANE-ROTATOR MODEL In order to describe the DW dynamics below the Walker breakdown, we consider a system of plane rota- tors. Let us reduce the primary (LLG) dynamical system to its single component. Saturating the magnetization dynamics to the easy plain ( my= 0), we neglect the spin rotation about the x-axis and z-axis since the relevant torque components are equal to zero. For H= (Hx,0,0), inserting mx=M1−a2 1 1+a2 1, my= 0, mz= 2Ma1 1+a2 1(21) (wherea1takes real values) into the y-component of (1), one arrives at a nonlinear diffusion equation /parenleftbigg −α∂a1 ∂t−γHxa1−δβ∂a1 ∂x+J∂2a1 ∂x2/parenrightbigg/parenleftbig 1+a2 1/parenrightbig −2Ja1/parenleftbigg∂a1 ∂x/parenrightbigg2 −β1a1/parenleftbig 1−a2 1/parenrightbig = 0.(22) We use another ansatz describing the dynamics con- strained to the xy-plain (a hard plain) mx=M1−a2 2 1+a2 2, my= 2Ma2 1+a2 2, mz= 0.(23) Then, we insert it into the z-component of (1) and we arrive at /parenleftbigg −α∂a2 ∂t−γHxa2−δβ∂a2 ∂x+J∂2a2 ∂x2/parenrightbigg/parenleftbig 1+a2 2/parenrightbig −2Ja2/parenleftbigg∂a2 ∂x/parenrightbigg2 −(β1+β2)a2/parenleftbig 1−a2 2/parenrightbig = 0(24) which differs from (22) by a constant at the anisotropy term. With relevance to the case δ= 0, one finds the two-domain solution a1(2)=wek1(2)x−γHxt/α, |k1|=/radicalbigg β1 J,|k2|=/radicalbigg β1+β2 J,(25) which correspond to the Bloch DW (to the Neel DW). WhenHx∝negationslash= 0, the DW propagates with the velocity c1(2)=γ|Hx| |k1(2)|α. (26) The applicability of the plane-rotator model is limited by the Walker-breakdown condition. The magnetic field |Hx|cannot exceed a critical value HW(see Fig. 2a) which corresponds to the spin deviation from the basic magnetization plane (a canting) at the center of the DW about a limit angle equal or smaller than π/4. We esti- mate an upper limit of the Walker critical field consid- ering the x-component of the LLG equation at the DW center ∂mx ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=x01(2)≈/bracketleftbiggβ2 Mmymz−δ∂mx ∂x/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle x=x01(2),(27)7 wherex01(2)≡ −log(w)/k1(2). Letϕdenotes the an- gle of the spin deviation (a canting) at DW center. In- serting (21) and transforming mymz→m2 zsin(ϕ)cos(ϕ) in (27) or inserting (23) and transforming mzmy→ m2 ysin(ϕ)cos(ϕ) in (27), one arrives at ∂a1(2) ∂t=β2 2sin(2ϕ)a1(2)−δ∂a1(2) ∂x(28) and finally, assuming |ϕ| ≤π/4, at |Hx| ≤HW≤maxHW≡αβ2 2γ+αδ γ|k2|.(29) This expression corresponds to the one given in [30, 37, 38]. However,wenoticethat, fortypicalnanowireswhose width-to-thickness ratio is bigger than 20 (double-atomic or triple-atomic layers of a submicrometer width), mea- suringHW, one has estimated the canting angle to take a value of a few degrees at most, [14, 30, 39]. ForH= 0,δ∝negationslash= 0, the two-domainsolution to(22)-(24) takes the form a1(2)=wek1(2)(x−δβt/α), |k1|=/radicalbigg β1 J,|k2|=/radicalbigg β1+β2 J.(30) It is seen that only the non-adiabatic part of the spin- transfer torque contributes to (22),(24) since the current- dependent term is proportional to β. From (28), the current-induced Walker breakdown corresponds to the critical current intensity δW≤β2/[2|k1|(1+β/α)] (31) ifHx= 0, (see Fig. 2b). It has been observed that HW,δWdecreasewith decreasingthe nanowirewidth-to- thicknessratio, [39, 40], becausethis ratiodeterminesthe strength of the easy-plain anisotropy while HW,δW→0 withβ2→0, [41]. Notice that analytical calculations us- ing 2D XY model, experimental observations, and simu- lations of the spin ordering in nanostripes show this or- dering to vary alongthe cross-sectionwidth ofthe nanos- tripe in the DW area, thus, revealing a complex topo- logical structure [10, 42]. Therefore, our plane-rotator description is valid only for a qualitative analysis of the DW dynamics in the nanostripes. Neither finding nonstationary double-Bloch nor double-Neel solutions in the form of the Hirota expan- sion (including its second order) does not manage. In particular, inserting a1(2)=w1ek1(2)x+w2ek′ 1(2)x 1+v1(2)ek1(2)x+k′ 1(2)xe−γHxt/α,(32) into (22)-(24), for k1=−k′ 1=±/radicalbig β1/J,k2=−k′ 2= ±/radicalbig (β1+β2)/J, leads to the divergence of v1(2)as it fol- lows from the approach of section 3. In order to describe the collision of Bloch and Neel walls below the Walker breakdown, I propose to apply an effective 1D model as- suming the magnetization precession to be overdamped,|H□|c HW1(2) x /c100c /c100Wa b FIG. 2. a) A scheme of the longitudinal-field dependence of the DW velocity for: a wire with single-axis anisotropy (solid line), a wire with double-axis anisotropy (dashed li ne). b) A scheme of the current-intensity dependence of the DW velocity for: a wire with single-axis anisotropy (solid lin e), a wire with double-axis anisotropy; β > α(dashed line), β < α (dotted line), β= 0 (dash-dotted line). thus, taking the lhs of (1) to be equal to zero. Inclusion of the constraint |m|=Mleads to the modified (by ne- glectingthefirsttermsonthelhs)system(4)-(5). Solving it, we predict the field-induced DW collision below the Walker breakdown to result in their reflection similar to the one described in section 3. The reflection is accom- panied by the change of the Bloch wall into the Neel wall and vice versa. Let us emphasize that there is no spon- taneous DW motion below the Walker breakdown when neglect magnetostatic effects [43]. The technological challenge of increasing the DW speed is especially important below the Walker break- down, where the driving-field is relatively weak. Refer- ring to this purpose, we mention an attempt utilizing an increase of the nanostripe-edge roughness, thus, an in- crease of the damping constant α, [44]. This approach fails since, accordingto simulationsof [39], the maximum of the field-induced DW velocity is insensitive to αbe- low the breakdown. It is because c1(2)∝α−1|Hx|while |Hx| ≤HW∝α. On the other hand, since βgrows with α, (thenon-adiabaticpartofthespin-transfertorqueisof a dissipative origin), the velocity of the current-induced DW motion c=δβ/α (33) can be insensitive to the increase of the nanostripe-edge roughness as well. One has attributed some reported DW-velocity increase due to the nanostripe-edge rough- ness to a decrease ofits effective cross-sectionwidth. An- other attempt utilized an increase of HWdue to the in- creaseofthe anisotropyconstant β2. It hasbeen donevia thenanowiredepositiononaspecificcrystallinesubstrate [45]. However, the most efficient method of influencing the maximum DW velocity below the Walker breakdown is the application of the transverse magnetic field, which is considered in the next section [46].8 V. DOMAIN WALL IN PERPENDICULAR TO EASY AXIS FIELD Let us define H±≡Hy±iHz. ForHx= 0, we search for a two-domain solution to (1) using a different ’multi- linearization’ than used in the previous sections f2/bracketleftbig −iDt+JD2 x+δ(β−i)Dx+αDt/bracketrightbig f∗ 1·g1 +γH+ 2f∗ 1(f∗ 1f2+g∗ 2g1)−/parenleftbigg β1+β2 2/parenrightbigg f2f∗ 1g1 −β2 2f∗2 1g∗ 2= 0, g∗ 2/bracketleftbig −iDt−JD2 x+δ(β−i)Dx+αDt/bracketrightbig f∗ 1·g1 −γH− 2g1(f∗ 1f2+g∗ 2g1)+/parenleftbigg β1+β2 2/parenrightbigg g∗ 2g1f∗ 1 +β2 2g2 1f2= 0, f2g1D2 xf∗ 1·f∗ 1−f∗ 1g∗ 2D2 xg1·g1= 0.(34) Let us focus our attention on the case α,β,δ= 0 for simplity. Then one has f1=f2=f,g1=g2=g, while in the general case the relations (6), (7) apply. We analyze the two cases of the external-field direction; the one parallel to the easy plane H+= iHz, and the one perpendicular to the easy plane H+=Hy. In the case of H+= iHz, we apply the ansatz f∗ 1=f2=q1+s1ek1x−l1t, g1=−g∗ 2= i/parenleftbig s1+q1ek1x−l1t/parenrightbig , (35) wherek1,q1,s1denote real constants, (the parameter l1can take complex values when α∝negationslash= 0). The solution in the form (35) describes the wall between two domains whose spins are deviated from the easy axis onto the external-field direction about an angle which grows with |H+|. Inserting this ansatz into (34), one finds s1=β1−/radicalbig β2 1−γ2H2z γHzq1. (36) Considering the solutions which are static in the absence of the electric current, l1= 0 forδ= 0, one arrives at |k1|=/radicalBigg β2 1−γ2H2z β1J. (37) In the case of H+=Hy, the ansatz relating to the deviationofthe domainmagnetizationfromthe easyaxis onto the external-field direction takes the form f∗ 1=f2=q2+s2ek2x−l2t, g1=g∗ 2=s2+q2ek2x−l2t. (38) with real k2,q2,s2. From (34), we find s2=β1+β2−/radicalBig (β1+β2)2−γ2H2y γHyq2.(39)The static solutions correspond to |k2|=/radicalBigg (β1+β2)2−γ2H2y (β1+β2)J. (40) The transverse external field does not drive the DW motion even in the presence of the magnetic dissipation (α∝negationslash= 0). When the current through the wire and the dis- sipation are applied, under the transverse magnetic field, the DW moves with the velocity cgiven by (14), which is independent of the value of this field. Then the solution to (34) satisfies the bilinearized LLG system (Eqs. (3) with additional H+-dependent terms) at the time points of the discrete set t=πn/Iml1, wheren= 0,±1,±2,..., sincef1=f2=f,g1=g2=gat these points. In- cluding an additional to H+longitudinal component of the magnetic field Hxdrives the DW motion. For the realistic case HW∼ |Hx|<|H+| ≪ |β1/γ|, neglect- ing small contributions to the Hx-dependent part of the torque, one finds the velocity of such a DW propagation (13) or (26) above and below the Walker breakdown, re- spectively, with |k1(2)|given by (37), (40). This velocity nonlinearly increases with |H+|, [47]. Searching for c1(2), in the case |Hx|< HW, additionally, I have taken the lhs of (1) equal to zero as discussed in section 4. The ma- nipulation of c1(2)via the application of the transverse magnetic field is potentially useful for speeding up the processing with a magnetically-encoded information. We also notice that the transverse-field dependence of |k1(2)| enables influencing the magnitude of the critical current of the Walker breakdown δW, following (31). VI. CONCLUSIONS We have analytically studied the DW dynamics in the presence of the external magnetic field and the electric current along the magnetic wire within the LLG ap- proach. It has demanded overcoming the difficulty aris- ing from breaking the time-reversal symmetry by inclu- sion of the magnetic dissipation. We have removed this asymmetry of the dynamical system by introducing ad- ditional (virtual) dynamical variables, which is a similar trick to the Lagrangian approach to the damped har- monic oscillator. Determining a connection of the addi- tional dynamical variables to the evolution of the mag- netization vector in specific ranges of time, we have ana- lyzed the dynamics of a single DW and of a pair of DWs. The magnetic-field-induced velocities of the DWs, the formulas (13) and (26), and the current-induced veloci- ties (14) and (33) are found to correspond to the ones of the Walker approach above and below the breakdown, respectively. According to [28], static three-domain so- lutions to 1D LLG equation describe pairs of Neel and Bloch walls. For the purposes of the qualitative dynam- ics analysis of a number of DWs below the Walker break- down, especially of the Neel-Bloch pairs, we have pro- posed a dynamical equation which differs from the LLG9 one by neglecting the lhs in (1). Below and above the breakdown, the neighboring Neel and Bloch walls move in the presence of the longitudinal external field in the opposite directions. Their collision results in the DW reflection accompanied by the reorientation of the Neel wall into the Bloch wall and vice versa. In other words, the DWs pass trough each other without changing their widths and velocities, however, the head-to-head DW structure changes into the tail-to-tail one and vice versa. Our method is useful for the analysis of two-domain systems under the transverse (with respect to the easy axis) external field, which enables a verification of nu- mericalandexperimentalresults[46–48]. 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2011-09-12

Analytical multi-domain solutions to the dynamical (Landau-Lifshitz-Gilbert) equation of a one-dimensional ferromagnet including an external magnetic field and spin-polarized electric current are found using the Hirota bilinearization method. A standard approach to solve the Landau-Lifshitz equation (without the Gilbert term) is modified in order to treat the dissipative dynamics. I establish the relations between the spin interaction parameters (the constants of exchange, anisotropy, dissipation, external-field intensity, and electric-current intensity) and the domain-wall parameters (width and velocity) and compare them to the results of the Walker approximation and micromagnetic simulations. The domain-wall motion driven by a longitudinal external field is analyzed with especial relevance to the field-induced collision of two domain walls. I determine the result of such a collision (which is found to be the elastic one) on the domain-wall parameters below and above the Walker breakdown (in weak- and strong-field regimes). Single-domain-wall dynamics in the presence of an external transverse field is studied with relevance to the challenge of increasing the domain-wall velocity below the breakdown.

Externally-driven transmission and collisions of domain walls in ferromagnetic wires

1109.2465v1

ACCELERATION CONTROL IN NONLINEAR VIBRATING SYSTEMS BASED ON DAMPED LEAST SQUARES V.N. Pilipchuk Wayne State University Detroit, MI 48202 November 2, 2021 Abstract A discrete time control algorithm using the damped least squares is introduced for acceleration and energy exchange controls in nonlinear vi- brating systems. It is shown that the damping constant of least squares and sampling time step of the controller must be inversely related to insure that vanishing the time step has little eect on the results. The algorithm is illustrated on two linearly coupled Dung oscillators near the 1:1 in- ternal resonance. In particular, it is shown that varying the dissipation ratio of one of the two oscillators can signicantly suppress the nonlinear beat phenomenon. 1 Introduction The damped least squares is a simple but eective analytical manipulation that helps to avoid singularity in practical minimization and control algorithms. It is also known as Levenberg-Marquardt method [11]. In order to illustrate the idea in simple terms, let us consider the minimization problem kEAuk2!min (1) whereE2Rnis a given vector, the notation k:::kindicates the Euclidean norm inRn,Ais typically a Jacobian matrix of nrows andmcolumns, and u2Rmis an unknown minimization vector. Although a formal solution of this problem is given by u= (ATA)1ATE, the matrix product ATAmay appear to be singular so that no unique solution is possible. This fact usually points to multiple possibilities of achieving the same result unless specic conditions are imposed on the vector u. The idea of damped least squares is to avoid such conditioning by adding one more quadratic form to the left hand side of expression (1) as follows kEAuk2+kuk2!min (2) 1arXiv:1110.2811v2 [math.OC] 14 Oct 2011whereis a positive scalar number, which is often called damping constant ; note that the term `damping' has no relation to the physical damping or energy dissipation eects in vibrating systems usually characterized by damping ratios . Now the inverse matrix includes the damping constant which can provide the uniqueness of solution given by u= (ATA+I)1ATE (3) whereIisnnidentity matrix. Dierent arguments are discussed in the literature regarding the use of damped least squares and best choice for the damping parameter [1], [2], [3], [4], [6], [7], [9], [10], [15], [16], [17], [23], [24]. In particular, it was noticed that the parameter may aect convergence properties of the corresponding algorithms. The parameter can be used also for other reason such as shifting the solution uinto desired area in Rm. In this case, the meaning of is rather close to that of Lagrangian multiplier imposing constraints on control inputs. In case of dynamical systems, when all the quantities in (2) may depend on time, a continuous time analogue of (2) can be written in the integral form min uZT 0(kEAuk2+kuk2)dt (4) where the interval of integration is manipulated as needed, for instance, Tcan be equal to sampling time of the controller [12]. However, in the present work, a discrete time algorithm based on the damped least squares solution (3), which is used locally at every sample time tn, is intro- duced. Such algorithm appears to be essentially discrete namely using dierent time stephmay lead to dierent results. Nevertheless, if the parameters and hare coupled by some condition then the control input and system response show no signicant dependence on the time step. A motivation for the present work is as follows. In order to comply with the standard tool of dynamical systems dealing with dierential equations, the methods of control are often formulated in continuous time by silently assuming that a discrete time analogous is easy to obtain one way or another whenever it is needed for practical reasons. For instance, data acquisition cards and on-board computers of ground vehicles usually acquire and process data once per 0.01 sec. Typically, based on the information, which is known about the system dynamic states and control inputs by the time instance tn, the computer must calculate control adjustments for the next active time instance, tn+1. The corresponding computational time should not therefore exceed tn+1tn= 0:01 sec. Generally speaking, it is possible to memorize snapshots of the dynamic states and control inputs at some of the previous times f...,tn2,tn1g. However, increasing the volume of input data may complicate the code and, as a result, slow down the calculation process. Therefore, let us assume that updates for the control inputs are obtained by processing the system states, controls, and target states given only at the current time instance, tn. The corresponding algorithm can be built on the system model described by its dierential equations of motion and some 2rule for minimizing the deviation (error) of the current dynamic states from the target. Recall that, in the present work, such a rule will be dened according to the damped least squares (2). Illustrating physical example of two linearly coupled Dung oscillators is considered. It is shown that the corresponding algorithm, which is naturally designed and eectively working in discrete time, may face a problem of transition to the continuous time limit. 2 Problem formulation Consider the dynamical system x=f(x;_x;t;u ) (5) wherex=x(t)2Rnis the system position (conguration) vector, the overdot indicates derivative with respect to time t, the right-hand side f2Rnrepresents a vector-function that may be interpreted as a force per unit mass of the system, andu=u(t)2Rmis a control vector, whose dimension may dier from that of the positional vector xso that generally n6=m. In common words, the purpose of control u(t) is to keep the acceleration x(t) of system (5) as close as possible to the target x(t). The term `close' will be interpreted below through a specically designed target function of the following error vector E(t) = x(t)x(t) (6) As discussed in Introduction, for practical implementations, the problem must be formulated in terms of the discrete time ftkgas follows. Let xk= x(tk), _xk= _x(tk), anduk=u(tk) are observed at some time instance tk. The corresponding target acceleration, x k= x(tk), is assumed to be known. Then, taking into account (5) and (6), gives the following error at the same time instance Ek= x kf(xk;_xk;tk;uk) (7) Now the purpose of control is to minimize the following target function Pk=1 2ET kWkEk (8) =1 2[x kf(xk;_xk;tk;uk)]TWk[x kf(xk;_xk;tk;uk)] whereWkisnndiagonal weight matrix whose elements are positive or at least non-negative functions of the system states, Wk=W(xk;_xk;tk). Note that all the quantities in expression (8) represent a snapshot of the system att=tkwhile including no data from the previous time step tk1. Since the control vector ukcannot be already changed at time tkthen quantity Pkis out of control at time tk. In other words expression (8) summarizes all what is observed now, at the time instance tk. The question is how to adjust the control vector ufor the next step tk+1based on the information included in 3(8) while the system state at t=tk+1is yet unknown, and no information from the previous times f...,tn2,tn1gis available. Let us represent such an update for the control vector in the form uk+1=uk+uk (9) wereukis an unknown adjustment of the control input. Replacingukin (8) by (9) and taking into account that f(xk;_xk;tk;uk+1) =f(xk;_xk;tk;uk) +Akuk+O(kukk2) (10) Ak=@f(xk;_xk;tk;uk)=@uk gives Pk=1 2(EkAkuk)TWk(EkAkuk) (11) whereAkis the Jacobian matrix of nrows andmcolumns. Although the replacement ukbyuk+1in (10) may look articial, this is how the update rule for the control vector uis actually dened here. Namely, if ukdid not provide a minimum for Pk(x k;xk;_xk;tk;uk), then let us minimize Pk(x k;xk;_xk;tk;uk+uk) with respect to ukand then apply the adjusted vector (9) at least the next next time, tn+1. Assuming that the variation ukis small, in other words, ukis still close enough to the minimum, expansion (10) is applied. Now the problem is formulated as a minimization of the quadratic form (11) with respect to the adjustment uk. However, what often happens practically is that function (11) has no unique minimum so that equation dPk duk= 0 (12) has no unique solution. In addition, even if the unique solution does exist, it may not satisfy some conditions imposed on the control input due to the physical specics of actuators. As a result, some constraint conditions may appear to be necessary to impose on the variation of control adjustment, uk. However, the presence of constraints would drastically complicate the problem. Instead, the target function (11) can be modied in order to move solution ukinto the allowed domain. For that reason, let us generalize function (11) as Pk=1 2(EkAkuk)TWk(EkAkuk) +1 2(Bk+Ckuk)Tk(Bk+Ckuk) (13) where k= (xk;_xk;tk) is a diagonal regularization matrix, Bk=B(xk;_xk;tk) is a vector-function of nelements, and Ck=C(xk;_xk;tk) is a matrix of nrows andmcolumns. Note that the structure of new function (13) is a generalization of (2). Sub- stituting (13) in (12), gives a linear set of equations in the matrix form whose solutionukbrings relationship (9) to the form 4uk+1=uk+ (AT kWkAk+CT kkCk)1(AT kWkEkCT kkBk) (14) The entire discrete time system is obtained by adding a discrete version of the dynamical system (5) to (14) . Assuming that the time step is xed, tk+1tk=h, a simple discrete version can be obtained by means of Euler explicit scheme as follows xk+1=xk+hvk vk+1=vk+hf(xk;vk;tk;uk) (15) Finally, equations (14) and (15) represent a discrete time dynamical system, whose motion should follow the target acceleration x k= x(tk). It will be shown in the next section that the structure of equation (14) does not allow for the transition to continuous limit of the entire dynamic system (14) through (15), unless some specic assumption are imposed on the parameters in order to guarantee that uk=O(h) ash!0. 3 The illustrating example The algorithm, which is designed in the previous section, is applied now to a two-degrees-of-freedom nonlinear vibrating system for an active control of the energy exchange (nonlinear beats) between the two oscillators. The problem of passive control of energy ows in vibrating systems is of great interest [22], and it is actively discussed from the standpoint of nonlinear beat phenomena [14]. The beating phenomenon takes place when frequencies of the corresponding linear oscillators are either equal or at least close enough to each other. For illustrating purposes, let us consider two unit-mass Dung oscillators of the same linear stiness Kcoupled by the linear spring of stiness . The system position is described by the vector-function of coordinates, x(t) = (x1(t);x2(t))T. Introducing the parameters = ( +K)1=2and"= =( +K), brings the dierential equations of motion to the form _x1=v1 _x2=v2 _v1=2 v1 2x1+"( 2x2x3 1)f1(x1;x2;v1) (16) _v2=2u v2 2x2+"( 2x1x3 2)f2(x1;x2;v2;u) whereis a positive parameter, anduare damping ratios1of the rst and the second oscillators, respectively; the damping ratio u, which is explicitly shown as an argument of the function f2(x1;x2;v2;u), will be considered as a control input. 1As mentioned in Introduction, the damping (dissipation) ratio should not be confused with the damping coecient . 5The problem now is to nd such variable damping ratio u=u(t) under which the second oscillator accelerates as close as possible to the given (target) acceleration, x 2(t). Following the discussion of the previous section, let us consider the prob- lem in the discrete time ftkg. In order to avoid confusion, the iterator k will be separated from the vector component indexes by coma, for instance, xk= (x1;k;x2;k)T. Since only the second mass acceleration is of interest and the system under consideration includes only one control input u, then, assum- ing the weights to be constant, gives Wk=0 0 0 1 ,Ak=@ @ukf1;k f2;k wheref1;kf1(x1;k;x2;k;v1;k) andf2;kf2(x1;k;x2;k;v2;k;uk), and other matrix terms become scalar quantities, say,  k=,Bk=b, andCk= 1. The unities inWkandCkcan always be achieved by re-scaling the target function and parameters andb. Note that re-scaling the target function by a constant factor has no eect on the solution of equation (12). As a result, the target function (13) takes the form Pk=1 2 x 2;kf2;k@f2;k @ukuk2 + 2(b+uk)2(17) In this case, equation (12) represents a single linear equation with respect to the scalar control adjustment, uk. Substituting the corresponding solution in (14) and taking into account (15), gives the discrete time dynamical system uk+1=uk(f2;kx 2;k)(@f2;k=@uk) +b (@f2;k=@uk)2+(18) and x1;k+1=x1;k+hv1;k x2;k+1=x2;k+hv2;k v1;k+1=v1;k+hf1;k (19) v2;k+1=v2;k+hf2;k Let us assume now that the target acceleration x 2is zero, in other words, the purpose of control is to minimize acceleration of the second oscillator at any sample time tkas much as possible. Let us set still arbitrary parameter balso to zero. Then the target function (17) and dynamical system (18) and (19) take the form Pk=1 2 f2(x1;k;x2;k;v2;k;uk) +@f2(x1;k;x2;k;v2;k;uk) @ukuk2 + 2(uk)2(20) 6uk+1=uk+2 v2;k 4 2v2 2;k+f2(x1;k;x2;k;v2;k;uk) x1;k+1=x1;k+hv1;k x2;k+1=x2;k+hv2;k (21) v1;k+1=v1;k+hf1(x1;k;x2;k;v1;k) v2;k+1=v2;k+hf2(x1;k;x2;k;v2;k;uk) where the functions f1andf2are dened in (16). As follows from the rst equation in (21), transition to the continuous time limit for the entire system (21) would be possible under the condition that 2 v2;k 4 2v2 2;k+=O(h), ash!0 (22) Condition (22) can be satised by assuming that = O(h). Such an as- sumption, however, makes little if any physical sense. As an alternative choice, the condition =O(h1) can be imposed by setting, for instance, h=0 (23) where0remains nite as h!0. However, condition (23) essentially shifts the weight on control to the second term of the target function (17) so that the function asymptotically takes the form Pk'0 2h(uk)2, ash!0 (24) Such a target function leads to the solution uk= 0, which eectively elim- inates the control equation. In other words, the iterative algorithm seems to be essentially discrete. As a result, the control input uk, generated by the rst equation in (21), depends upon sampling time interval h. Let us illustrate this observation by implementing the iterations (21) under the xed set of parame- ters,"= 0:1, = 1:0,= 1:5,= 0:025, and initial conditions, u0= 0:025, x1;0= 1:0,x2;0= 0:1,v1;0=v2;0= 0. The values to vary are two dierent sam- pling time intervals, h= 0:01 andh= 0:001, and three dierent values of the damping constant, = 0:1,= 1:0, and= 10:0. For comparison reason, Fig. 1 shows time histories of the system coordinates under the xed control vari- ableu=. This (no control) case corresponds to free vibrations of the model (16) whose dynamics represent a typical beat-wise decaying energy exchange between the two oscillators. As mentioned at the beginning of this section, the beats are due to the 1:1 resonance in the generating system ( "= 0,u== 0); more details on non-linear features of this phenomenon, the related analytical tools, and literature overview can be found in [20] and [14]. In particular, the standard averaging method was applied to the no damping case of system (16) in [20]. 7Now the problem is to suppress the beat phenomenon by preventing the energy ow from the rst oscillator into the second oscillator. As follows from Figs. 2 through 5, such a goal can be achieved by varying the damping ratio of the second oscillator, fukg, during the vibration process according to the algorithm2(21). First, the diagrams in Figs. 2 and 3 conrm that the sampling time interval hrepresents an essential parameter of the entire control loop. In particular, decreasing the sampling interval from h= 0:01 toh= 0:001 eec- tively increases the strength of the control; compare fragments (b) in Figs. 2 and 3. However, if such decrease of the sampling time is accompanied by the increase of according to condition (23), then the strength of control remains practically unchanged; compare now fragments (b) in Figs. 2 and 4. As follows from fragments (a) in Figs. 2 and 4, the above modication of both parameters, hand, also brings some dierence in the system response during the interval 80<t< 150, but this is rather due to numerical eect of the time step. Finally, analyzing the diagrams in Figs. 3 and 5, shows that reducing the parameter  as many as ten times under the xed time step hleads to a signicant increase of the control input fukgwith a minor eect on the system response though. Therefore the parameter can be used for the purpose of satisfying some con- straint conditions on the control inputs fukgin case such conditions are due to physical limits of the corresponding actuators. In addition, let us show that parametermay aect the convergence of algorithm (21) based on the following convergence criterion [18]: For a xed point zto be a point of attraction of the algorithm zk+1=G(zk) a sucient condition is that the Jacobian matrix of Gat the point zhas all its eigenvalues numerically less than 1, and a necessary condition is that they are numerically at most 1. The geometric rate of convergence is the numerically largest eigenvalue of this Jacobian. Applying this criterion to the algorithm (21) at zero point, gives that one of the eigenvalues is always zero, q0= 0, whereas another four eigenvalues, qi (i= 1;:::;4) are proportional to the time step, qi=hpi, where the coecients piare given by the roots of algebraic equation p4+ 2 p3+ 2 2p2+ 2 3p+ (1"2) 4= 0 (25) As follows from (25), the damping coecient has no in uence on the con- vergence condition near the equilibrium point, and the convergence can always be achieved under a small enough time step h. Nevertheless, the damping coef- cient may appear to aect the convergence away from the equilibrium point. In this case, analytical estimates for eigen values of the Jacobian become tech- nically complicated unless "= 0, when four of the ve eigenvalues vanish as h!0, except one eigenvalue, which is estimated by q= 1 + 4 2v2 21 (26) 2Note that, although the algorithm is designed to suppress accelerations of the second oscillator, acceleration and energy levels of vibrating systems are related. 8This root gives q!q0= 0 asv2!0. However, when v26= 0, equation (26) gives the estimate 0 <q1 as1>0. Therefore, only the necessary convergence condition is satised for = 0. 4 Conclusions In this work, a discrete time control algorithm for nonlinear vibrating systems using the damped least squares is introduced. It is shown that the corresponding damping constant and sampling time step hmust be coupled by the condition h=constant in order to preserve the result of calculation when varying the time step. In particular, the above condition prohibits a direct transition to the continuos time limit. This conclusion and other specics of the algorithm are illustrated on the nonlinear two-degrees-of-freedom vibrating system in the neighborhood of 1:1 resonance. It is shown that the dissipation ratio of one of the two oscillators can be controlled in such way that prevents the energy ex- change (beats) between the oscillators. From practical standpoint, controlling the dissipation ratio can be implemented by using devices based on the physical properties of magnetorheological uids (MRF) [8], [19]. In particular, dier- ent MRF dampers are suggested to use for semi-active ride controls of ground vehicles and seismic response reduction. 9References [1] Chan, S. K., and Lawrence, P. D., \General inverse kinematics with the error damped pseudoinverse," Proc. IEEE International Conference on Robotics and Automation , 834-839, (1988). [2] Chiaverini, S., \Estimate of the two smallest singular values of the Jacobian matrix: Applications to damped least-squares inverse kinematics," Journal of Robotic Systems , 10, 991-1008, (1988). [3] Chiaverini, S. Siciliano, B., and Egeland, O., \Review of damped least- squares inverse kinematics with experiments on an industrial robot ma- nipulator," IEEE Transactions on Control Systems Technology , 2, 123-134, (1994). [4] Chung, C. Y., and Lee, B. H., \Torque optimizing control with singularity- robustness for kinematically redundant robots," Journal of Intelligent and Robotic Systems , 28, 231-258, (2000). [5] Colbaugh, R., Glass, K., and Seraji, H., \Adaptive tracking control of manipulators: theory and experiments," Robotics & Computer-Integrated Manufacturing, 12(3), 209-216, (1996). [6] Deo, A. S., and Walker, I. D., \Robot subtask performance with singularity robustness using optimal damped least squares," Proc. IEEE International Conference on Robotics and Automation , 434-441, (1992). [7] Deo, A. S., and Walker, I. D., \Adaptive non-linear least squares for in- verse kinematics," Proc. IEEE International Conference on Robotics and Automation , 186-193, (1993). [8] Dyke, S.J., Spencer Jr., B.F., Sain, M.K., and Carlson, J.D., \Seismic response reduction using magnetorheological dampers," Proceedings of the IFAC World Congress, San Francisco, California, June 30-July 5 , Vol. L, 145-150, (1996). [9] Hamada, C., Fukatani, K., Yamaguchi, K., and Kato, T., \Development of vehicle dynamics integrated management," SAE Paper No. 2006-01-0922. [10] Hattori, Y., \Optimum vehicle dynamics control based on tire driving and braking forces," R&D Review of Toyota CRDL , 38 (5), 23-29, (2003). [11] Levenberg, K., \A Method for the Solution of Certain Non-Linear Prob- lems in Least Squares," The Quarterly of Applied Mathematics , 2, 164{168, (1944). [12] Lee, J.H., and Yoo, W.S., \Predictive control of a vehicle trajectory using a coupled vector with vehicle velocity and sideslip angle," International Journal of Automotive Technology , 10 (2), 211-217, (2009). 10[13] Maciejewski, A. A., and Klein, C. A., \The singular value decomposi- tion: Computation and applications to robotics," International Journal of Robotic Research , 8 , 63-79, (1989). [14] Manevitch, L.I., Gendelman, O.V., Tractable Models of Solid Mechanics: Formulation, Analysis and Interpretation , Springer - Verlag, Berlin Heidel- berg, 2011, 297 p. [15] Mayorga, R. V., Milano, N., and Wong, A. K. C., \A simple bound for the appropriate pseudoinverse perturbation of robot manipulators," Proc. IEEE International Conference on Robotics and Automation , Vol. 2, 1485- 1488, (1990). [16] Mayorga, R. V., Wong, A. K. C., and Milano, N., \A fast procedure for manipulator inverse kinematics evaluation and pseudoinverse robustness," IEEE Transactions on Systems, Man, and Cybernetics , 22, 790-798, (1992). [17] Nakamura, Y., and Hanafusa, H., \Inverse kinematics solutions with sin- gularity robustness for robot manipulator control," Journal of Dynamic Systems, Measurement, and Control , 108 , 163-171, (1986). [18] Ostrowski, A M, Solution of Equations and Systems of Equations , Aca- demic Press Inc., 1960, 352 p. [19] Phule, P.P., \Magnetorheological (MR) uids: Principles and applica- tions," Smart Materials Bulletin , February, 7-10, (2001). [20] Pilipchuk, V.N., Nonlinear Dynamics: Between Linear and Impact Limits , Springer, 2010, 360 p. [21] Seraji, H., Colbaugh, R., \Improved conguration control for redundant robots," J. Robotic Systems , 7-6, 897-928, (1990). [22] Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Ker- schen, G., Lee, Y.S., Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems , Springer-Verlag, Berlin Heidelberg, 2009, 1032 p. [23] Wampler, C. W., \Manipulator inverse kinematic solutions based on vector formulations and damped least squares methods," IEEE Transactions on Systems, Man, and Cybernetics , 16, 93-101, (1986). [24] Wampler, C. W., and Leifer, L. J., \Applications of damped least-squares methods to resolved-rate and resolved-acceleration control of manipula- tors," Journal of Dynamic Systems, Measurement, and Control , 110, 31-38, (1988). 11Figure 1: No control beat dynamics with the decaying energy exchange between two Dung's oscillators; u== 0:025. 12Figure 2: Beat suppression under the time increment h= 0:01 and weight parameter= 1:0: (a) the system response, (b) control input - the damping ratio of second oscillator. 13Figure 3: Beat suppression under the reduced time increment h= 0:001 and the same weight parameter = 1:0: (a) the system response, (b) control input - the damping ratio of second oscillator. 14Figure 4: Beat suppression under the reduced time increment h= 0:001 but increased weight parameter = 10:0: (a) the system response, (b) control input - the damping ratio of second oscillator. 15Figure 5: Beat suppression under the reduced time increment h= 0:001 and vanishing weight parameter = 0:1: (a) the system response, (b) control input - the damping ratio of second oscillator. 16

2011-10-12

A discrete time control algorithm using the damped least squares is introduced for acceleration and energy exchange controls in nonlinear vibrating systems. It is shown that the damping constant of least squares and sampling time step of the controller must be inversely related to insure that vanishing the time step has little effect on the results. The algorithm is illustrated on two linearly coupled Duffing oscillators near the 1:1 internal resonance. In particular, it is shown that varying the dissipation ratio of one of the two oscillators can significantly suppress the nonlinear beat phenomenon.

Acceleration Control in Nonlinear Vibrating Systems based on Damped Least Squares

1110.2811v2

arXiv:1111.4295v2 [cond-mat.mes-hall] 8 Jun 2012JournalofthePhysicalSocietyofJapan FULLPAPERS Charge and Spin Transport in MagneticTunnel Junctions: Mic roscopic Theory DaisukeMiura∗and AkimasaSakuma DepartmentofAppliedPhysics,Tohoku University Sendai980-8579 We study the charge and spin currents passing through a magne tic tunnel junction (MTJ) on the basis of a tight-binding model. The currents are evaluat ed perturbatively with respect to thetunnelHamiltonian.Thechargecurrenthastheform A[M1(t)×˙M1(t)]·M2+B˙M1(t)·M2, whereM1(t) andM2denote the directions of the magnetization in the free layer and fixed layer,respectively.Theconstant Avanisheswhenoneorbothlayers areinsulators,whilethe constantBdisappearswhenbothlayersareinsulatorsorthesameferro magnets.Thefirstterm intheexpressionforchargecurrentrepresentsdissipatio ndrivenbytheeffectiveelectricfield inducedbythedynamicmagnetization.Inaddition,fromani nvestigationofthespincurrent, we obtain the microscopic expression for the enhanced Gilbe rt damping constant ∆α. We showthat∆αisproportionaltothetunnelconductanceand dependsonthe biasvoltage. KEYWORDS: spintronics, magnetictunneljunction,spin cur rent, spindynamics 1. Introduction Magnetictunnel junctions(MTJs), which consist of a thin tu nnel barrier sandwiched be- tween two ferromagnetic layers,1–5are promising for their use in magnetic random access memory (MRAM).6However, the primary disadvantage of conventional MRAM des igns, which employ a current-induced field to write data, is that th e writing current increases withthedevicedensity.Thus,therehas been considerablei nterestin exploitingspin-transfer torque(STT)7,8instead.9–13InsuchanSTTMRAMdevice,thecriticalcurrentisproportio nal to the product of the volume and the Gilbert damping constant αof the free layer, making lowαan importantcriterionforelectrodematerials. Tothisend,severalstudieshaveexploredthedynamicsandt hedistributionofthemagne- tizationsin STT MRAMby usingtheLandau–Lifshitz–Gilbert (LLG)equation withan STT term.14–18However,othertorques(spintorques)alsoactonthedynami cmagnetizationinthe ∗E-mailaddress:dmiura@solid.apph.tohoku.ac.jp 1/13J.Phys. Soc. Jpn. FULLPAPERS freelayer,whichforminreactiontotheoutwardflowofspins fromthelayer:Mizukami etal. experimentally showed that αincreases with the thickness of the nonmagnetic metal (NM) layer in NM/Py/NM films, and that this enhancement continues up to thickness es of several hundrednanometers.19Theirexperimentsupportstheimportanceofspintorquesin themag- netizationdynamicsofmesoscopicdevicessuch asSTTMRAMs .Further, thisexperimental findingwassupportedimmediatelybyTserkovnyak etal.’s20,21theoryofspinpumpingbased onscatteringtheory,withadditionaltheoreticalconfirma tionbyUmetsu etal.onthebasisof theKuboformula.22,23 Several studies have also investigated charge transport in the presence of magnetization dynamicsin magneticmultilayers.It is known that dynamicm agnetizations inducean e ffec- tiveelectromagneticfield.24,25Oheet al.simulated the effectiveelectric field induced by the motion of the magnetic vortex core in a magnetic disk,26and the field was observed experi- mentally.27Furthermore, Zhang et al.phenomenologicallyderived the LLG equation having the STT term induced by this e ffective electric field.28And Moriyama et al.observed the dc voltage across generated by the precession of the magnetiza tion in an Al/AlOx/Ni80Fe20/Cu tunnel junction.29The origins of this voltage have been discussed from a theore tical stand- point(scatteringtheory).30–32Inaddition,chargeandspincurrentsinferromagnetswithm ag- netizations that slowly vary in space and time have been stud ied microscopically.33–35These studies employed the s-d model in continuous space and treat ed the perturbation within the frameworkoftheKeldysh–Greenfunction.36,37 Similarly,ouraimistodescribethechargeandspintranspo rtinMTJsinthepresenceofa voltageacrossthebarrierandthedynamicalmagnetization inthefreelayer.Thissituationjust correspondstoanSTTMRAMcellduringthewritingstage.Int hispaper,wemicroscopically describe the charge and spin currents passing through an MTJ . However, in contrast with previous works that relied on models in continuous space, we calculate the currents on the basisofatight-bindingscheme.Thismakesiteasiertoacco untforthepropertiesofmaterials and the space dependence of the magnetization in magnetic mu ltilayers, such as MTJs, with stronglyinhomogeneousmagneticstructures.Inthecalcul ations,weconsiderthevoltageand the dynamics of the magnetization in Berry’s adiabatic appr oximation under the assumption thattheeffectiveexchangefieldislargerthanthevoltageanddynamics .Ourmodelshowsthat the charge current induced by the dynamical magnetization h as the form A[ML(t)×˙ML(t)]· MR+B˙ML(t)·MR, whereML(t) andMRdenote the directions of the magnetization in the freelayerandfixedlayer,respectively.Thefirsttermtends totheformgivenbyTserkovnyak et al.,31which expressedthedc current dueto theprecession of ML(t)aboutMRas aspecial 2/13J.Phys. Soc. Jpn. FULLPAPERS i jtij ... L RTLR ... ... ML(t) MR ... Left hand side layer Right hand side layer Tunnel barrier Fig. 1. Schematicofone-dimensionalmagnetictunneljunction. TLRis thetunnelingamplitudeand tijrepre- sents the hoppingmatrix between sites iandjlocated at either side of the interface. ML(t) andMRdenote the directionsoftheeffectiveexchangefieldsforthe left(L)andright(R)handside layer,respectively. case; in this sense, our result is a generalization of their w ork. Furthermore, from the results concerningspintransport,wesuccessfullyderivetheenha ncedGilbertdampingandpropose amicroscopicexpressionforit. 2. Model and Formalism 2.1 Model Hamiltonian We consider the motion of electrons in an e ffective exchange field. Furthermore, assume that the ferromagnetic layer on the left-hand side (LHS) of t he MTJ is the free layer; that is, the direction of the field at time tin this layer, ML(t), rotates time-dependently (see Fig. 1). Thus, the direction of the field on the right-hand side (RH S) (fixed layer), MR, is time- independent. Note that we ignore the inner structure of the t unnel barrier and account for its properties via the simple tunnel amplitude TLRbetween sites L and R, which denote the surfaceson theLHSandRHS, respectively.In thismodel,the totalHamiltonianfortheMTJ isthesumoftheonedimensionaltight-bindingHamiltonian sintheferromagneticlayers, HL(t) :=/summationdisplay i,j∈LHSc† i/bracketleftBig −tijˆ1−δijJLML(t)·ˆσ/bracketrightBig cj, (1) HR:=/summationdisplay i,j∈RHSc† i/bracketleftBig −tijˆ1−δijJRMR·ˆσ/bracketrightBig cj, (2) andthetunnelHamiltonian, HT:=−TLRc† LcR+H.c., (3) 3/13J.Phys. Soc. Jpn. FULLPAPERS wherec† iσ(ciσ) is an operator that creates (annihilates) the σspin electron at site i, andtijis thehopping integral between sites iandj. The constant JL(JR) represents the strength of the interactionbetweenthespinofanelectronandthee ffectiveexchangefieldontheLHS(RHS) layer;and ˆσisthePaulimatrix,wherehat ‘ˆ’denotesa2 ×2 matrixinspin-space. 2.2 Adiabaticapproximation Assuming JL≫/planckover2pi1|dML(t)/dt|,weadopt Berry’s adiabaticapproximation38forHL(t): ci(t)≃ˆUL(t)eiγ(t)ˆσzdifori∈LHS, (4) HL(t)→Had L:=/summationdisplay i,j∈LHSd† i/bracketleftBig −tijˆ1−δijJLˆσz/bracketrightBig dj, (5) whereci(t)isintheHeisenbergrepresentationwithrespectto HL(t),ˆUL(t)isarotationmatrix satisfyingtheequation ˆU† L(t)ML(t)·ˆσˆUL(t)=ˆσz, andγ(t)is Berry’s phasedefined by γ(t) :=i/integraldisplay dt/bracketleftBigg ˆU† L(t)dˆUL(t) dt/bracketrightBigg ↑↑. (6) Withtheapproximation(4), wereplace HTwith Had T(t) :=−TLRd† Le−iγ(t)ˆσzˆU† L(t)ˆURdR+H.c., (7) whereˆURis a rotation matrix satisfying the equation ˆU† RMR·ˆσˆUR=ˆσz, anddi:= ˆU† Rcifori∈RHS. Finally, our total Hamiltonian is H(t) :=Had L+HR+Had T(t), where HR=/summationtext i,j∈RHSd† i/bracketleftBig −tijˆ1−δijJRˆσz/bracketrightBig dj. Thus, a nonequilibrium statistical average of the form/angbracketleftBig diσ(t)d† i′σ′(t′)/angbracketrightBig can bederivedperturbativelywithrespectto Had T(t)usingtheKeldysh–Green functiontechnique. 2.3 Chargeand spincurrents Thechargecurrent Ie(t)andspincurrent Is(t)passingthroughtheMTJaredefined by Ie(t) :=2ℜi /planckover2pi1TRL/angbracketleftBig d† R(t)ˆU† RˆUL(t)eiγ(t)ˆσzdL(t)/angbracketrightBig [1/s], (8) Is(t) :=2ℜi /planckover2pi1TRL/angbracketleftBig d† R(t)ˆU† RˆσˆUL(t)eiγ(t)ˆσzdL(t)/angbracketrightBig [1/s], (9) where∝angbracketleft···∝angbracketrightdenotesastatisticalaveragein H(t).36,37 By introducingthelesserfunction, /bracketleftBigˆG< LR(t,t′)/bracketrightBig σσ′:=i /planckover2pi1/angbracketleftBig/bracketleftBig d† R(t′)ˆU† R/bracketrightBig σ′/bracketleftBigˆUL(t)eiγ(t)ˆσzdL(t)/bracketrightBig σ/angbracketrightBig , eqs.(8)and (9)can bewrittenin theform Ie(t)=2ℜTRLtrˆG< LR(t,t), (10) 4/13J.Phys. Soc. Jpn. FULLPAPERS Is(t)=2ℜTRLtr ˆσˆG< LR(t,t). (11) Inthefirst orderin Had T(t),wehave ˆG< LR(t,t)≃−TLR/integraldisplay dt′ˆUL(t)eiγ(t)ˆσzˆgL(t−t′)e−iγ(t)ˆσzˆU† L(t)ˆA(t,t′)ˆURˆgR(t′−t)ˆU† R/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ,(12) where<denotesthelessercomponentofKeldysh–Greenfunctions,36,37and ˆA(t,t′) :=ˆUL(t)eiγ(t)ˆσze−iγ(t′)ˆσzˆU† L(t′). (13) Moreover,weintroducetheunperturbedKeldysh–Greenfunc tionsdefined by /bracketleftbigˆgL(t)/bracketrightbig σσ′:=−i /planckover2pi1/angbracketleftBig TdLσ(t)d† Lσ′/angbracketrightBig 0=−i /planckover2pi1/angbracketleftBig TdLσ(t)d† Lσ/angbracketrightBig 0δσσ′, (14) /bracketleftbigˆgR(t)/bracketrightbig σσ′:=−i /planckover2pi1/angbracketleftBig TdRσ(t)d† Rσ′/angbracketrightBig 0=−i /planckover2pi1/angbracketleftBig TdRσ(t)d† Rσ/angbracketrightBig 0δσσ′, (15) where T is the time-ordering operator on the Keldysh contour , and∝angbracketleft···∝angbracketright0denotes an equilib- rium statistical average in Had L+HR. Since ˆgL(t) is the diagonal matrix in spin-space, ˆ gL(t) andBerry’s phasefactors commute.Thus, ˆG< LR(t,t)reduces to ˆG< LR(t,t)=−TLR/integraldisplaydE 2π/planckover2pi1/integraldisplaydE′ 2π/planckover2pi1e−iE′t//planckover2pi1ˆU† L(t)ˆgL(E)ˆUL(t)ˆA(t,E′)ˆU† RˆgR(E−E′)ˆUR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle< ,(16) where we employ the Fourier transform of a function f(t) with respect to t, defined by the relation f(E) :=/integraldisplay dteiEt//planckover2pi1f(t). (17) E′ineq.(16)representstheenergythatanelectronobtainsfr omthedynamicsofthemagne- tization;weconsideritinthefirst order: ˆG< LR(t,t)≃−TLR/integraldisplaydE 2π/planckover2pi1ˆU† L(t)ˆgL(E)ˆUL(t)ˆU† RˆgR(E)ˆUR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle< −TLR/integraldisplaydE 2π/planckover2pi1ˆU† L(t)ˆgL(E)ˆUL(t)/planckover2pi1 idˆA(t,t′) dt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet′=tˆU† RdˆgR(E) dEˆUR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle< . (18) Then,usingtherelations /planckover2pi1 idˆA(t,t′) dt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet′=t=/planckover2pi1 2ˆσ·ML(t)×dML(t) dt, (19) ˆU† L(t)ˆgL(E)ˆUL(t)=ˆ1¯gL(E)+ˆσ·ML(t)∆gL(E), (20) ˆU† RˆgR(E)ˆUR=ˆ1¯gR(E)+ˆσ·MR∆gR(E), (21) where ¯gL(R)(E) :=1 2tr ˆgL(R)(E)=/bracketleftbigˆgL(R)(E)/bracketrightbig ↑↑+/bracketleftbigˆgL(R)(E)/bracketrightbig ↓↓ 2, (22) 5/13J.Phys. Soc. Jpn. FULLPAPERS ∆gL(R)(E) :=1 2tr ˆσzˆgL(R)(E)=/bracketleftbigˆgL(R)(E)/bracketrightbig ↑↑−/bracketleftbigˆgL(R)(E)/bracketrightbig ↓↓ 2, (23) wecan decompose ˆG< LR(t,t)intotwoterms: ˆG< LR(t,t)=ˆ1G< LR(t,t)+ˆσ·G< LR(t,t). (24) Herewedefine G< LR(t,t) :=−TLR/integraldisplaydE 2π/planckover2pi1/bracketleftBigg ¯gL(E)¯gR(E)+ML(t)·MR∆gL(E)∆gR(E) +/planckover2pi1 2ML(t)×dML(t) dt·MR¯gL(E)d∆gR(E) dE−i/planckover2pi1 2dML(t) dt·MR∆gL(E)d∆gR(E) dE/bracketrightBigg< , (25) G< LR(t,t) :=−TLR/integraldisplaydE 2π/planckover2pi1/bracketleftBigg ML(t)∆gL(E)¯gR(E)+MR¯gL(E)∆gR(E)+iML(t)×MR∆gL(E)∆gR(E) +/planckover2pi1 2ML(t)×dML(t) dt¯gL(E)d¯gR(E) dE−i/planckover2pi1 2dML(t) dt∆gL(E)d¯gR(E) dE +i/planckover2pi1 2/braceleftBigg ML(t)×dML(t) dt/bracerightBigg ×MR¯gL(E)d∆gR(E) dE+/planckover2pi1 2dML(t) dt×MR∆gL(E)d∆gR(E) dE/bracketrightBigg< . (26) Ie(t)andIs(T)are expressedinterms of G< LR(t,t)andG< LR(t,t)as follows: Ie(t)=4ℜTRLG< LR(t,t), (27) Is(t)=4ℜTRLG< LR(t,t). (28) Finally,takingthelessercomponents,weobtainthefollow ingin thelow-temperaturelimit: Ie(t)=2π|TLR|2/braceleftBigg ¯ρL(µ)∆ρR(µ)ML(t)×dML(t) dt·MR −/integraldisplayµ dE/bracketleftBigg ∆ρL(E)d∆χR(E) dE−d∆χL(E) dE∆ρR(E)/bracketrightBiggdML(t) dt·MR/bracerightBigg , (29) Is(t)=4π|TLR|2 /planckover2pi1/integraldisplayµ dE/bracketleftbig∆ρL(E)∆χR(E)+∆χL(E)∆ρR(E)/bracketrightbigML(t)×MR +4π|TLR|2 /planckover2pi1/braceleftBigg ¯ρL(µ)¯ρR(µ)/planckover2pi1 2ML(t)×dML(t) dt −/integraldisplayµ dE/bracketleftBigg ∆ρL(E)d¯χR(E) dE−d∆χL(E) dE¯ρR(E)/bracketrightBigg/planckover2pi1 2dML(t) dt +/integraldisplayµ dE/bracketleftBigg ¯ρL(E)d∆χR(E) dE−d¯χL(E) dE∆ρR(E)/bracketrightBigg/planckover2pi1 2/bracketleftBigg ML(t)×dML(t) dt/bracketrightBigg ×MR +∆ρL(µ)∆ρR(µ)/planckover2pi1 2dML(t) dt×MR/bracerightBigg , (30) 6/13J.Phys. Soc. Jpn. FULLPAPERS Hereµis the chemical potential of the system. ¯ ρL(R)(E) and∆ρL(R)(E) are the spin-averaged local density of states (DOS) and the spin polarization of th e local DOS, respectively, at the LHS(RHS) layersurface, defined by ¯ρL(R)(E) :=−1 πℑ¯gr L(R)(E), (31) ∆ρL(R)(E) :=−1 πℑ∆gr L(R)(E), (32) wheregr’s are retarded Green’s functions from the calculations tak ing the lesser component. Furthermore,theχ’s aredefined as thereal parts oftheretarded Green’sfuncti ons, ¯χL(R)(E) :=1 πℜ¯gr L(R)(E), (33) ∆χL(R)(E) :=1 πℜ∆gr L(R)(E). (34) 3. DiscussionandSummary 3.1 Chargecurrent The form A[ML(t)×˙ML(t)]·MR+B˙ML(t)·MRof the charge current (eq. (29)) driven by the magnetizationdynamics is consistent with previous w orks; The first term tends to the form given by Tserkovnyak et al.31in the special case where ML(t) precesses around MR, as discussed in§3.3. And Xiao et al.30have derived the same form for the charge current passing through the MTJ on the basis of scattering theory in t he continuum space, whereas, wecalculatethecurrentonthebasisofthetight-bindingmo del.Newinsightswhichwefound out in this study are as follows.If the electronic structure in the two layers of the MTJ is the same [i.e.,∆ρL(E)=∆ρR(E) and∆χL(E)=∆χR(E)] or both layers are insulators, we have B=0. However, if either layer is metallic, finite Bshould be measured because the real partoftheretarded Green’s functionremainsfinite,which r eflects virtualtransitionsthrough forbiddenbands.Theterm[ ML(t)×˙ML(t)]·MRineq.(29)representsthechargecurrentdriven bytheeffectiveelectricfield(i.e.,thespinelectricfield),asment ionedin§3.3.Inotherwords, theeffectiveelectrochemicalpotentialofthefreelayerischang ed bythedynamicsof ML(t), andtheresultantdi fferenceinelectrochemicalpotentialsbetweenthetwolayer smanifestsas a bias voltage.31This situation may be realized when a barrier exists between the electrode and lead, or when the di ffusion constant of the free layer is small enough to maintain t he changed effective chemical potential. Otherwise, this charge current will flow back to the reservoirconnected to thefree layerwithouttunnelingthr oughthebarrier oftheMTJ. 7/13J.Phys. Soc. Jpn. FULLPAPERS 3.2 Spincurrent The term ML(t)×MRin eq. (30) represents the static e ffective Heisenberg coupling be- tweenML(t)andMR.Thatis,theequationofmotionfor ML(t)describedbythisspincurrent corresponds to the equationdML(t) dt=Jeff /planckover2pi1|SL(t)|ML(t)×MR[SL(t) is defined by eq. (42)]. This affordsaHeisenberg couplingenergy of −JeffML(t)·MR, where Jeff:=−4π|TLR|2/integraldisplayµ dE/bracketleftbig∆ρL(E)∆χR(E)+∆χL(E)∆ρR(E)/bracketrightbig(35) =1 π/integraldisplayµ dEℑGr↑ LR(E)∆R(E)Gr↓ RL(E)∆L(E), (36) Grσ ij(E) :=gr iσ(E)Tijgr jσ(E), (37) ∆i(E) :=gr i↑(E)−1−gr i↓(E)−1. (38) ∆i(E) describes the exchange splitting at site i, and this result agrees with the expression presentedby Liechtenstein et al.39 Let usconsidertheterm ML×dML(t) dtineq. (30): 2π|TLR|2¯ρL(µ)¯ρR(µ)ML(t)×dML(t) dt=/planckover2pi1 2e2¯ΓML(t)×dML(t) dt, (39) wheree>0is theelementary charge and ¯ΓisthetunnelconductanceoftheMTJ, ¯Γ:=4π|TLR|2e2 /planckover2pi1¯ρL(µ)¯ρR(µ). (40) This term describes the spin pumping in the MTJ and a ffords the following microscopic expressionfortheenhanced Gilbertdampingconstant: ∆α=/planckover2pi1 2e2¯Γ |SL(t)|, (41) whereSL(t)isthetotalspinpolarizationoftheelectrons intheLHSla yer, SL(t) :=2/summationdisplay i∈LHS/integraldisplayµ dE∆ρi(E)ML(t). (42) Equation (41) agrees with the corrected Gilbert damping con stant derived by Zhang et al.28 phenomenologically after considering the e ffect of the spin electric field induced by the dy- namic magnetization. In addition, in the present formulati on, from the fact that ∆αvanishes if one ignores Berry’s phase (6),40it follows that one of the origins of spin pumping is the spinelectric field. Asaconsequenceofthis, ∆αis proportionalto theconductance ¯Γ. Thesizedependenceof ∆αcan bedescribed as follows: ∆α∝1 λ, (43) whereλisthicknessofthefreelayer,because |SL(t)|isroughlyproportionaltothevolumeof 8/13J.Phys. Soc. Jpn. FULLPAPERS thefree layer, and ¯Γtothecross-sectionalarea ofthebarrier. 3.3 Analysisofeffectivefield For a more transparent physical interpretation of the curre nts, we rewrite eqs. (29) and (30)as follows: −eIe(t)=/summationdisplay σ=±1/bracketleftBig ΓR σε1 σ(t)+γL σε2 σ(t)/bracketrightBig ·MR, (44) −eIs(t)=/bracketleftBiggeJeff /planckover2pi1ML(t)−∆Γ/planckover2pi1 2edML(t) dt/bracketrightBigg ×MR +/summationdisplay σ=±1σ/braceleftBig ΓR σε1 σ(t)+/bracketleftBig γL σε2 σ(t)·MR/bracketrightBig ML(t)−/bracketleftBig γR σ+ML(t)·MRγL σ/bracketrightBig ε2 σ(t)/bracerightBig ,(45) wherethe“conductances” aredefined by ΓR σ:=2π|TLR|2e2¯ρL(µ)ρRσ(µ) /planckover2pi1, (46) ∆Γ:=4π|TLR|2e2∆ρL(µ)∆ρR(µ) /planckover2pi1, (47) γL σ:=−2π|TLR|2e2 /planckover2pi1/integraldisplayµ dE/bracketleftBigg ρLσ(E)d∆χR(E) dE−dχLσ(E) dE∆ρR(E)/bracketrightBigg , (48) γR σ:=−2π|TLR|2e2 /planckover2pi1/integraldisplayµ dE/bracketleftBigg ρRσ(E)d∆χL(E) dE−dχRσ(E) dE∆ρL(E)/bracketrightBigg , (49) andtheeffectivedrivingfields can bedefined by ε1 σ(t) :=−σ/planckover2pi1 2eML(t)×dML(t) dt, (50) ε2 σ(t) :=−σ/planckover2pi1 2edML(t) dt. (51) The conductances represented by a capital letter denote the “Fermi surface terms,” whereas thoserepresentedbyasmallletterdenotethe“Fermiseater ms.”Thespin-dependente ffective voltageε1 σ(t)·MRin eq. (44) just corresponds to the spin electric field betwee n the layers. To compare the expressionsobtained in continuous space and in discrete space, let us define the correspondences M(r,t) :=ML(t) andM(r+∆r,t) :=MR, where∆rdenotes the barrier thickness.Thenwefind ε1 σ(t)·MR≃∆ri/parenleftBig −σ/planckover2pi1 2e/parenrightBig∂M(r,t) ∂t×∂M(r,t) ∂xi·M(r,t),whichiswell-knownas thespinelectricfield.When ML(t)steadilyprecessaboutthedirectionof MRwithaconstant coneangleθand aconstantfrequency ω,thevoltageis time-independent: ε1 σ(t)·MR=−σ/planckover2pi1ω 2esin2θ, (52) Thisaffords an estimate/planckover2pi1ω/2e∼20µV at 10 GHz. The Fermi sea term in eq. (44) vanishes in this case. This result is in good agreement with that of Xia oet al.30and Tserkovnyak et 9/13J.Phys. Soc. Jpn. FULLPAPERS al.31Notethatin general theFermisea termiscertainly theaccur rent. Next, let us consider the spin current (45). The terms includ ingΓR σε1 σ(t)+/bracketleftBig γL σε2 σ(t)· MR/bracketrightBig ML(t) describe the spin transport due to the spin σcomponent of the charge current. By consideringε2 σ(t) as a driving force, we can interpret the term/bracketleftBig γR σ+ML(t)·MRγL σ/bracketrightBig ε2 σ(t) asthe“tunnelingmagnetoresistance(TMR) e ffect”inspintransport. 3.4 Effects ofbiasvoltage Finally, we consider the charge and spin transport in the pre sence of a bias voltage V(t) across the MTJ. In Berry’s adiabatic approximation under th e assumption JL≫e|V(t)|, the effectsofV(t)can beincludedbyreplacing eq. (4)with ci(t)≃e−ie /planckover2pi1/integraltext dtV(t)ˆUL(t)eiγ(t)ˆσzdifori∈LHS. (53) Inthefirst orderindV(t) dt, theeffectiveexchangeconstantand conductancesdi fferas follows: Jeff→Jeff+(γL ↑−γL ↓)/planckover2pi1 eV(t)+∆Γ/planckover2pi12 2ed dµln/bracketleftBigg∆ρL(µ) ∆ρR(µ)/bracketrightBiggdV(t) dt, (54) ΓR σ→ΓR σ1−d dµln/bracketleftBigg¯ρL(µ) ρRσ(µ)/bracketrightBigg eV(t)−/integraldisplayµ dE¯ρL(E)d3χRσ(E) dE3−d3¯χL(E) dE3ρRσ(E) ¯ρL(µ)ρRσ(µ)e/planckover2pi1 2dV(t) dt,(55) ∆Γ→∆Γ1−d dµln/bracketleftBigg∆ρL(µ) ∆ρR(µ)/bracketrightBigg eV(t)−/integraldisplayµ dE∆ρL(E)d3∆χR(E) dE3−d3∆χL(E) dE3∆ρR(E) ∆ρL(µ)∆ρR(µ)e/planckover2pi1 2dV(t) dt, (56) γL σ→γL σ−2π|TLR|2e2 /planckover2pi1/integraldisplayµ dE/bracketleftBigg ρLσ(E)d2∆χR(E) dE2+d2χLσ(E) dE2∆ρR(E)/bracketrightBigg eV(t) +2π|TLR|2e2 /planckover2pi1/bracketleftBiggdρLσ(µ) dµd∆ρR(µ) dµ−d2ρLσ(µ) dµ2∆ρR(µ)−ρLσ(µ)d2∆ρR(µ) dµ2/bracketrightBigge/planckover2pi1 2dV(t) dt,(57) γR σ→γR σ−2π|TLR|2e2 /planckover2pi1/integraldisplayµ dE/bracketleftBigg ρRσ(E)d2∆χL(E) dE2+d2χRσ(E) dE2∆ρL(E)/bracketrightBigg eV(t) +2π|TLR|2e2 /planckover2pi1/bracketleftBiggdρRσ(µ) dµd∆ρL(µ) dµ−d2ρRσ(µ) dµ2∆ρL(µ)−ρRσ(µ)d2∆ρL(µ) dµ2/bracketrightBigge/planckover2pi1 2dV(t) dt.(58) In addition,atermdescribingtheTMRe ffect, 1 e/bracketleftBig¯Γ+∆ΓML(t)·MR/bracketrightBig V(t)+1 −e/bracketleftbig¯γ+∆γML(t)·MR/bracketrightbig/planckover2pi1 2dV(t) dt(59) appears inthechargecurrent, where ¯γ:=4π|TLR|2e2 /planckover2pi1/integraldisplayµ dE/bracketleftBigg ¯ρL(E)d2¯χR(E) dE2+d2¯χL(E) dE2¯ρR(E)/bracketrightBigg , ∆γ:=4π|TLR|2e2 /planckover2pi1/integraldisplayµ dE/bracketleftBigg ∆ρL(E)d2∆χR(E) dE2+d2∆χL(E) dE2∆ρR(E)/bracketrightBigg . 10/13J.Phys. Soc. Jpn. FULLPAPERS Forthespincurrent, aterm describingtheSTT e ffect, 1 e/bracketleftBigg (ΓL ↑−ΓL ↓)V(t)−(γR ↑+γR ↓)/planckover2pi1 2edV(t) dt/bracketrightBigg ML(t)+1 e/bracketleftBigg (ΓR ↑−ΓR ↓)V(t)+(γL ↑+γL ↓)/planckover2pi1 2edV(t) dt/bracketrightBigg MR isadded, where ΓL σ:=2π|TLR|2e2ρLσ(µ)¯ρR(µ) /planckover2pi1. (60) ThenfortheGilbertdamping,since ¯Γ=ΓR ↑+ΓR ↓,∆αchanges as follows: ∆α→∆α1−d dµln/braceleftBigg¯ρL(µ) ¯ρR(µ)/bracerightBigg eV(t)−/integraldisplayµ dE¯ρL(E)d3¯χR(E) dE3−d3¯χL(E) dE3¯ρR(E) ¯ρL(µ)¯ρR(µ)e/planckover2pi1 2dV(t) dt.(61) Thisresultindicatesthat when writingdatato an STT MRAM ce ll,thedampingofthemag- netization dynamics is influenced by not only the spin pumpin g but also the bias voltage. However, the effect of the bias voltage on ∆αvanishes when both electrodes have the same electronicstructure. In summary, we derived, at the microscopic level, the charge and spin currents passing through an MTJ in response to arbitrary motion of the magneti zation in the free layer. The chargecurrentconsistsofbothFermisurfaceandFermiseat erms.TheFermisurfacetermis driven by the spin electric field and manifests as a dc current for steady precession of ML(t) in the direction of MR, whereas the Fermi sea term is due to virtual transitions and essen- tially manifests as the ac current. With regard to spin trans port, we focused particularly on theenhancedGilbertdamping(orthespinpumpinge ffect)andthusobtainedthemicroscopic expression for the enhanced Gilbert damping constant ∆α=/planckover2pi1 2e2¯Γ |SL(t)|. Under a bias voltage, the DOSs of the two layers in the MTJ are shifted. Thus, the bia s voltage changes the ef- fective exchange constant and the conductances, thus produ cing modulation of ∆α. All the conductances consist of the tunneling amplitude TLRand the local DOS on the surfaces of the layers; the real part of a retarded Green’s function can b e obtained from the imaginary part (namely, the local DOS) via the Kramers–Kronig relatio nship. In this formulation, the properties of the barrier layer material are considered in t he local DOS, which can be easily obtainedbyfirst-principlescalculations. 11/13J.Phys. Soc. Jpn. FULLPAPERS References 1) T.Miyazaki andN.Tezuka: J. Magn.Magn.Mater. 139(L231)1995. 2) J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey: Phy s. Rev. Lett. 74(1995) 3273. 3) S. Yuasa, A. Fukushima, T. Nagahama, K. Ando, and Y. Suzuki : Jpn. J. Appl. Phys. 43 (2004)L588. 4) S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando : Nat. Mater. 3(2004) 862. 5) S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughe s, M. Samant, and S.-H. Yang:Nat. Mater. 3(2004)862. 6) R. Scheuerlein, W. Gallagher, S. Parkin, A. Lee, S. Ray, R. Robertazzi, and W. Reohr: Solid-State Circuits Conference, 2000. Digest of Technica l Papers. ISSCC. 2000 IEEE International,2000,pp.128 –129. 7) J. C. Slonczewski:J.Magn.Magn. Mater. 159(1996)L1. 8) L. Berger: Phys.Rev.B 54(1996)9353. 9) F. J. Albert, J. A. Katine, R. A. Buhrman, and D. C. Ralph: Ap pl. Phys. Lett. 77(2000) 3809. 10) Y. Huai, D. Apalkov, Z. Diao, Y. Ding, A. Panchula, M. Paka la, L.-C. Wang, and E. Chen: Jpn.J.Appl.Phys. 45(2006)3835. 11) J. Sun:J. Magn.Magn.Mater. 202(1999)157 . 12) Y. Liu,Z. Zhang,P. P. Freitas, and J. L.Martins:Appl.Ph ys.Lett.82(2003)2871. 13) Y. Huai,M. Pakala, Z.Diao, and Y. Ding:Appl.Phys. Lett. 87(2005)222510. 14) J. Miltat,G. Albuquerque,A.Thiaville,and C. Vouille: J.Appl.Phys. 89(2001)6982. 15) Y.Liu,Z.Zhang,J.Wang,P.P.Freitas,andJ.L.Martins: J.Appl.Phys. 93(2003)8385. 16) J. C. Slonczewski:Phys.Rev.B 71(2005)024411. 17) Y. Zhang,Z. Zhang,Y. Liu,B. Ma,and Q. Y. Jin:J.Appl.Phy s.99(2006)08G515. 18) Y. Zhang,Z. Zhang,Y. Liu,B. Ma,and Q. Y. Jin:Appl.Phys. Lett.90(2007)112504. 19) S. Mizukami,Y.Ando, andT. Miyazaki:Jpn.J.Appl.Phys. 40(2001)580. 20) Y. Tserkovnyak,A. Brataas, and G. E.W. Bauer: Phys.Rev. Lett.88(2002)117601. 21) Y. Tserkovnyak,A. Brataas, and G. E.W. Bauer: Phys.Rev. B66(2002)224403. 12/13J.Phys. Soc. Jpn. FULLPAPERS 22) N. Umetsu,D. Miura, andA. Sakuma: J.Phys.:Conf. Ser. 266(2011)012084. 23) N. Umetsu,D. Miura, andA. Sakuma: J.Appl.Phys. 111(2012)07D117. 24) G. E.Volovik:J. Phys.C 20(1987)L83. 25) P.-Q. Jin,Y.-Q. Li, andF.-C. Zhang:J. Phys.A 39(2006)7115. 26) J. Oheand S. Maekawa: J.Appl.Phys. 105(2009)07C706. 27) J. Ohe,S. E.Barnes, H.-W.Lee, andS. Maekawa:Appl.Phys .Lett.95(2009)123110. 28) S. Zhang andS. S.-L. Zhang:Phys.Rev.Lett. 102(2009)086601. 29) T.Moriyama,R.Cao,X.Fan,G.Xuan,B.K.Nikoli´ c,Y.Tse rkovnyak,J.Kolodzey,and J. Q.Xiao: Phys.Rev.Lett. 100(2008)067602. 30) J. Xiao,G. E.W. Bauer, and A.Brataas: Phys.Rev.B 77(2008)180407. 31) Y. Tserkovnyak,T.Moriyama,and J.Q. Xiao:Phys.Rev. B 78(2008)020401. 32) S.-H.Chen,C.-R.Chang,J.Q.Xiao,andB.K.Nikoli´ c:Ph ys.Rev.B 79(2009)054424. 33) H. Kohno,G. Tatara, and J. Shibata:J. Phys.Soc. Jpn. 75(2006)113706. 34) A. Takeuchiand G. Tatara: J. Phys.Soc. Jpn. 77(2008)074701. 35) A. Takeuchi,K. Hosono,and G.Tatara: Phys.Rev. B 81(2010)144405. 36) H.HaugandA.-P.Jauho: QuantumKineticsinTransportandOpticsofSemiconductors (Springer-Verlag, 2008)second,substantiallyreviseded . 37) J. Rammer: Quantum Field Theory of Non-equilibrium States (Cambridge University Press, 2007). 38) M.V. Berry: Proc. R. Soc. London,Ser. A 392(1984)45. 39) A.I.Liechtenstein,M.I.Katsnelson,V.P.Antropov,an dV.A.Gubanov:J.Magn.Magn. Mater.67(1987)65 . 40) D. Miuraand A. Sakuma:J. Appl.Phys. 109(2011)07C909. 13/13

2011-11-18

We study the charge and spin currents passing through a magnetic tunnel junction (MTJ) on the basis of a tight-binding model. The currents are evaluated perturbatively with respect to the tunnel Hamiltonian. The charge current has the form $A[\bm M_1(t)\times\dot{\bm M}_1(t)]\cdot\bm M_2+B\dot{\bm M}_1(t)\cdot\bm M_2$, where $\bm M_1(t)$ and $\bm M_2$ denote the directions of the magnetization in the free layer and fixed layer, respectively. The constant $A$ vanishes when one or both layers are insulators, {while the constant $B$ disappears when both layers are insulators or the same ferromagnets.} The first term in the expression for charge current represents dissipation driven by the effective electric field induced by the dynamic magnetization. In addition, from an investigation of the spin current, we obtain the microscopic expression for the enhanced Gilbert damping constant $\varDelta \alpha$. We show that $\varDelta\alpha$ is proportional to the tunnel conductance and depends on the bias voltage.

Charge and Spin Transport in Magnetic Tunnel Junctions: Microscopic Theory

1111.4295v2

arXiv:1111.4655v1 [math.OC] 20 Nov 2011NULL CONTROLLABILITY OF THE STRUCTURALLY DAMPED WAVE EQUATION WITH MOVING POINT CONTROL PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON Abstract. We investigate the internal controllability of the wave equ ation with structural damping on the one dimensional torus. We assume that the cont rol is acting on a moving point or on a moving small interval with a constant velocity. We pro ve that the null controllability holds in some suitable Sobolev space and after a fixed positiv e time independent of the initial conditions. 1.Introduction In this paper we consider the wave equation with structural damping1 ytt−yxx−εytxx= 0 (1.1) wheretis time,x∈T=R/(2πZ) is the space variable, and εis a small positive parameter corresponding to the strength of the structural damping. Th at equation has been proposed in [21] as an alternative model for the classical spring-mass-dam per PDE. We are interested in the control properties of ( 1.1). The exact controllability of ( 1.1) with an internal control function supported in the wholedomain was studied in [ 12,14]. With a boundary control, it was proved in [ 22] that (1.1) is not spectrally controllable (hence not null controllab le), but that some approximate controllability may be obtained in some ap propriate functional space. Thebadcontrol propertiesfrom( 1.1)come fromtheexistenceof afiniteaccumulation pointin the spectrum. Such a phenomenon was noticed first by D. Russel l in [25] for the beam equation with internal damping, by G. Leugering and E. J. P. G. Schmidt in [15] for the plate equation with internal damping, and by S. Micu in [ 19] for the linearized Benjamin-Bona-Mahony (BBM) equation yt+yx−ytxx= 0. (1.2) Even if the BBM equation arises in a quite different physical co ntext, its control properties share important common features with ( 1.1). Remind first that the full BBM equation yt+yx−ytxx+yyx= 0 (1.3) is a popular alternative to the Korteweg-de Vries (KdV) equa tion yt+yx+yxxx+yyx= 0 (1.4) as a model for the propagation of unidirectional small ampli tude long water waves in a uniform channel. ( 1.3) is often obtained from ( 1.4) in the derivation of the surface equation by noticing Key words and phrases. Structuraldamping; waveequation; nullcontrollability; Benjamin-Bona-Mahony equa- tion; Korteweg-de Vries equation; biorthogonal sequence; multiplier; sine-type function. 1The terminology internal damping is also used by some authors. 12 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON that, in the considered regime, yx∼ −yt, so thatyxxx∼ −ytxx. The dispersive term −ytxx has a strong smoothing effect, thanks to which the wellposedne ss theory of ( 1.3) is dramatically easier than for ( 1.4). On the other hand, the control properties of ( 1.2) or (1.3) are very bad (compared to those of ( 1.4), see [23]) precisely because of that term. It is by now classical that an “intermediate” equation between ( 1.3) and (1.4) can be derived from ( 1.3) by working in a moving frame x=ct,c∈R. Indeed, letting z(x,t) =y(x−ct,t) (1.5) we readily see that ( 1.3) is transformed into the following KdV-BBM equation zt+(c+1)zx−czxxx−ztxx+zzx= 0. (1.6) It is then reasonable to expect the control properties of ( 1.6) to be better than those of ( 1.3), thanks to the KdV term −czxxxin (1.6). In [24], it was proved that the equation ( 1.6) with a forcing term supported in (any given) subdomain is locally exactly controllable in H1(T) provided that T >(2π)/c. Going back to the original variables, it means that the equa tion yt+yx−ytxx+yyx=b(x+ct)h(x,t) (1.7) with a moving distributed control is exactly controllable i nH1(T) in (sufficiently) large time. Actually, this control time has to be chosen in such a way that the support of the control, which is moving at the constant velocity c, can visit all the domain T. The concept of moving point control was introduced by J. L. Li ons in [17] for the wave equation. One important motivation for this kind of control is that the exact controllability of the wave equation with a pointwise control and Dirichlet bou ndary conditions fails if the point is a zero of some eigenfunction of the Dirichlet Laplacian, whi le it holds when the point is moving under some (much more stable) conditions easy to check (see e .g. [2]). The controllability of the wave equation (resp. of the heat equation) with a moving p oint control was investigated in [17,9,2] (resp. in [ 10,4]). See also [ 27] for Maxwell’s equations. As the bad control properties of ( 1.1) come from the BBM term −εytxx, it is natural to ask whether better control properties for ( 1.1) could be obtained by using a moving control, as for the BBM equation in [ 24]. The aim of this paper is to investigate that issue. Throughout the paper, we will take ε= 1 for the sake of simplicity. All the results can be extended without difficulty to any ε>0. Letysolve ytt−yxx−ytxx=b(x+ct)h(x,t). (1.8) Thenv(x,t) =y(x−ct,t) fulfills vtt+(c2−1)vxx+2cvxt−vtxx−cvxxx=b(x)˜h(x,t) (1.9) where˜h(x,t) =h(x−ct,t). Furthermore the new initial condition read v(x,0) =y(x,0), vt(x,0) =−cyx(x,0)+yt(x,0). (1.10) As for the KdV-BBM equation, the appearance of a KdV term (nam ely−cvxxxin (1.9)) results in much better control properties. We shall see that (i) there is no accumulation point in the spectrum of the free evolution equation ( ˜h= 0 in (1.9));CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 3 (ii) the spectrum splits into one part of “parabolic” type, a nd another part of “hyperbolic” type. It follows that one can expect at most a null controllability result in large time. We will see that this is indeed the case. Throughout the paper, we assume that c=−1 for the sake of simplicity. Let us now state the main results of the paper. We shall denote by (y0,ξ0) an initial condition (taken in some appropriate space) decomposed in Fourier ser ies as y0(x) =/summationdisplay k∈Zckeikx, ξ0(x) =/summationdisplay k∈Zdkeikx. (1.11) We shall consider several control problems. The first one rea ds ytt−yxx−ytxx=b(x−t)h(t), x∈T,t>0, (1.12) y(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T (1.13) wherehis the scalar control. Theorem 1.1. Letb∈L2(T)be such that βk=/integraldisplay Tb(x)e−ikxdx/\e}atio\slash= 0fork/\e}atio\slash= 0, β0=/integraldisplay Tb(x)dx= 0. For any time T >2πand any (y0,ξ0)∈L2(T)2decomposed as in (1.11), if /summationdisplay k/\e}atio\slash=0|βk|−1(|k|6|ck|+|k|4|dk|)<∞andc0=d0= 0, (1.14) then there exists a control h∈L2(0,T)such that the solution of (1.12)-(1.13)satisfiesy(.,T) = yt(.,T) = 0. By Lemma 2.3(see below) there exist simple functions bsuch that |βk|decreases like 1 /|k|3, so that ( 1.14) holds for ( y0,ξ0)∈Hs+2(T)×Hs(T) withs>15/2. The second problem we consider is ytt−yxx−ytxx=b(x−t)h(x,t), x∈T,t>0, (1.15) y(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T, (1.16) wherethe control function his here allowed to dependalso on x. For that internal controllability problem, the following result will be established. Theorem 1.2. Letb=1ωwithωa nonempty open subset of T. Then for any time T >2π and any (y0,ξ0)∈Hs+2(T)×Hs(T)withs>15/2there exists a control h∈L2(T×(0,T))such that the solution of (1.15)-(1.16)satisfiesy(.,T) =yt(.,T) = 0. We now turn our attention to some internal controls acting on a single moving point. The first problem we consider reads ytt−yxx−ytxx=h(t)δt, x∈T, t>0, (1.17) y(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T, (1.18)4 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON whereδx0represents the Dirac measure at x=x0. We can as well replace δtbydδt dxin (1.17), which yields another control problem: ytt−yxx−ytxx=h(t)dδt dx, x∈T, t>0, (1.19) y(x,0) =y0(x), yt(x,0) =ξ0(x), x∈T. (1.20) Then we will obtain the following results. Theorem 1.3. For any time T >2πand any (y0,ξ0)∈Hs+2(T)×Hs(T)withs>9/2, there exists a control h∈L2(0,T)such that the solution of (1.17)-(1.18)satisfiesy(T,.)−[y(T,.)] = yt(T,.) = 0, where[f] = (2π)−1/integraltext2π 0f(x)dxis the mean value of f. Theorem 1.4. For any time T >2πand any (y0,ξ0)∈Hs+2(T)×Hs(T)withs >7/2and such that/integraltext Ty0(x)dx=/integraltext Tξ0(x)dx= 0, there exists a control h∈L2(0,T)such that the solution of(1.19)-(1.20)satisfiesy(T,.) =yt(T,.) = 0. The paper is organized as follows. Section 2is devoted to the proofs of the above theorems: in subsection 2.1we investigate the wellposedness and the spectrum of ( 1.9) forc=−1; in sub- section2.2the null controllability of ( 1.12)-(1.13), (1.17)-(1.18) and (1.19)-(1.20) are formulated as moment problems; Theorem 1.1is proved in subsection 2.4thanks to a suitable biorthogonal family which is shown to exist in Proposition 2.2; Theorem 1.2is deduced from Theorem 1.1 in subsection 2.5; finally, the proofs of Theorems 1.3and1.4, that are almost identical to the proof of Theorem 1.1, are sketched in subsection 2.6. The rather long proof of Proposition 2.2is postponed to Section 3. It combines different results of complex analysis about enti re functions of exponential type, sine-type functions, atomization of m easures, and Paley-Wiener theorem. 2.Proof of the main results 2.1.Spectral decomposition. The free evolution equation associated with ( 1.9) reads vtt−2vxt−vtxx+vxxx= 0. (2.1) Letvbe as in ( 2.1), and letw=vt. Then (2.1) may be written as /parenleftbigg v w/parenrightbigg t=A/parenleftbigg v w/parenrightbigg :=/parenleftbigg w 2wx+wxx−vxxx/parenrightbigg . (2.2) The eigenvalues of Aare obtained by solving the system /braceleftbigg w=λv, 2λvx+λvxx−vxxx=λ2v.(2.3) Expanding vas a Fourier series v=/summationtext k∈Zvkeikx, we see that ( 2.3) is satisfied provided that for eachk∈Z (λ2+(k2−2ik)λ−ik3)vk= 0. (2.4) Forvk/\e}atio\slash= 0, the only solution of ( 2.4) reads λ=λ± k=−(k2−2ik)±√ k4−4k2 2· (2.5)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 5 −180 −160 −140 −120 −100 −80 −60 −40 −20 0−10 −5 0510 Λk+ Λk− Λ2 Figure 1. Spectrum of ( 2.1) splits into a hyperbolic part (Λ+ kin blue), a para- bolic part (Λ− kin green) and a finite dimensional part (Λ 2in red). Note that λ± 0= 0, λ± 2=−2+2i, λ± −2=−2−2i while λ+ k/\e}atio\slash=λ− lfork,l∈Z\{0,±2}withk/\e}atio\slash=l. For|k| ≥3,λ± k=−k2±k2(1−2k−2+O(k−4)) 2+ik. Hence λ+ k=−1+ik+O(k−2) as|k| → ∞, (2.6) λ− k=−k2+1+ik+O(k−2) as|k| → ∞. (2.7) The spectrum Λ = {λ± k;k∈Z}may be split into Λ = Λ+∪Λ−∪Λ2where Λ+=/braceleftbig λ+ k;k∈Z\{0,±2}/bracerightbig , Λ−=/braceleftbig λ− k;k∈Z\{0,±2}/bracerightbig , Λ2={0,−2±2i} denote the hyperbolic part, the parabolic part, and the set of double eigenvalues, respectively. It is displayed on Figure 1. (See also [ 13] for a system whose spectrum may also be decomposed into a hyperbolic part and a parabolic part.) An eigenvector associated with the eigenvalue λ± k,k∈Z, is/parenleftbigg eikx λ± keikx/parenrightbigg , and the correspond- ing exponential solution of ( 2.1) reads v± k(x,t) =eλ± kteikx. Fork∈ {0,±2}, we denote λk=λ+ k=λ− k,vk(x,t) =eλkteikx, and introduce ˜vk(x,t) :=teλkteikx.6 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON Then we easily check that ˜ vksolves (2.1) and /parenleftbigg ˜vk ˜vkt/parenrightbigg (x,0) =/parenleftbigg0 eikx/parenrightbigg . Any solution of ( 2.1) may be expressed in terms of the v± k’s, thevk’s, and the ˜ vk’s. Introduce first the Hilbert space H=H1(T)×L2(T) endowed with the scalar product /a\}bracketle{t(v1,w1),(v2,w2)/a\}bracketri}htH=/integraldisplay T[(v1v2+v′ 1v′ 2)+w1w2]dx. Pick any/parenleftbigg v0 w0/parenrightbigg =/parenleftbigg/summationtext k∈Zckeikx /summationtext k∈Zdkeikx/parenrightbigg ∈H. (2.8) Fork∈Z\{0,±2}, we write /parenleftbigg ckeikx dkeikx/parenrightbigg =a+ k/parenleftbigg eikx λ+ keikx/parenrightbigg +a− k/parenleftbigg eikx λ− keikx/parenrightbigg (2.9) with a+ k=dk−λ− kck λ+ k−λ− k, (2.10) a− k=dk−λ+ kck λ− k−λ+ k· (2.11) Fork∈ {0,±2}, we write /parenleftbigg ckeikx dkeikx/parenrightbigg =ak/parenleftbigg eikx λkeikx/parenrightbigg +˜ak/parenleftbigg0 eikx/parenrightbigg (2.12) with ak=ck,˜ak=dk−λkck. (2.13) It follows that the solution ( v,w) of /parenleftbigg v w/parenrightbigg t=A/parenleftbigg v w/parenrightbigg ,/parenleftbigg v w/parenrightbigg (0) =/parenleftbigg v0 w0/parenrightbigg (2.14) may be decomposed as /parenleftbigg v(x,t) w(x,t)/parenrightbigg =/summationdisplay k∈Z\{0,±2}{a+ keλ+ kt/parenleftbigg eikx λ+ keikx/parenrightbigg +a− keλ− kt/parenleftbigg eikx λ− keikx/parenrightbigg } +/summationdisplay k∈{0,±2}{akeλkt/parenleftbigg eikx λkeikx/parenrightbigg +˜akeλkt/parenleftbigg teikx (1+λkt)eikx/parenrightbigg }. (2.15) Proposition 2.1. Assume that (v0,w0)∈Hs+1(T)×Hs(T)for somes≥0. Then the solution (v,w)of(2.14)satisfies (v,w)∈C([0,+∞);Hs+1(T)×Hs(T)).CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 7 Proof.Assume first that ( v0,w0)∈C∞(T)×C∞(T). Decompose ( v0,w0) as in (2.8), and leta± k fork∈Z\ {0,±2}, andak,˜akfork∈ {0,±2}, be as in ( 2.10)-(2.11) and (2.13), respectively. Then, from the classical Fourier definition of Sobolev space s, we have that ||(v0,w0)||Hs+1(T)×Hs(T)∼/parenleftigg/summationdisplay k∈Z(|k|2+1)s/parenleftbig (|k|2+1)|ck|2+|dk|2/parenrightbig/parenrightigg1 2 ∼ /summationdisplay k∈Z\{0,±2}|k|2s(k2|a+ k|2+k4|a− k|2)+/summationdisplay k∈{0,±2}(|ak|2+|˜ak|2) 1 2 . For the last equivalence of norms, we used ( 2.6)-(2.7) and (2.9)-(2.11). Since |eλ+ kt|+|eλ− kt| ≤Cfor|k|>2, t≥0, we infer that /summationdisplay k∈Z\{0,±2}|k|2s/parenleftig k2|a+ keλ+ kt|2+k4|a− keλ− kt|2/parenrightig ≤C/summationdisplay k∈Z\{0,±2}|k|2s/parenleftbig k2|a+ k|2+k4|a− k|2/parenrightbig <∞, hence ||(v,w)||L∞(R+, Hs+1(T)×Hs(T))≤C||(v0,w0)||Hs+1(T)×Hs(T)· (2.16) The result follows from ( 2.15) and (2.16) by a density argument. /square 2.2.Reduction to moment problems. 2.2.1.Internal control. We investigate the following control problem vtt−2vxt−vtxx+vxxx=b(x)h(t), (2.17) whereb∈L2(T), suppb⊂ω⊂Tandh∈L2(0,T). The adjoint equation to ( 2.17) reads ϕtt−2ϕxt+ϕtxx−ϕxxx= 0. (2.18) Note thatϕ(x,t) =v(2π−x,T−t) is a solution of ( 2.18) ifvis a solution of ( 2.17) forh≡0. Pick any (smooth enough) solutions vof (2.17) andϕof (2.18), respectively. Multiplying each term in ( 2.17) byϕand integrating by parts, we obtain /integraldisplay T[vtϕ+v(−ϕt+2ϕx−ϕxx)]/vextendsingle/vextendsingle/vextendsingle/vextendsingleT 0dx=/integraldisplayT 0/integraldisplay Thbϕdxdt. (2.19) Pick firstϕ(x,t) =eλ± −k(T−t)eikx=eλ± k(T−t)eikxfork∈Z. Then (2.19) may be written /a\}bracketle{tvt(T),eikx/a\}bracketri}ht+(λ± k−2ik+k2)/a\}bracketle{tv(T),eikx/a\}bracketri}ht−eλ± kTγ± k =/integraldisplayT 0h(t)eλ± k(T−t)dt/integraldisplay Tb(x)e−ikxdx, (2.20) where/a\}bracketle{t.,./a\}bracketri}htstands for the duality pairing /a\}bracketle{t.,./a\}bracketri}htD′(T),D(T), and γ± k=/a\}bracketle{tvt(0),eikx/a\}bracketri}ht+(λ± k−2ik+k2)/a\}bracketle{tv(0),eikx/a\}bracketri}ht.8 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON If we now pick ϕ(x,t) = (T−t)eλk(T−t)eikxfork∈ {0,±2}, then (2.19) yields /a\}bracketle{tv(T),eikx/a\}bracketri}ht−/braceleftig TeλkT/a\}bracketle{tvt(0),eikx/a\}bracketri}ht+[1+T(λk−2ik+k2)]eλkT/a\}bracketle{tv(0),eikx/a\}bracketri}ht/bracerightig =/integraldisplayT 0(T−t)h(t)eλk(T−t)dt/integraldisplay Tb(x)e−ikxdx k∈ {0,±2}.(2.21) Setβk=/integraltext Tb(x)e−ikxdxfork∈Z. The control problem can be reduced to a moment problem. Assume that there exists some function h∈L2(0,T) such that βk/integraldisplayT 0eλ± k(T−t)h(t)dt=−eλ± kTγ± k∀k∈Z, (2.22) βk/integraldisplayT 0(T−t)eλk(T−t)h(t)dt =−TeλkT/a\}bracketle{tvt(0),eikx/a\}bracketri}ht−[1+T(λk−2ik+k2)]eλkT/a\}bracketle{tv(0),eikx/a\}bracketri}ht ∀k∈ {0,±2}.(2.23) Then it follows from ( 2.20)-(2.23) that /a\}bracketle{tvt(T),eikx/a\}bracketri}ht+(λ± k−2ik+k2)/a\}bracketle{tv(T),eikx/a\}bracketri}ht= 0∀k∈Z, (2.24) /a\}bracketle{tv(T),eikx/a\}bracketri}ht= 0∀k∈ {0,±2}. (2.25) Sinceλ+ k/\e}atio\slash=λ− kfork∈Z\{0,±2}, this yields v(T) =vt(T) = 0. (2.26) 2.2.2.Point control. Let us consider first the control problem vtt−2vxt−vtxx+vxxx=h(t)dδ0 dx· (2.27) Then the right hand side of ( 2.19) is changed into/integraltextT 0h(t)/a\}bracketle{tdδ0 dx,ϕ/a\}bracketri}htdt. Forϕ(x,t) =eλ± k(T−t)eikx, we have /a\}bracketle{tdδ0 dx,ϕ/a\}bracketri}ht=−/a\}bracketle{tδ0,∂ϕ ∂x/a\}bracketri}ht=ikeλ± k(T−t) hence the right hand sides of ( 2.20) and (2.21) are changed into/integraltextT 0(ik)eλ± k(T−t)h(t)dtand/integraltextT 0(ik)(T−t)eλk(T−t)h(t)dt, respectively. Let βk=ikfork∈Z. Note that β0= 0 and that (2.20)-(2.21) fork= 0 read /a\}bracketle{tvt(T),1/a\}bracketri}ht−/a\}bracketle{tvt(0),1/a\}bracketri}ht= 0, (2.28) /a\}bracketle{tv(T),1/a\}bracketri}ht −T/a\}bracketle{tvt(0),1/a\}bracketri}ht −/a\}bracketle{tv(0),1/a\}bracketri}ht= 0. (2.29) Thus, the mean values of vandvtcannot be controlled. Let us formulate the moment problem to be solved. Assume that /a\}bracketle{tv(0),1/a\}bracketri}ht=/a\}bracketle{tvt(0),1/a\}bracketri}ht= 0, (2.30)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 9 and that there exists some h∈L2(0,T) such that ik/integraldisplayT 0eλ± k(T−t)h(t)dt=−eλ± kTγ± k∀k∈Z\{0}, (2.31) ik/integraldisplayT 0(T−t)eλk(T−t)h(t)dt =−TeλkT/a\}bracketle{tvt(0),eikx/a\}bracketri}ht−[1+T(λk−2ik+k2)]eλkT/a\}bracketle{tv(0),eikx/a\}bracketri}ht ∀k∈ {±2}.(2.32) Then we infer from ( 2.20)-(2.21) (with the new r.h.s.) and ( 2.28)-(2.32) that v(T) =vt(T) = 0. Finally, let us consider the control problem vtt−2vxt−vtxx+vxxx=h(t)δ0· (2.33) Then the computations above are valid with the new values of βkgiven by βk=/a\}bracketle{tδ0,eikx/a\}bracketri}ht= 1, k∈Z. It will be clear from the proof of Theorem 1.1that/a\}bracketle{tvt(T),1/a\}bracketri}htcan be controlled, while /a\}bracketle{tv(T),1/a\}bracketri}ht cannot. To establish Theorem 1.3, we shall have to find a control function h∈L2(0,T) such that /integraldisplayT 0eλ± k(T−t)h(t)dt=−eλ± kTγ± k∀k∈Z, (2.34) /integraldisplayT 0(T−t)eλk(T−t)h(t)dt =−TeλkT/a\}bracketle{tvt(0),eikx/a\}bracketri}ht−[1+T(λk−2ik+k2)]eλkT/a\}bracketle{tv(0),eikx/a\}bracketri}ht ∀k∈ {±2}.(2.35) 2.3.A Biorthogonal family. To solve the moments problems in the previous section, we nee d to construct a biorthogonal family to the functions eλ± kt,k∈Z, andteλkt,k∈ {±2}. More precisely, we shall prove the following10 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON Proposition 2.2. There exists a family {ψ± k}k∈Z\{0,±2}∪{ψk}k∈{0,±2}∪{˜ψk}k∈{±2}of functions inL2(−T/2,T/2)such that /integraldisplayT/2 −T/2ψ± k(t)eλ± ltdt=δl kδ− +k,l∈Z\{0,±2}, (2.36) /integraldisplayT/2 −T/2ψ± k(t)eλltdt=/integraldisplayT/2 −T/2ψ± k(t)teλptdt= 0k∈Z\{0,±2}, l∈ {0,±2}, p∈ {±2},(2.37) /integraldisplayT/2 −T/2ψl(t)eλ± ktdt=/integraldisplayT/2 −T/2˜ψp(t)eλ± ktdt= 0l∈ {0,±2}, k∈Z\{0,±2}, p∈ {±2},(2.38) /integraldisplayT/2 −T/2ψl(t)eλktdt=δk l,/integraldisplayT/2 −T/2ψl(t)teλptdt= 0l,k∈ {0,±2}, p∈ {±2}, (2.39) /integraldisplayT/2 −T/2˜ψp(t)eλktdt= 0,/integraldisplayT/2 −T/2˜ψp(t)teλqtdt=δq pp,q∈ {±2}, k∈ {0,±2}, (2.40) ||ψ+ k||L2(−T/2,T/2)≤C|k|4k∈Z\{0,±2}, (2.41) ||ψ− k||L2(−T/2,T/2)≤C|k|2e−T 2k2+2√ 2π|k|k∈Z\{0,±2}, (2.42) whereCdenotes some positive constant. In Proposition 2.2,δl kandδ− +denote Kronecker symbols ( δl k= 1 ifk=l, 0 otherwise, while δ− += 1 if we have the same signs in the l.h.s of ( 2.36), 0 otherwise). The proof of Proposition 2.2is postponed to Section 3. We assume Proposition 2.2true for the time being and proceed to the proofs of the main results of the paper. 2.4.Proof of Theorem 1.1.Pick any pair ( y0,ξ0)∈L2(T)2fulfilling ( 1.14). From ( 1.10) with c=−1, we have that v(0) =y0,vt(0) =dy0 dx+ξ0, so that γ± k=/a\}bracketle{tdy0 dx+ξ0,eikx/a\}bracketri}ht+(λ± k−2ik+k2)/a\}bracketle{ty0,eikx/a\}bracketri}ht, =/a\}bracketle{tξ0,eikx/a\}bracketri}ht+(λ± k−ik+k2)/a\}bracketle{ty0,eikx/a\}bracketri}ht, k∈Z. Let γk=γ± kfork∈ {0,±2}. The result will be proved if we can construct a control functi onh∈L2(0,T) fulfilling ( 2.22)- (2.23). Let us introduce the numbers α± k=−β−1 keλ± kT 2γ± k, k∈Z\{0,±2}, αk=−β−1 keλkT 2γk, k∈ {±2}, ˜αk=−β−1 k/parenleftbiggT 2eλkT 2γk+eλkT 2/a\}bracketle{ty0,eikx/a\}bracketri}ht/parenrightbigg , k∈ {±2}, and ψ(t) =/summationdisplay k∈Z\{0,±2}α+ kψ+ k(t)+/summationdisplay k∈Z\{0,±2}α− kψ− k(t)+/summationdisplay k∈{±2}[αkψk(t)+ ˜αk˜ψk(t)].CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 11 Finally leth(t) =ψ(T 2−t). Note that h∈L2(0,T) with ||h||L2(0,T)=||ψ||L2(−T 2,T 2) ≤C /summationdisplay k∈{±2}(|dk|+|ck|)+/summationdisplay k∈Z\{0,±2}|βk|−1(|dk|+|k|2|ck|)|k|4 +/summationdisplay k∈Z\{0,±2}|βk|−1(|dk|+|ck|)|k|2e−T|k|2+2√ 2π|k| <∞, by (1.14). Then it follows from ( 1.14) and (2.36)-(2.40) that fork∈Z\{0,±2} βk/integraldisplayT 0eλ± k(T−t)h(t)dt=βkeλ± kT 2/integraldisplayT/2 −T/2eλ± kτψ(τ)dτ=βkeλ± kT 2α± k=−eλ± kTγ± k. and also that βk/integraldisplayT 0eλk(T−t)h(t)dt=−eλkTγkfork∈ {0,±2}, βk/integraldisplayT 0(T−t)eλk(T−t)h(t)dt=−TeλkTγk−eλkT/a\}bracketle{ty0,eikx/a\}bracketri}htfork∈ {0,±2}, as desired. /square 2.5.Proof of Theorem 1.2.Setǫ= (T−2π)/2,v(x,t) =y(x+t,t) andξ(x,t) =yt(x,t). We first steer to 0 the components of vandvtalong the mode associated to the double eigenvalue λ0= 0. Denote γ(t) =/integraltext Tv(x,t)dxandη(t) =/integraltext Tvt(x,t)dx. According to ( 1.11),γ(0) = 2πc0, η(0) = 2πd0and dγ dt=η,dη dt=/integraldisplay ω˜h(x,t)dx. Take aC∞scalar function ̟(t) on [0,ǫ] with̟(0) = 1 and ̟(ǫ) = 0 and such that the support ofd̟/dtlies inside [0 ,ǫ]. Consider another C∞function of x,¯b(x) with support inside ωand such that/integraltext ω¯b(x)dx= 1. Then the C∞control ˜h(x,t) =¯b(x)¯h(t) with¯h(t) =d2 dt2((c0+d0t)̟(t)) steers (γ,η) from (c0,d0) at timet= 0 to (0,0) at timet=ǫ. Its support lies inside [0 ,ǫ]. Since γ(ǫ) =/integraltext Ty(x,ǫ)dxandη(ǫ) =/integraltext Tξ(x,ǫ)dx, we can assume that c0=d0= 0 up to a time shift ofǫ. Sinceωis open and nonempty, it contains a small interval [ a,a+ 2σπ] whereσ >0 is a quadratic irrational ; i.e., an irrational number which is a root of a quadratic equ ation with integral coefficients. Set for t∈[ǫ,T] h(x,t) =/parenleftbig 1[a,a+σπ](x−t)−1[a+σπ,a+2σπ](x−t)/parenrightbig/tildewideh(t)12 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON where/tildewidehdenotes a control input independent of x. Thenb(x−t)h(x,t) =/tildewideb(x−t)/tildewideh(t) where /tildewideb(x) =1[a,a+σπ](x)−1[a+σπ,a+2σπ](x) satisfies/integraltext T/tildewideb(x)dx= 0. Moreover there exists by Lemma 2.3(see below) a number C >0 such that for all k∈Z∗ /tildewideβk=/integraldisplay T/tildewideb(x)e−ikxdx≥C |k|3. According to Theorem 1.1we can find/tildewideh∈L2(ǫ,T) steeringy(.,ǫ) andξ(.,ǫ) toy(.,T) = ξ(.,T) = 0 as soon as /summationdisplay k/\e}atio\slash=0k6|/tildewideck|+k4|/tildewidedk| |/tildewideβk|<∞, with y(x,ǫ) =/summationdisplay k∈Z/tildewideckeikx, ξ(x,ǫ) =/summationdisplay k∈Z/tildewidedkeikx. LetWdenote the space of the couples (ˆ y,ˆξ)∈L2(T)2such that ||(ˆy,ˆξ)||W:=|ˆc0|+|ˆd0|+/summationtext k/\e}atio\slash=0(|k|9|ˆck|+|k|7|ˆdk|)<∞, where ˆy(x) =/summationtext k∈Zˆckeikxandˆξ(x) =/summationtext k∈Zˆdkeikx. Clearly,W endowed with the norm ||·||W, is a Banach space. Standard estimations based on the spectr al decomposition used to prove Proposition 2.1show that if the initial value ( y0,ξ0) lies inW, then the solution of ( 1.12)-(1.13) (withh≡0) remains in W. Therefore, since/summationtext k/\e}atio\slash=0(|k|9|ck|+ |k|7|dk|)<∞and since the control is C∞with respect to x∈Tandt∈[0,ǫ], we also have/summationtext k/\e}atio\slash=0|k|9(|/tildewideck|+|k|7|/tildewidedk|)<∞(see e.g. [ 5,20]). Since (y0,ξ0)∈Hs+2(T)×Hs(T) withs>15/2, we have by Cauchy-Schwarz inequality for ς= 2s−15>0 that /summationdisplay k/\e}atio\slash=0(|k|9|ck|+|k|7|dk|)≤2(/summationdisplay k/\e}atio\slash=0|k|−1−ς)1 2(/summationdisplay k/\e}atio\slash=0|k|19+ς|ck|2+|k|15+ς|dk|2)1 2<∞./square Lemma 2.3. Letσ∈(0,1)be a quadratic irrational, and let ˜b,˜βkbe defined as above. Then ˜β0= 0and there exists C >0such that for all k∈Z∗,|˜βk| ≥C |k|3. Proof.Being a quadratic irrational, σis approximable by rational numbers to order 2 and to no higher order [ 8, Theorem 188]); i.e., there exists C0>0 such that for any integers pandq, q/\e}atio\slash= 0,/vextendsingle/vextendsingle/vextendsingleσ−p q/vextendsingle/vextendsingle/vextendsingle≥C0 q2. On the other hand, |˜βk|=4 |k|sin2(π 2kσ) fork/\e}atio\slash= 0. Pick any k/\e}atio\slash= 0, take p∈Zsuch that 0 ≤π 2kσ−pπ < πand use the elementary inequality sin2θ≥4θ2 π2valid for θ∈[−π 2,π 2]. Then two cases occur. (i) If 0≤π 2kσ−pπ≤π 2, then sin2(π 2kσ) = sin2(π 2kσ−pπ)≥4 π2(π 2kσ−pπ)2=k2/parenleftig σ−2p k/parenrightig2 ≥C2 0 k2; (ii) If−π 2≤π 2kσ−(p+1)π≤0, then sin2(π 2kσ−(p+1)π)≥4 π2(π 2kσ−(p+1)π)2=k2/parenleftig σ−2(p+1) k/parenrightig2 ≥C2 0 k2. The lemma follows with C= 4C2 0. /squareCONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 13 2.6.Proofs of Theorem 1.3and Theorem 1.4.The proofs are the same as for Theorem 1.1, with the obvious estimate /summationdisplay |k|>2|k|p(|k|2|ck|+|dk|)≤Cε /summationdisplay |k|>2{|k|2p+5+ε|ck|2+|k|2p+1+ε|dk|2 1 2 forp∈ {3,4},ε>0. /square 3.Proof of Proposition 2.2 This section is devoted to the proof of Proposition 2.2. The method of proof is inspired from the one in [ 6,7,18]. We first introduce an entire function vanishing precisely at theiλ± k’s, namely the canonical product P(z) =z(1−z iλ2)(1−z iλ−2)/productdisplay k∈Z\{0,±2}(1−z iλ+ k)/productdisplay k∈Z\{0,±2}(1−z iλ− k)· (3.1) Next, following [ 1,7], we construct a multiplier mwhich is an entire function that does not vanish at the λ± k’s, such that P(z)m(z) is bounded for zreal whileP(z)m(z) has (at most) a polynomial growth in zas|z| → ∞on each line Im z=const.Next, fork∈Z\ {0,±2}we construct a function I± kfromP(z) andm(z) and we define ψ± kas the inverse Fourier transform ofI± k. The other ψk’s are constructed in a quite similar way. The fact that ψ± kis compactly supported in time is a consequence of Paley-Wiener theorem. 3.1.Functions of type sine. To estimate carefully P(z), we use the theory of functions of type sine (see e.g. [ 16, pp. 163–168] and [ 26, pp. 171–179]). Definition 3.1. An entire function f(z)of exponential type πis said to be of type sine if (i)The zerosµkoff(z)are separated; i.e., there exists η>0such that |µk−µl| ≥η k/\e}atio\slash=l; (ii)There exist positive constants A,BandHsuch that Aeπ|y|≤ |f(x+iy)| ≤Beπ|y|∀x∈R,∀y∈Rwith|y| ≥H. (3.2) Some of the most important properties of an entire function o f type sine are gathered in the following Proposition 3.2. (see[16, Remark and Lemma 2 p. 164] ,[26, Lemma 2 p. 172] ) Letf(z)be an entire function of type sine, and let {µk}k∈Jbe the sequence of its zeros, where J⊂Z. Then (1)For anyε>0, there exist some constants Cε,C′ ε>0such that Cεeπ|Imz|≤ |f(z)| ≤C′ εeπ|Imz|if dist{z,{µk}}>ε. (2)There exist some constants C1,C2such that 0<C1<|f′(µk)|<C2<∞ ∀k∈J. Finally, we shall need the following result.14 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON Theorem 3.3. (see[16, Corollary p. 168 and Theorem 2 p. 157] Letµk=k+dkfork∈Z, withµ0= 0,µk/\e}atio\slash= 0fork/\e}atio\slash= 0, and(dk)k∈Zbounded, and let f(z) =z/productdisplay k∈Z\{0}(1−z µk) = lim K→∞z/productdisplay k∈{−K,...,K}\{0}(1−z µk)· Thenfis a function of type sine if, and only if, the following three properties are satisfied: (1) inf k/\e}atio\slash=l|µk−µl|>0; (2)There exists some constant M >0such that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay k∈Z(dk+τ−dk)k k2+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤M∀τ∈Z; (3) limsup y→+∞log|f(iy)| y=π,limsup y→−∞log|f(−iy)| |y|=π. Corollary 3.4. Assume that µk=k+dk, whered0= 0anddk=d+O(k−1)as|k| → ∞for some constant d∈C, and thatµk/\e}atio\slash=µlfork/\e}atio\slash=l. Thenf(z) =z/producttext k∈Z\{0}(1−z µk)is an entire function of type sine. Proof.We check that the conditions (1), (2) and (3) in Theorem 3.3are fulfilled. (1) Fromµk−µl=k−l+O(k−1,l−1) and the fact that µk−µl/\e}atio\slash= 0 fork/\e}atio\slash=l, we infer that (1) holds. (2) Let us write dk=d+ekwithek=O(k−1). Then for all τ∈Z /parenleftigg/summationdisplay k∈Z(dk+τ−dk)2/parenrightigg1 2 ≤2/parenleftigg/summationdisplay k∈Z|ek|2/parenrightigg1 2 <∞. Therefore, for any τ∈Z, by Cauchy-Schwarz inequality /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay k∈Z(dk+τ−dk)k k2+1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftigg/summationdisplay k∈Z(dk+τ−dk)2/parenrightigg1 2/parenleftigg/summationdisplay k∈Z(k k2+1)2/parenrightigg1 2 ≤2/parenleftigg/summationdisplay k∈Z|ek|2/parenrightigg1 2/parenleftigg/summationdisplay k∈Z(k k2+1)2/parenrightigg1 2 =:M <∞. (3) We first notice that f(z) =z∞/productdisplay k=1(1−z µk)(1−z µ−k)· Letz=iy, withy∈R. Then (1−z µk)(1−z µ−k) = 1−y2+i(µk+µ−k)y µkµ−k with µkµ−k=−k2+O(k), µk+µ−k= 2d+O(k−1).CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 15 It follows that for any given ε∈(0,1), there exist k0∈N∗and some numbers C1,C2>0 such that 1+(1−ε)y2−C1|y| |µkµ−k|≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1−z µk)(1−z µ−k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1+y2+C2|y| |µkµ−k|(3.3) fory∈Randk≥k0>0. Let n(r) := #{k∈N∗;|µkµ−k| ≤r}. Since|µkµ−k| ∼k2ask→ ∞andµk/\e}atio\slash= 0 fork/\e}atio\slash= 0, we obtain that √r−C3≤n(r)≤√r+C3forr>0, (3.4) n(r) = 0 for 0 <r<r 0, (3.5) for some constants C3>0,r0>0. It follows that limsup |y|→∞log|f(iy)| |y|≤limsup |y|→+∞|y|−1∞/summationdisplay k=1log/vextendsingle/vextendsingle/vextendsingle/vextendsingle(1−iy µk)(1−iy µ−k)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤limsup |y|→∞|y|−1∞/summationdisplay k=1log(1+y2+C2|y| |µkµ−k|), where we used the fact that lim |y|→∞|y|−1log/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−iy µ±k/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0 for 1 ≤k≤k0. On the other hand, setting ρ=y2+C2|y| ≥0, we have that ∞/summationdisplay k=1log(1+ρ |µkµ−k|) =/integraldisplay∞ 0log(1+ρ t)dn(t) =ρ/integraldisplay∞ 0n(t) t(t+ρ)dt =/integraldisplay∞ 0n(ρs) s(s+1)ds ≤√ρ/integraldisplay∞ 0ds√s(s+1)+C3/integraldisplay∞ r0/ρds s(s+1) ≤ |y|/radicalbig 1+C2|y|−1π+C3log/parenleftbig 1+r−1 0(y2+C2|y|)/parenrightbig . Thus limsup |y|→∞log|f(iy)| |y|≤π. Using again ( 3.3), we obtain by the same computations that limsup y→+∞log|f(iy)| y≥π,and limsup y→−∞log|f(iy)| |y|≥π. The proof of (3) is completed. /square16 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON In what follows, arg zdenotes the principal argument of any complex number z∈C\R−; i.e., argz∈(−π,π), and logz= log|z|+iargz,√z=/radicalbig |z|eiargz 2. We introduce, for k∈Z\{0}, µk= sgn(k)/radicalig −λ− k=k/radicaligg 1+√ 1−4k−2 2−ik−1=:k+dk, k∈Z with dk=−i 2+O(k−1). andµ0= 0. Let P1(z) =z/productdisplay k∈Z\{0}(1+z iλ+ k), (3.6) P2(z) =z/productdisplay k∈Z\{0}(1+z iλ− k), (3.7) P3(z) =z2/productdisplay k∈Z\{0}(1+z2 λ− k), (3.8) andP4(z) =z/productdisplay k∈Z\{0}(1−z µk). (3.9) It follows from ( 2.7) that the convergence in ( 3.7) is uniform in zon each compact set of C, so thatP2is an entire function. Note also that P2(z) =iP3(e−iπ 4√z), (3.10) P3(z) =−P4(z)P4(−z), (3.11) P(z) =P1(−z)P2(−z) z(1−z iλ2)(1−z iλ−2)· (3.12) Applying Corollary 3.4toP1, noticing that −iλ+ k=k+i+O(k−2) withλ+ k/\e}atio\slash=λ− lfork/\e}atio\slash=l, andλ+ 0= 0, we infer that P1(z) is an entire function of sine type. Thus, for givenε>0 there are some positive constants C4,C5,C6such that C4eπ|y|≤ |P1(x+iy)| ≤C5eπ|y|,dist (x+iy,{−iλ+ k})>ε (3.13) |P′ 1(−iλ+ k)| ≥C6, k∈Z. (3.14) Next, applying Corollary 3.4toP4, noticing that µk=k−i 2+O(k−1)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 17 withµk/\e}atio\slash=µlifk/\e}atio\slash=landµ0= 0, we infer that P4(z) is also an entire function of sine type. In particular, it is of exponential type π |P4(z)| ≤Ceπ|z|, z∈C. (3.15) Therefore, we have for any ε>0 and for some positive constants C7,C8,C9 C7eπ|y|≤ |P4(x+iy)| ≤C8eπ|y|,dist (x+iy,{µk})>ε (3.16) |P′ 4(µk)| ≥C9, k∈Z. (3.17) In particular, P3is an entire function of exponential type 2 πwith C2 7e2π|y|≤ |P3(x+iy)| ≤C2 8e2π|y|dist (±(x+iy),{µk})>ε. (3.18) Combined to ( 3.10), this yields |P2(z)| ≤Ce2π√ |z|z∈C. (3.19) Substituting e−iπ 4√ztox+iyin (3.18) yields C2 7exp(2π|Im(e−iπ 4√z)|)≤ |P2(z)| ≤C2 8exp(2π|Im(e−iπ 4√z)|) dist(±e−iπ 4√z,{µk})>ε. (3.20) From (3.20) (applied for xlarge enough) and the continuity of P2onC, we obtain that |P2(x)| ≤Ce√ 2π√ |x|. (3.21) We are now in a position to give bounds for the canonical produ ctPin (3.1). Proposition 3.5. The canonical product Pin(3.1)is an entire function of exponential type at mostπ. Moreover, we have for some constant C >0 |P(x)| ≤C(1+|x|)−3e√ 2π√ |x|, x∈R, (3.22) |P′(iλ+ k)| ≥C−1|k|−3e√ 2π√ |k|k∈Z\{0,±2}, (3.23) |P′(iλ− k)| ≥C−1|k|−7eπk2k∈Z\{0,±2}. (3.24) Proof.Note first that dist( R,{−iλ+ k;k/\e}atio\slash= 0})>0 from ( 2.5). Since (1 +is z)P1(z) is also an entire function of sine type for s≫1, with dist( R,{−iλ+ k;k/\e}atio\slash= 0} ∪ {is})>0, we infer from Proposition 3.2that for some constant C >0 |P1(x)| ≤C∀x∈R. Combined to ( 3.12) and (3.21), this yields ( 3.22). Let us turn to ( 3.23). Note first that for k∈Z\{0,±2} P′(iλ+ k) =P′ 1(−iλ+ k)P2(−iλ+ k) (−iλ+ k)(1−λ+ k λ2)(1−λ+ k λ−2)· (3.25) Clearly, for some δ>0,|λ+ k−λ− l|>δfor allk∈Z\{0,±2},l∈Z, and |Im (e−iπ 4/radicalig −iλ+ k)|=|Im/parenleftbig1−i√ 2/radicalbig k+i+O(k−2)/parenrightbig |=/radicalbigg |k| 2+O(|k|−1 2).18 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON With (3.20), this gives |P2(−iλ+ k)| ≥Ce√ 2π√ |k|. (3.26) It follows then from ( 3.14), (3.25), and (3.26) that |P′(iλ+ k)| ≥Ce√ 2π√ |k| |k|3 for some constant C >0 independent of k∈Z\{0,±2}. On the other hand P′(iλ− k) =P′ 2(−iλ− k)P1(−iλ− k) (−iλ− k)(1−λ− k λ2)(1−λ− k λ−2)· (3.27) By (2.7) and (3.13), we have that |P1(−iλ− k)| ≥Ceπk2, k∈Z\{0,±2}· From (3.10)-(3.11), we have that P′ 2(z) =eiπ 4 2√z/bracketleftig P′ 4(e−iπ 4√z)P4(−e−iπ 4√z)−P4(e−iπ 4√z)P′ 4(−e−iπ 4√z)/bracketrightig . Forz=−iλ− k,e−iπ 4√z=/radicalig −λ− k= sgn(k)µk, hence P′ 2(−iλ− k) =1 2µkP′ 4(µk)P4(−µk). Since|µk+µl|>δ>0 fork∈Z\{0}andl∈Z, we have from ( 3.16) that|P4(−µk)| ≥cwhile, by (3.17),|P′ 4(µk)|>c>0. It follows that for some constant C >0 |P′ 2(−iλ− k)| ≥C |k|∀k∈Z\{0}. Therefore, |P′(iλ− k)| ≥Ceπk2 |k|7, k∈Z\{0}. /square We seek for an entire function m(the so-called multiplier ) such that |m(x)| ≤C(1+|x|)e−√ 2π√ |x|, x∈R, |m(iλ+ k)| ≥C−1|k|−3e−√ 2π√ |k|, k∈Z\{0}, |m(iλ− k)| ≥C−1eaπk2−2√ 2π√ |k|, k∈Z\{0}. We shall use the same multiplier as in [ 7], providing additional estimates required to evaluate it at the points iλ− kfork∈Z. Let s(t) =at−b√ t, t> 0 (3.28)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 19 where the constants a>0 andb>0 will be chosen later. Note that sis increasing for t>/parenleftbigb 2a/parenrightbig2 and thats(B) = 0 where B= (b/a)2. Let ν(t) =/braceleftbigg 0t≤B, s(t)t≥B.(3.29) Introduce first g(z) =/integraldisplay∞ 0log(1−z2 t2)dν(t) =/integraldisplay∞ Blog(1−z2 t2)ds(t)z∈C\R,(3.30) U(z) =/integraldisplay∞ 0log|1−z2 t2|dν(t) =/integraldisplay∞ Blog|1−z2 t2|ds(t)z∈C. (3.31) Note thatgis holomorphic on C\RandUis continuous on C, withU(z) = Reg(z). Next we atomize the measure µin the above integrals, setting ˜g(z) =/integraldisplay∞ 0log(1−z2 t2)d[ν(t)]z∈C\R, (3.32) ˜U(z) =/integraldisplay∞ 0log|1−z2 t2|d[ν(t)]z∈C, (3.33) where [x] denotes the integral part of x. Again, ˜gis holomorphic on C\Rand˜Uis continuous onCwith˜U(z) = Reg(z). Actually, exp˜ gis an entire function. Indeed, if {τk}k≥0denotes the sequence of discontinuity points for t/ma√sto→[ν(t)], thenτk∼k/aask→ ∞and ˜g(z) =/summationdisplay k≥0log(1−z2 τ2 k), z∈C\R· (3.34) Therefore, e˜g(z)=/productdisplay k≥0(1−z2 τ2 k), (3.35) the product being uniformly convergent on any compact set in C. We shall pick later m(z) = exp(˜g(z−i)) witha=T 2π−1 andb=√ 2. The strategy, which goes back to [ 1], consists in estimating carefully U, and nextU−˜U. Let forx>0 w(x) =−π√x+xlog/vextendsingle/vextendsingle/vextendsingle/vextendsinglex+1 x−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle−√xlog/vextendsingle/vextendsingle/vextendsingle/vextendsingle√x+1√x−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle+2√xarctan(√x).(3.36) Note thatw∈L∞(R+), for lim x→∞w(x) =−2 andw(0+) = 0. Lemma 3.6. [7]It holds U(x)+bπ/radicalbig |x|=−aBw(|x|)∀x∈R (3.37) Our first aim is to extend that estimate to the whole domain C. Lemma 3.7. There exists some positive constant C=C(a,b)such that −C−bπ(1+1√ 2)/radicalbig |y| ≤U(z)+bπ/radicalbig |x|−aπ|y| ≤C, z =x+iy∈C.(3.38)20 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON Proof.We follow the same approach as in [ 7]. We first use the following identity from [ 7, (36)] (note thatUis even) U(z) =|Imz|(πa+1 π/integraldisplay∞ −∞U(t) |z−t|2dt). (3.39) To derive ( 3.38), it remains to estimate the integral term in ( 3.39) forz=x+iy∈C. We may assume without loss of generality that y>0. From Lemma 3.6, we can write U(t) =−bπ/radicalbig |t|−aBw(|t|) wherew∈L∞(R+). Then, with t=ys, /vextendsingle/vextendsingle/vextendsingle/vextendsingley π/integraldisplay∞ −∞aBw(|t|) (x−t)2+y2dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ ||w||L∞(R+)aBy π/integraldisplay∞ −∞ds y((x y−s)2+1) =aB||w||L∞(R+)=:C. (3.40) On the other hand, still with t=ys, and using explicit computations in [ 7] of some integral terms, y π/integraldisplay∞ −∞(−bπ)/radicalbig |t| (x−t)2+y2dt=−b√y/integraldisplay∞ −∞/radicalbig |s| (x y−s)2+1ds =−b√y π/radicalbigg 2/radicalig 1+x2 y2−2x y+π/radicalbigg 2/radicalig 1+x2 y2+2x y =−bπ√ 2(/radicalig/radicalbig x2+y2−x+/radicalig/radicalbig x2+y2+x). Routine computations give /radicalbig 2|x| ≤/radicalig/radicalbig x2+y2−x+/radicalig/radicalbig x2+y2+x≤/radicalbig 2|x|+(√ 2+1)√y∀x∈R,∀y>0. Therefore −bπ/radicalbig |x|−bπ(1+1√ 2)√y≤y π/integraldisplay∞ −∞(−bπ)/radicalbig |t| (x−t)2+y2dt≤ −bπ/radicalbig |x|. Combined to ( 3.39) and (3.40), this yields ( 3.38). /square In order to obtain estimates for ˜U(z), we need to give bounds from above and below for ˜U(z)−U(z) =/integraldisplay∞ 0log|1−z2 t2|d([ν](t)−ν(t)). We need the following lemma, which is inspired from [ 11, Vol. 2, Lemma p. 162] Lemma 3.8. Letν:R+→R+be nondecreasing and null on (0,B). Then for z=x+iywith y/\e}atio\slash= 0, we have −log+|x| |y|−log+x2+y2 B2−log2≤I=/integraldisplay∞ 0log|1−z2 t2|d([ν](t)−ν(t))≤log+|x| |y|·(3.41)CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 21 Proof.The proof of the upper bound is the same as in [ 11]. It is sketched here just for the sake of completeness. Pick any z=x+iywithy/\e}atio\slash= 0. Integrate by part in Ito get I=/integraldisplay∞ 0(ν(t)−[ν(t)])∂ ∂tlog|1−z2 t2|dt. Letζ=z2 t2. If Rez2≤0 (i.e. if |x| ≤ |y|), then the distance |1−ζ|is decreasing w.r.t. t (t∈(0,+∞)), so that I≤0. If Rez2>0, then|1−ζ|decreases to the minimal value|Imz2| |z2| taken att=t∗:=|z|2 |x|, and then it increases. Since 0 ≤ν(t)−[ν(t)]≤1, we have that I≤/integraldisplay∞ t∗∂ ∂tlog|1−z2 t2|dt= log|z2| |Imz2|= log(|x| 2|y|+|y| 2|x|)≤log|x| |y|· Let us pass to the lower bound. If Re z2≤0, I≥/integraldisplay∞ B∂ ∂tlog|1−z2 t2|dt=−log|1−z2 B2|· Assume now that Re z2>0. Ift∗=|z2| |x|≤B,I≥0. Ift∗>B, then I≥/integraldisplayt∗ B∂ ∂tlog|1−z2 t2|dt=−log|z2| |Imz2|−log|1−z2 B2|· Note that log|1−z2 B2| ≤log(1+|z B|2)≤log+x2+y2 B2+log2. Therefore I≥ −log+|x| |y|−log+x2+y2 B2−log2. /square Gathering together Lemma 3.7and Lemma 3.8, we obtain the Proposition 3.9. There exists some positive constant C=C(a,b)such that for any complex numberz=x+iywithy/\e}atio\slash= 0, −C−bπ(1+1√ 2)/radicalbig |y|−log+|x| |y|−log+(x2+y2 B2)−log2≤˜U(z)+bπ/radicalbig |x|−aπ|y| ≤C+log+|x| |y|. (3.42) Pick now a=T 2π−1>0, b=√ 2,andm(z) = exp˜g(z−i) (3.43) Note that |m(z)|= exp˜U(z−i). The needed estimates for the multiplier mare collected in the following22 PHILIPPE MARTIN, LIONEL ROSIER, AND PIERRE ROUCHON Proposition 3.10. mis an entire function on Cof exponential type at most aπ. Furthermore, the following estimates hold for some constant C >0: |m(x)| ≤C(1+|x|)e−√ 2π√ |x|, x∈R (3.44) |m(iλ+ k)| ≥C−1|k|−3e−√ 2π√ |k|, k∈Z\{0} (3.45) |m(iλ− k)| ≥C−1eaπk2−2√ 2π|k|, k∈Z\{0}. (3.46) Proof.(3.44) follows at once from ( 3.42) (withy=−1). We infer from ( 2.5) that fork∈Z\{0} Im (iλ± k)≤ −1 2. (3.47) It follows then from ( 2.6) and (3.42) that |m(iλ+ k)|= exp˜U(−k−2i(1+O(k−2)))≥C|k|−3e−√ 2π√ |k|(k/\e}atio\slash= 0). Finally, from ( 2.7) and (3.42), we infer that |m(iλ− k)|= exp˜U(−k−i(k2+O(k−2))) ≥Cexp(−√ 2π/radicalbig |k|+aπk2−(√ 2+1)π|k|−4log|k|) ≥Cexp(aπk2−2√ 2π|k|). /square We are in a position to define the functions in the biorthogona l family. Pick first any k∈ Z\{0,±2}, and set I± k(z) =P(z) P′(iλ± k)(z−iλ± k)·m(z) m(iλ± k)·(1−z iλ2)(1−z iλ−2) (1−λ± k λ2)(1−λ± k λ−2)· Clearly,I± kis an entire function of exponential type at most π(1+a) =T/2. Furthermore, we have that I± k(iλ± l) =δl kδ− +∀l∈Z, (3.48) whereδ− +is 1 if the two signs in the l.h.s. are the same, and 0 otherwise . Moreover, (I± k)′(iλ±2) = 0. (3.49) On the other hand, by ( 2.6), (3.22), (3.23), (3.44) and (3.45), we have that |I+ k(x)| ≤C|k|4 |x−iλ+ k|≤C|k|4 1+|k+x|· ThusI+ k∈L2(R) with ||I+ k||L2(R)≤C|k|4. (3.50) Finally, by ( 2.7), (3.22), (3.24), (3.44) , and (3.46), we have that |I− k(x)| ≤C|k|3 |x−iλ− k|e−(a+1)πk2+2√ 2π|k|≤C|k|3 |x+k|+k2e−T 2k2+2√ 2π|k|.CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 23 Thus ||I− k||L2(R)≤C|k|2e−T 2k2+2√ 2π|k|. (3.51) It remains to introduce the functions I0(z),I2(z),I−2(z),˜I2(z), and˜I−2(z). We set I0(z) =P(z) P′(0)z·m(z) m(0)·(1−z iλ2)(1−z iλ−2), ˜I2(z) =−iP(z) P′(iλ2)·m(z) m(iλ2)·1−z iλ−2 1−λ2 λ−2, ˜I−2(z) =−iP(z) P′(iλ−2)·m(z) m(iλ−2)·1−z iλ2 1−λ−2 λ2, K2(z) =i˜I2(z) z−iλ2, I2(z) =K2(z)−iK′ 2(iλ2)˜I2(z), K−2(z) =i˜I−2(z) z−iλ−2, I−2(z) =K−2(z)−iK′ 2(iλ−2)˜I−2(z). Then we have that I0(0) = 1, I0(iλ± k) = 0k∈Z\{0}, I′ 0(iλ±2) = 0, (3.52) ˜I2(iλ± k) = 0k∈Z,˜I′ 2(iλ2) =−i,˜I′ 2(iλ−2) = 0, (3.53) ˜I−2(iλ± k) = 0k∈Z,˜I′ −2(iλ−2) =−i,˜I′ −2(iλ2) = 0, (3.54) I2(iλ± k) = 0k∈Z\{2}, I2(iλ2) = 1, I′ 2(iλ±2) = 0, (3.55) I−2(iλ± k) = 0k∈Z\{−2}, I−2(iλ−2) = 1, I′ −2(iλ±2) = 0. (3.56) Moreover,I0,˜I2,˜I−2, I2, andI−2are entire functions of exponential type at most π(1+a) and they belong all to L2(R). Letψ± k,ψk, and˜ψkdenote the inverse Fourier transform of I± k,Ik, and˜Ikfork∈Z\{0,±2}, k∈ {0,±2}andk∈ {±2}, respectively. Then, by Paley-Wiener theorem, the functio nsψ± k,ψk and˜ψkbelong toL2(R), and are supported in [ −T/2,T/2]. On the other hand, if I(z) =ˆψ(z) =/integraltext∞ −∞ψ(t)e−itzdtwithψ∈L2(R),suppψ⊂[−T/2,T/2], then /integraldisplayT 2 −T 2ψ(t)eλtdt=I(iλ) and −i/integraldisplayT 2 −T 2tψ(t)eλtdt=I′(iλ). Thus (2.36)-(2.40) follow from ( 3.48)-(3.49) and (3.52)-(3.56), while ( 2.41)-(2.42) follow from (3.50)-(3.51). 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Anal. 141(1998), no. 4, 297–329. [14] G. Leugering, Optimal controllability in viscoelasticity of rate type , Math. Methods Appl. Sci. 8(1986), 368–386. [15] G. Leugering, E. J. P. G. Schmidt, Boundary control of a vibrating plate with internal damping , Mathematical Methods in the Applied Sciences, 11(1989), 573–586. [16] B. Ya. Levin, Lectures on Entire Functions, Translatio ns of Mathematical Monographs, American Mathe- matical Society, Vol. 150, 1996. [17] J.-L. Lions, Pointwise control for distributed systems , in Control and estimation in distributed parameter systems, edited by H. T. Banks, SIAM, 1992. [18] W. A. J. Luxemburg, J. Korevaar, Entire functions and M¨ untz-Sz´ asz type approximation , Transactions of the American Mathematical Society 157(1971), 23–37. [19] S. Micu, On the controllability of the linearized Benjamin-Bona-Ma hony equation , SIAM J. Control Optim. 39(2001), 1677–1696.CONTROL OF THE WAVE EQUATION WITH STRUCTURAL DAMPING 25 [20] A. Pazy, Semigroups of linear operators and applications to partial differential equations , Applied Mathemat- ical Sciences, vol. 44, Springer-Verlag, 1983. [21] M. Pellicer, J. Sol` a-Morales, Analysis of a viscoelastic spring-mass model , J. Math. Anal. Appl. 294(2), (2004) 687–698. [22] L. Rosier, P. Rouchon, On the controllability of a wave equation with structural da mping, Int. J. Tomogr. Stat.5(2007), no. W07, 79–84. [23] L. Rosier, B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equatio n: recent progresses , Jrl Syst & Complexity 22(2009), 647–682. [24] L. Rosier, B.-Y. Zhang, Unique continuation property and control for the Benjamin- Bona-Mahony equation , preprint. [25] D. L. Russell, Mathematical models for the elastic beam and their control- theoretic implications , in H. Brezis, M. G. Crandall and F. Kapper (eds), Semigroup Theory and Applications , Longman, New York (1985). [26] R. M. Young, An Introduction to Nonharmonic Fourier Series , Academic Press, 1980. [27] X.Zhang, Exact internal controllability of Maxwell’s equations , Appl.Math. Optim.41 (2000), no. 2, 155–170. Centre Automatique et Syst `emes, Mines ParisTech, 60 boulevard Saint-Michel, 75272 Pa ris Cedex, France E-mail address :philippe.martin@mines-paristech.fr Institut Elie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 70239, 54506 Vandœuvre-l `es-Nancy Cedex, France E-mail address :rosier@iecn.u-nancy Centre Automatique et Syst `emes, Mines ParisTech, 60 boulevard Saint-Michel, 75272 Pa ris Cedex, France E-mail address :pierre.rouchon@mines-paristech.fr

2011-11-20

We investigate the internal controllability of the wave equation with structural damping on the one dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.

Null controllability of the structurally damped wave equation with moving point control

1111.4655v1

arXiv:1201.3553v1 [cond-mat.mes-hall] 17 Jan 2012Magnetic vortex echoes: application to the study of arrays o f magnetic nanostructures F. Garcia1, J.P. Sinnecker2, E.R.P. Novais2, and A.P. Guimar˜ aes2∗ 1Laborat´ orio Nacional de Luz S´ ıncrotron, 13083-970, Camp inas, SP, Brazil and 2Centro Brasileiro de Pesquisas F´ ısicas, 22290-180, Rio de Janeiro, RJ, Brazil (Dated: December 13, 2018) We propose theuse ofthe gyrotropic motion of vortexcores in nanomagnets toproduce amagnetic echo, analogous to the spin echo in NMR. This echo occurs when an array of nanomagnets, e.g., nanodisks, is magnetized with an in-plane ( xy) field, and after a time τa field pulse inverts the core magnetization; the echo is a peak in Mxyatt= 2τ. Its relaxation times depend on the inhomogeneity, on the interaction between the nanodots and on the Gilbert damping constant α. Its feasibility is demonstrated using micromagnetic simul ation. To illustrate an application of the echoes, we have determined the inhomogeneity and measured t he magnetic interaction in an array of nanodisks separated by a distance d, finding a d−ndependence, with n≈4. PACS numbers: 75.70.Kw,75.78.Cd,62.23.Eg,76.60.Lz The interest in magnetic vortices and their properties and applications has grown steadily in the last years[1– 4]. Vortices have been observed, for example, in disks and ellipses having sub-micron dimensions[5]. More re- cently, the question of the intensity of the coupling be- tweenneighbordiskswith magneticvortexstructureshas attracted an increasing interest[6–9]. At thevortexcorethe magnetizationpointsperpendic- ularlyto the plane: this characterizesits polarity, p= +1 for the + zdirection and p=−1 for−z. The direc- tion of the moments in the vortex defines the circulation: c=−1 for clockwise (CW) direction, and +1 for CCW. If removed from the equilibrium position at the center of the nanodisk by, for example, an in-plane field, and then left to relax, a vortex core will perform a gyrotropic motion, with angular frequency ω, given for thin disks[3] byωG≈(20/9)γMsβ(β=h/Ris the aspect ratio)[10]. We propose in this paper that, manipulating the dy- namic properties of the vortex in an analogous way as it is done in Nuclear Magnetic Resonance (NMR), a new phenomenon results, the magnetic vortex echo (MVE), similar to the spin echo observed in NMR[11]. This new echo may provide information on fundamental properties of arrays of nanodisks, e.g., their inhomogeneity and in- teractions. Despite the fact that applications of vortices necessarilyinvolvearrays,mostoftherecentpublications deal with the analysis of individual nanodisks or arrays with a few elements. Therefore, the possibility of char- acterizing large arrays is of much interest. In this paper we have examined the motion of vortex cores in an ar- ray of nanometric disks under the influence of a pulsed magnetic field, using micromagnetic simulation. Let us consider an array of nanodisks where the vor- tex cores precess with a distribution of angular frequen- cies centered on ω0, of width ∆ ω, arising from any type of inhomogeneity (see note [10]). We assume that the frequencies vary continuously, and have a Gaussian dis- tribution P(ω) with mean square deviation ∆ ω. To sim- plify we can assume that the polarization of every vortex FIG. 1. (Color online) Diagram showing the formation of magnetic vortex echoes; the disks are described from a refer - ence frame that turns with the average translation frequenc y ω0: a) disks with in-plane magnetizations Mialong the same direction (defined by the white arrows); b) after a time τ the disks on the left, center and right have turned with fre- quencies, respectively, lower, higher and equal to ω0; c) the polarities of the vortex cores are reversed, and the ωiof the vortex cores (and of the Mi) change sign, and d) after a sec- ond interval τthe cores (and Mi) are again aligned, creating the echo. is the same: pi= +1. This is not necessary for our argu- ment, but, if required, the system can be prepared; see ref. [12] and the references therein. Since the direction of rotation of the magnetic vortex cores after removal of the in-plane field is defined exclu- sively by p, all the cores will turn in the same direction; as the vortex core turns, the in-plane magnetization of the nanodot also turns.2 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s45/s49/s48/s49/s45/s49/s48/s49/s45/s49/s48/s49/s45/s49/s48/s49 /s32/s32 /s116/s32/s40/s110/s115/s41/s32/s61/s32/s49/s48/s32/s110/s109 /s44/s32 /s61/s32/s50/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s46/s48/s48/s53/s77/s32/s40/s49/s48/s45/s49/s52 /s65/s47/s109/s41 /s32/s61/s32/s50/s48/s32/s110/s109/s44/s32 /s61/s32/s50/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s32/s61/s32/s50/s48/s32/s110/s109 /s44/s32 /s61/s32/s49/s48/s32/s110/s115 /s44/s32 /s61/s32/s52/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48/s46/s48/s48/s49 /s100/s99/s98/s97/s32/s61/s32/s49/s48/s32/s110/s109 /s44/s32 /s61/s32/s51/s48/s32/s110/s115 /s44/s32 /s32/s61/s32/s48 /s32 FIG. 2. (Color online) Micromagnetic simulation of magneti c vortex echoes, for 100 nanodisks, with d= infinity, and a) σ= 10nm, τ= 30ns, α= 0; b) σ= 20nm, τ= 10ns andτ= 40ns (two pulses), and α= 0.001; c)σ= 20nm, τ= 20ns, α= 0; d)σ= 10nm, τ= 20ns, α= 0.005. The inversion pulses ( Bz=−300mT) are also shown (in red). /s48/s46/s48/s48/s48 /s48/s46/s48/s48/s49 /s48/s46/s48/s48/s50 /s48/s46/s48/s48/s51 /s48/s46/s48/s48/s52 /s48/s46/s48/s48/s53/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48 /s32/s32/s49/s47/s84 /s50/s32/s40/s110/s115/s45/s49 /s41 /s68/s97/s109/s112/s105/s110/s103/s32/s99/s111/s110/s115/s116/s97/s110/s116/s32 FIG. 3. (Color online) Variation of the inverse of the re- laxation times T2(diamonds) obtained by fitting the curves of echo intensity versus time interval τtoM0exp(−τ/T2), as a function of α, for simulations made with D= 250nm, σ= 10nm, separation infinite; the continuous line is a linear least squares fit. Let us assume that all the vortex cores have been dis- placed from their equilibrium positions alongthe positive xaxis[13], by a field B. The total in-plane magnetiza- tion (that is perpendicular to the displacement of the core) will point along the yaxis, therefore forming an angleθ0= 0 att= 0. Using the approach employed in the description of magnetic resonance (e.g., see [14, 15]),one can derive the total in-plane magnetization: My(t) =My(0)/integraldisplay∞ −∞e/bracketleftBig −1 2(ω−ω0)2 ∆ω2/bracketrightBig ∆ω√ 2πcos(ωt)dω(1) an integral that is[16] the Fourier transform of the func- tionP(ω); usingT∗ 2= 1/∆ω: My(t) =My(0)exp/parenleftbigg −1 2t2 T∗2 2/parenrightbigg cos(ω0t) (2) This result shows that the total magnetization tends to zero, as the different contributions to My(t) get gradu- ally out of phase. This decay is analogous to the free induction decay (FID) in NMR; its characteristic time is T∗ 2= 1/∆ω. After a time τ, the angle rotated by each vortex core will beωτ; if att=τwe invert the polarities of the vor- tices in the array, using an appropriate pulse, the motion of the cores will change direction (i.e., ω→ −ω), and one obtains: My(t−τ) =My(0)/integraldisplay∞ −∞e−1 2(ω−ω0)2 ∆ω2 ∆ω√ 2πcos[ω(τ−t)]dω(3) The magnetization at a time t > τis then: My(t) =My(0)e[−1 2(t−2τ)2 T∗2 2]e(−t−τ T2)cos(ω0t) (4) This means that the magnetization component My(t) in- creases for τ < t < 2τ, reaching a maximum at a time t= 2τ: this maximum is the magnetic vortex echo , anal- ogous to the spin echo observed in magnetic resonance, which has important applications in NMR, including in Magnetic Resonance Imaging (MRI)[14, 15] (Fig. 1). In the case of the NMR spin echo, the maximum arises from the refocusing of the in-plane components of the nuclear magnetization. In Eq. 4 we have included a relaxation term contain- ing the time constant T2- also occurring in NMR -, to account for a possible decay of the echo amplitude with time; its justification will be given below. In the arrayof nanodisks, there will be in principle two contributionsto the defocusing ofthe magnetization, i.e., two mechanisms for the loss of in-plane magnetization memory: 1) the spread in values of βandH(see note [10]), producing an angular frequency broadening term ∆ω, and 2) irreversible processes that are characterized by a relaxation time T2: thus 1/T∗ 2= ∆ω+1/T2. The second contribution is the homogeneous term whose inverse, T2, is the magnetic vortex transverse re- laxationtime, analogousto thespin transverserelaxation time (or spin-spin relaxation time) T2in magnetic reso- nance. The irreversible processes include a) the interac- tion between the disks, which amounts to random mag- netic fields that will increase or decrease ωof a given3 disk, producing a frequency spread of width ∆ ω′= 1/T′ 2, and b) the loss in magnetization (of rate 1 /Tα) arising from the energy dissipation related to the Gilbert damp- ing constant α. Identifying Tαto the NMR longitudinal relaxation time T1, one has [14]: 1 /T2= 1/T′ 2+1/2Tα. Therefore the relaxation rate 1 /T∗ 2is given by: 1 T∗ 2= ∆ω+1 T2= ∆ω+1 T′ 2+1 2Tα(5) 1/T∗ 2is therefore the total relaxation rate of the in-plane magnetization, composed of a) ∆ ω, the inhomogeneity term, and b) 1 /T2, the sum ofall the other contributions, containing1 /T′ 2, duetotheinteractionbetweenthedisks, and 1/Tα, the rate of energy decay. The vortex cores will reach the equilibrium position at r= 0 after a time t∼Tα, therefore there will be no echo for 2 τ≫Tα. The vortex echo maximum at t= 2τ, from Eq. 4, is My(2τ)∝exp(−τ/T2); one should therefore note that the maximum magnetization recovered at a time 2 τde- creases exponentially with T2, i.e., this maximum is only affected by the homogeneous part of the total decay rate given by Eq. 5. In other words, the vortex echo cancels the loss in Mydue to the inhomogeneity ∆ ω, but it does not cancel the decrease in Mydue to the interaction be- tween the nanodisks (the homogeneous relaxation term 1/T′ 2), or due to the energy dissipation (term 1 /2Tα). Note also that if one attempted to estimate the inho- mogeneityofan arrayofnanodotsusing anothermethod, for example, measuring the linewidth of a FMR spec- trum, one would have the contribution of this inhomo- geneity together with the other terms that appear in Eq. 5, arising from interaction between the dots and from the damping. On the other hand, measuring the vor- tex echo it would be possible to separate the intrinsic inhomogeneity from these contributions, since T2can be measured separately, independently of the term ∆ ω.T2 can be measured by determining the decay of the echo amplitude for different values of the interval τ. The Fourier transform of either the vortex free induc- tiondecayorthetimedependenceoftheecho My(t) gives the distribution of gyrotropic frequencies P(ω). For the experimental study of vortex echoes, the se- quence of preparation (at t=0) and inversion fields (at t=τ) should of course be repeated periodically, with a periodT≫Tα. As in pulse NMR, this will produce echoes on every cycle, improving the S/N ratio of the measured signals. Also note that the time T∗ 2can be ob- tained either from the initial decay (FID, Eq. 2) or from the echo (Eq. 4), but T2can only be obtained from the MVE. In order to demonstrate the MVE, we have performed micromagnetic simulations of an assembly of 100 mag- netic nanodisks employing the OOMMF code[17]. The simulated system was a square array of 10 ×10 disks, thickness 20nm, with distance dfrom center to center. In order to simulate the inhomogeneity of the system,we have introduced a Gaussian distribution of diame- ters, centered on 250nm and mean square deviation σ; σ= 10nm corresponds to ∆ ω≈1.6×108s−1. The disks were placed at random on the square lattice. The initial state of the disks ( p= +1 and c=−1) was pre- pared by applying an in-plane field of 25mT; the po- larity was inverted with a Gaussian pulse of amplitude Bz=−300mT, with width 100ps. The results for the cased=∞were simulations made on the disks one at a time, adding the individual magnetic moments µi(t). We have successfully demonstrated the occurrence of the magnetic vortex phenomenon, and have shown its potential as a characterization technique. The simula- tions have confirmed the occurrence of the echoes at the expected times ( t= 2τ). For different values of σ, the T∗ 2time, and consequently the duration of the FID and the width of the echo are modified (Fig. 2a, 2c); increas- ingαresults in a faster decay of the echo intensity as a function of time (Fig. 2a, 2d). We have also obtained multiple echoes, by exciting the system with two pulses (Fig. 2b)[18]. Fig. 3 shows the dependence of T2onαforσ= 10nm; essentially the same result is obtained for σ= 20nm, sinceT2does not depend on ∆ ω(Eq. 5). Taking a linear approximation, 1 /T2=Aα, and since for d=infinity there is no interaction between the disks, 1 /T2= 1/2Tα, and therefore: 1 Tα= 2Aα (6) From the least squares fit (Fig. 3), A= 1.6×1010s−1. This relation can be used to determine experimentally α, measuring T2with vortex echoes, for an array of well- separated disks. Recently some workers have analyzed the important problem of the interaction between disks exhibiting mag- netic vortices, obtaining that it varies with a d−ndepen- dence: Vogel and co-workers [6], using FMR, obtained for a 4×300 array a dependence of the form d−6, the same found by Sugimoto et al. [8] using a pair of disks excited with rf current. Jung et al. [7] studying a pair of nanodisks with time-resolved X-ray spectroscopy, found n= 3.91±0.07 and Sukhostavets et al. [9], also for a pair of disks, in this case studied by micromagnetic simulation, obtained n= 3.2 and 3.7 for the xandy interaction terms, respectively. As a first approximation one can derive the depen- dence of the contribution to 1 /T∗ 2related to the distance between the disks as: T∗ 2=B+Cd−n(7) From our simulations, and using Eq. 7 we found, from the best fit, that this interaction varies as d−n, withn= 3.9±0.1, in a good agreement with [7] and reasonable agreement with Sukhostavets et al.[9].4 /s48 /s49 /s50 /s51 /s52 /s53 /s54/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48/s54/s46/s53 /s32/s32/s84 /s50/s42/s32/s40/s110/s115/s41 /s100/s45/s52 /s32/s40/s49/s48/s45/s49/s49 /s110/s109/s45/s52 /s41/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50 /s32/s32/s77 /s97 /s103 /s110 /s101 /s116/s105/s99 /s32/s109 /s111 /s109 /s101 /s110 /s116/s32/s120 /s32/s49 /s48 /s49/s52/s32 /s40/s65 /s32/s109 /s50 /s41 /s84/s105/s109/s101/s32/s40/s110/s115/s41 FIG. 4. (Color online) Variation of the relaxation times T∗ 2versusd−4for an array of 10 ×10 magnetic nanodisks with a distribution of diameters centered on D= 250nm (σ= 10nm), damping constant α= 0.001 and separation dbetween their centers; the continuous line is a linear least squares fit. The inset shows a vortex echo simulation for the array, with d= 500nm, τ= 30ns, α= 0.001. Determining 1 /T′ 2has allowed us to obtain the inten- sity of the interaction between the disks as a function of separation dbetween them. Substituting Eq. 6 and Eq. 7 in Eq. 5, we can obtain the expression for the interaction as a function of d: 1 T′ 2=1 B+Cd−n−Aα−∆ω≈dn |C|−Aα−∆ω; (8) (approximation valid for dsmall). In Fig. 4 we show the results of the simulations with σ= 10nm and α= 0.001. Assuming n= 4 and making a linear squares fit, we obtained B= 6.15×10−9s,C=−4.03×10−35s m4. A new phenomenon, the magnetic vortex echo, anal- ogous to the NMR spin echo, is proposed and demon- strated here through micromagnetic simulation. Appli- cationsofthe magneticvortexechoincludesthe measure- ment of the inhomogeneity, such as, distribution of di- mensions, aspect ratios, defects, and perpendicular mag- netic fields and so on, in a planar array of nanodisks or ellipses; it may be used to study arrays of nanowires or nanopillars containing thin layers of magnetic material. These properties cannot be obtained directly, for exam- ple, from the linewidth of FMR absorption. The MVE is a tool that can be used to evaluate the interactionbetween the elements ofalargearrayofnano- magnets with vortex ground states. It can also be used to determine the Gilbert damping constant αin thesesystems. The authors would like to thank G.M.B. Fior for the collaboration; we are also indebted to the Brazilianagen- cies CNPq, CAPES, FAPERJ, FAPESP. ∗Author to whom correspondence should be addressed: apguima@cbpf.br [1] A. P. Guimar˜ aes, Principles of Nanomagnetism (Springer, Berlin, 2009) [2] C. L. Chien, F. Q. Zhu, and J.-G. Zhu, Physics Today 60, 40 (2007) [3] K. Y. Guslienko, K.-S. Lee, and S.-K. Kim, Phys. Rev. Lett.100, 027203 (2008) [4] F. Garcia, H. Westfahl, J. Schoenmaker, E. J. Carvalho, A. D. Santos, M. Pojar, A. C. Seabra, R. Belkhou, A. Bendounan, E. R. P. Novais, and A. P. Guimar˜ aes, Appl. Phys. Lett. 97, 022501 (2010) [5] E. R. P. Novais, P. Landeros, A. G. S. Barbosa, M. D. Martins, F. Garcia, and A. P. Guimar˜ aes, J. Appl. Phys. 110, 053917 (2011) [6] A. Vogel, A. Drews, T. Kamionka, M. Bolte, and G. Meier, Phys. Rev. Lett. 105, 037201 (2010) [7] H. Jung, K.-S. Lee, D.-E. Jeong, Y.-S. Choi, Y.-S. Yu, D.-S. Han, A. Vogel, L. Bocklage, G. Meier, M.- Y. Im, P. Fischer, and S.-K. Kim, Sci. Rep. 59, 1 (2011/08/10/online) [8] S. Sugimoto, Y. Fukuma, S. Kasai, T. Kimura, A. Bar- man, and Y. Otani, Phys. Rev. Lett. 106, 197203 (2011) [9] O.V.Sukhostavets,J. M.Gonzalez, andK.Y.Guslienko, Appl. Phys. Express 4, 065003 (2011) [10] The sources of inhomogeneity are the spread in radii, in thickness or the presence of defects. An external perpen- dicular field Haddsacontributionto ω[19],ω=ωG+ωH, withωH=ω0p(H/Hs), where pis the polarity and Hs the field that saturates the nanodisk magnetization. A distribution ∆ His another source of the spread ∆ ω [11] E. L. Hahn, Phys. Rev. 80, 580 (1950) [12] R. Antos, M. Urbanek, and Y. Otani, J. Phys.: Conf. Series200, 042002 (2010) [13] If the disks have different circulations ( c=±1) the cores will be displaced in opposite directions, butthe effect will be the same, since the Miwill all point along the same direction. [14] C. P. Slichter, Principles of Magnetic Resonance, 3. ed. (Springer, Berlin, 1990) [15] A. P. Guimar˜ aes, Magnetism and Magnetic Resonance in Solids(John Wiley & Sons, New York, 1998) [16] T. Butz, Fourier Transformation for Pedestrians (Springer, Berlin, 2006) [17] Available from http://math.nist.gov/oommf/ [18] These echoes, however, are not equivalent to the stimu- lated echoes observed in NMR with two 90opulses[11] [19] G. de Loubens, A. Riegler, B. Pigeau, F. Lochner, F. Boust, K. Y. Guslienko, H. Hurdequint, L. W. Molenkamp, G. Schmidt, A. N. Slavin, V. S. Tiberke- vich, N.Vukadinovic,andO.Klein,Phys.Rev.Lett. 102, 177602 (2009)

2012-01-17

We propose the use of the gyrotropic motion of vortex cores in nanomagnets to produce a magnetic echo, analogous to the spin echo in NMR. This echo occurs when an array of nanomagnets, e.g., nanodisks, is magnetized with an in-plane (xy) field, and after a time \tau a field pulse inverts the core magnetization; the echo is a peak in M_{xy} at t=2\tau. Its relaxation times depend on the inhomogeneity, on the interaction between the nanodots and on the Gilbert damping constant \alpha. Its feasibility is demonstrated using micromagnetic simulation. To illustrate an application of the echoes, we have determined the inhomogeneity and measured the magnetic interaction in an array of nanodisks separated by a distance d, finding a d^{-n} dependence, with n\approx 4.

Magnetic vortex echoes: application to the study of arrays of magnetic nanostructures

1201.3553v1

Current -induced motion of a transverse magnetic domain wall in the presence of spin Hall effect Soo-Man Seo1, Kyoung-Whan Kim2, Jisu Ryu2, Hyun -Woo Lee2,a), and Kyung -Jin Lee1,3,4,b) 1Department of M aterials Science and Engineering, Korea University , Seo ul 136 -701, Korea 2PCTP and Department of Physics, Pohang University of Science and Technology , Kyungbuk 790 - 784, Korea 3Center for Nanoscale Science and Technology , National Institute of Standards and Technology, Gaithersburg, Maryland 20899 -8412, USA 4Maryland Nanocenter, University of Maryland, College Park, MD 20742, USA We theoretically study the current -induced dynamics of a transverse magnetic domain wall in bi-layer nanowire s consisting of a ferromagnet on top of a nonmagnet having strong spin-orbit coupling . Domain wall dynamics is characterized by two threshold current densities, WB thJ and REV thJ , where WB thJ is a threshold for the chirality switching of the domain wall and REV thJ is another threshold for the reversed domain wall motion caused by spin Hall effect . Domain wall s with a certain chirality may move opposite to the electron -flow direction with high speed in the current range WB thREV th JJ J for the system designed to satisfy the conditio ns WB REV th thJJ and , where is the Gilbert damping constant and is the nonadiabaticity of spin torque . Micromagnetic simulation s confirm the validity of analytical results. 1 a)Electronic mail: hwl@postech.ac.kr. b)Electronic mail: kj_lee@korea.ac.kr . Electric manipulati on of domain wall s (DW s) in magnetic nanowire s can be realized by the spin-transfer torque (STT) due to the coupling between local magnetic moment s of the DW and spin-polarized current s1,2. Numerous studie s on this subject have addressed its fundamental physic s3-5, and to explore its potential in application s such as data storage and logic devices6. Up until now, h owever, most studies have focused on the effect of the spin current that i s polarized by a ferromagnet ic layer . Another way to generate a spin curren t is the spin Hall effect (SHE)7,8. In ferromagnet (FM) |nonmagnet (NM) bi -layer system s, an in-plane charge current density (Jc) passing through the NM is converted into a perpendic ular spin current density (Js) owing to the SHE . The ratio of Js to Jc is parameterized by spin Hall angle . This spin current caused by SHE exerts a STT (= SHE -STT) on the FM and consequently modifies its m agnetization dynamics. During the last decade , most studies on the SHE have focused on measuring the spin Hall angle9-14. Recently the magnetization switching15 and the modulation of propagating spin waves by SHE -STT were investigated16-18. However, the effect of SHE -STT on current - induced DW dynamics has not been treated . In this Letter, we study DW dynamics including all current -induced STT s in a nanowire consisting of FM/NM bi-layers (Fig. 1) , where FM has a n in-plane magnetic anisotropy and NM has strong spin-orbit coupling ( SOC ) responsible for the SHE. A charge current passing through the FM generates conventional adiabatic and nonadiabatic STTs19-21, whereas a charge current flowing through the NM experiences SHE and generates SHE -STT on the FM . For the current running in the x axis, t he modified Landau-Lifshitz -Gilbert equation including all the STTs is given by ),ˆ ( y cxbxbt tJ SH J J m mmmmm mmm Hmm eff (1) where m is the unit vector along the magnetization, α is the Gilbert damping constant, bJ ) 2 (S F B eM PJg is the magnitude of adiabatic STT, β is the nonadiabatic ity of STT, J SHc ( 2 )SH N S FJ eM t is the magnitude of SHE -STT, θSH is an effective spin Hall angle for the bi -layer system , γ is the gyromagnetic ratio, g is the Land é g-factor, μB is the Bohr magneton, P is the spin polarization in the FM, e is the electron charge, MS is the saturation magnetization of the FM, and JF (JN) is the current density in the FM (NM). JF and JN are determined by a simple circuit model; i.e., ) /() (0 N N FF F N F F t t ttJ J and ) /() (0 N N FF N N F N t t ttJ J , where J0 is the total current density in the bi -layer nanowire , σF (σN) is the conductivity of the FM (NM), and tF (tN) is the thickness of the FM (NM). We assume that θSH is smaller than 1 as is usually the case experimentally . For a nanowire with an in-plane magnetic anisotropy, a net effective field is given by 2 22ˆ ,mH x Heff K x d SAHmMx (2) where A is the exchange stiffness constant, HK is the easy axis anisotropy field along the x axis, and dH is the magnetostatic field given by )() (~r )(3rmrr rH Nd MS d , where the components of the tensor N~ are given by 3 2 2/]/31[ r rx Nxx , 5/3 rxy Nxy [22]. Other components are defined in a similar way . For a one-dimensional DW as shown in Fig. 1, the spatial profile of the magnetization is described by )sin sin, cos sin, (cos m, where ) ( sech sin Xx , ) ( tanh cos Xx , )(tX is the DW position, )(t is the DW tilt angle , and λ is the DW width . By using the procedure developed by Thiele23, we obtain the equation s of motion for the two collective coordinates X and in the rigid DW limit, sin(2 ),2d JH Xbtt (3) ,eff JXbtt (4) where ) 2(S d d MK H , sin 1SH eff B , /2SH SH N F FB J t PJ , and Kd is the hard-axis anisotropy energy density . From Eqs. (3) and (4), one finds that the effect of SHE - STT on DW d ynamics is captured by replacing β by βeff. Assuming that FM is Permall oy (Py: Ni80Fe20) and NM is Pt, for the parameters of tF = 4 nm, tN = 3 nm, σF = σN, θSH = 0.1, β ≈ 0.01 to 0.03 [ 24], P = 0.7, and λ = 30 nm, we find SHB ≈ 18 to 56, which is not small . Therefore, βeff can be much larger than β unless sin is extremely small. Furthermore, it is possible that βeff is even negative if 1 sinSHB . To get an insight into the effect of SHE -STT on DW dynamics, we derive several analytical solutions from Eqs. (3) and (4). It is known that DW dynamics in a nanowire can be classified into two regimes; i.e., below and above the Walker breakdown25. Below the Walker breakdown, increases in the initial time stage and then becomes saturat ed to a certain value over time. In this limit ( 0 t as t ), we obtain ,) (22sin effd JHb (5) Threshold adiabatic STT for the Walker breakdown ( WB Jb ) is obtained from the maximum value of the right -hand -side of Eq. ( 5); i.e., ) (22sin maxeff dWB J H b . Note that WB Jb is not simply 2( )d effH because eff also includes . When BSH = 0, Eq. (5) reduces to 2dWB J H b , reproducing the previ ous result [26] in the absence of SHE. For WB JJbb (below t he Walker breakdown) and using the small -angle approximation , DW velocity (vDW) is given by ,) (1 SHJ dJ SH J DWBb HbB b v (6) where the sign “+” and “” in the parenthesis corresponds to the initial tilt angle s 0 and 00 , respectively. This 0 dependen ce of DWv originates from the fact that SHE - STT acts like a damping or an anti -damping term depending on 0 . When , J DW b v so that vDW does not depend on SHE -STT. However, this condition is hardly realized in the bi -layer system that we consider since the strong SOC in NM increases the intrinsic α of FM through the spin pumping effect27. When BSH = 0, J DW b v , consistent with the DW velocity in the absence of SHE26. Note that in our sign convention, a negative bJ corresponds to the electron -flow in + x direction and a positive DWv corresponds to the DW motion along the electron -flow direction. Therefore, when the term in the parenthesis of Eq. ( 6) is negative, the DW moves against the electron -flow direction instead of along it. Threshold adiabatic STT for this re versed DW motion ( REV Jb ) is given by .REV d J SHHbB (7) For WB JJbb (far above the Walker breakdown) , the time-average d values of sin and 2sin can be set to zero because of the precession of . In this limit, DWv is determined by Eq. (3) and becomes bJ so that the DW moves along the electron -flow direction and its motion does not depend on SHE -STT. Based on the above investigations , there are two interesting effects of SHE on current - induced DW dynamics. First, current -induced DW dynamics is determined by two thre sholds, WB Jb and REV Jb . Whe n WB J JREV J b b b , the DW can move against the electron -flow direction. Note that the existence of such bJ range implicitly assumes WB JREV J b b . When this inequality is not satisfied, the DW always moves along the electron -flow direction. For all cases, DWv can be larger than Jb depending on the parameters (see Eq. ( 6)). Second , vDW is asymmet ric against the initial tilt angle 0 for a fixed current polarity. A similar argument is also valid for a fixed 0 but with varying the current polarity; i.e., DWv is asymmetric with respect to the current polarity for a fixed 0 . This behavior follows because SHE -STT acts like a damping term for one sign of the current but acts like an anti - damping term for the other sign. Therefore, although the condi tion of REV WB J J Jb b b is satisfied, the reversed DW motion is expected to be observed only for one current polarity. To verify the analytical results , we perform a one-dimensional micromagnetic simulation by numerically solving Eq. (1). We con sider a Py/Pt bi -layer nano wire of (length × width × thickness ) = (2000 n m × 80 nm × 4 nm (Py) and 3 nm (Pt)) (Fig. 1) . Py m aterial parameters of MS = 800 kA/m, A = 1.3×10−11 J/m, P = 0.7, α = 0.02, and β = 0.01 to 0.03 are used . The crystalline anisotropy and the temperature are assumed to be zero. Conductivit ies of both layers are assumed to be the same as σPy = σPt = 6.5 (μΩm)1, and thus J0 = JF = JN. For all cases, the in itial DW tilt angle 0 is set to zero. Analytical and numerical results are compared in Fig. 2. DW velocity ( DWv ) and DW tilt angle ( DW ) as a function of the total current density of the bi -layer (J0) for three values of θSH (= +0.1, 0.0, 0.1) and β = 0.01 (thus > ) are shown in Fig. 2(a) and (b) , respectively . DWv is estimated from the terminal velocity . Here, we test both positive and negative values of θSH since the spin Hall angle can have e ither sign . Current dependence s of DWv (Fig. 2(a)) and DW (Fig. 2(b)) show close correlation , meaning that the DW tilting plays a crucial role for the effect of SHE on DW dynamics as demonstrated analytically . In Fig. 2(a) , the numerical results (symbols) are in agreement with the results obtained from Eq. (6) (lines) . For θSH = 0, DWv is linearly proportional to J0 and the DW always move s along the electron -flow direction . However, for 0.5×1012 ≤ J0 ≤ 1.0×1012 A/m2 with θSH = 0.1 (1.0×1012 A/m2 ≤ J0 ≤ 0.5×1012 A/m2 with θSH = 0.1 ), DWv has the same polarity as the current . Thus, the DW moves along the current -flow direction for these ranges of the current . The threshold for the reversed DW motion is consistent with the analytical solution of Eq. (7); i.e., REV Jb = ±26.6 m/s corresponding to J0 = ±0.52×1012 A/m2. The maximum DWv is obtained at J0 = ±1 .0×1012 A/m2 immediately before the DW experiences Walker breakdown and switches its chirality . As shown in the Fig. 2 (c), the normalized y-component of the magnetization at the DW center (my) abruptly change s from 1 to 1 for J0 = 1.0×1012 A/m2 and θSH = 0.1. This current density is consist ent with the threshold for Walker breakdown ( WB Jb ); i.e., WB Jb = ±53 m/s corresponding to J0 = ±1.04 5×1012 A/m2. At this current density, vDW is enhanced by a factor of 5 compared to the case for θSH = 0. Fig. 3 (a) and (b) show DWv and DW as a function of J0 for three values of θSH (= +0.1, 0.0, 0.1) and β = 0.0 3 (thus ). Similar ly to the cases for β = 0.0 1, DWv is closely correlated to DW and significantly enhance d near WB Jb . In this case, in contrast to the case for , reversed DW motion is not observed . It is because the sign of the () term in Eq. (6) is negative in this case , and thus the overall sign of DWv corresponds to the DW motion along the electron -flow direction. We find that the current -induced Oersted field has only a negligible effect on DWv (not shown). Thus, the numerical results confirm the validity of the analytical solutions; the DW moves along the current -flow directio n at the limited range of the current (i.e., REV WB J J Jb b b ) when . In addition this reversed DW motion appears only for one current polarity . Finally, we remark the effect of SHE on DW dynamics in the nanowire with a perpendicular anisotropy . It was experimentally reported that the DW moves along the current -flow direction with a high DWv (≈ 400 m/s ) in the perpendicularly magnetized nanowire consisting of Pt/Co/AlO x [28, 29 ]. We n ote that this D W dynamics cannot be explained by the SHE only. Considering the materials parameter s in Ref. [ 29] as MS = 1090 kA/m, K = 1.2×106 J/m3, A = 1.3×10−11 J/m, α = 0.2, P = 0.7, λ = 5 nm, and assuming θSH = 0.1 and β = 0.1, we find BSHλ = 18.8 that is comparable to the value for the Py/Pt bi-layer tested in this work . For Pt/Co/AlO x, however, REV Jb and WB Jb are respectively 1.5 and 3 m/s (corresponding to J0 = 0.4×1011 and 0.8×1011 A/m2). These thresholds are much smaller t han th ose of the Py/Pt bi-layer since Hd of DW in a perpendicular system is smaller than in an in-plane system (i.e., Hd = 848 mT for the system of Py/Pt in this w ork, 33 mT for the system in Ref. [29])22. Note that the maximum DW velocity moving along the current - flow direction ( REV DWv ) is obtained at WB JJbb . The WB Jb (= 3 m/s) in Pt/Co/AlO x system is too small to allow such a high REV DWv (≈ 400 m/s) . Indeed, the numerically obtained maximum REV DWv is 8.2 m/s at J0 = 0.71×1011 A/m2 (bJ = 2.64 m/s ) (not shown) , which is much smaller than the experimentally obtained value, 400 m/s . More importantly, in the Pt/Co/AlO x system, the reversed DW motion was observ ed at both current polarities31 whereas the SHE allows the reversed motion at only one current polarity. On the other hand, we theoretically demonstrated that the DW dynamics reported in Ref. [ 28, 29] can be explained by STTs caused by Rashba SOC32. We als o remark that one of us reported the effect of SOC on current -driven DW motion recently33. In Ref. [33], h owever, the effect of SOC within FM was investigated , in contrast to the present work where the effect of SOC in NM of the FM /NM bi -layer system is investigated . To conclude, we present the analytical model for current -induced DW motion in the presence of SHE . We demonstrate that DW dynamics is significantly affected by the SHE . In particular, for the case of , the SHE enables t he reversed DW motion with high speed at one current polarity when the system is designed to satisfy the condition of WB JREV J b b and the current density is selected to be in the range between the two thresholds. Our result demo nstrates that the engineering of SOC and thus the SHE provides an important opportunity for an efficient operation of spintronic devices. This work was supported by the NRF ( 2010 -0014109, 2010 -0023798, 2011 -0009278, 2011 -0028163 , 2011 -0030046 ) and the MKE/KEIT (2009 -F-004-01). K.J.L. acknowledges support under the Cooperative Research Agreement between the University of Maryland and the National Institute of Standards and Technology Center for Nanoscale Science and Technology, Award 70NANB10H193, through the University of Maryland. REFERENCE [1] J. C. Slonczewski, J. Magn. Mag. Mater. 159, L1 (1996) . [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). [4] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature (London) 428, 539 (2004). [5] M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. Heyderman, F. Nolting, and U. Rüdiger, Phys. Rev. Lett. 94, 106601 (2005). [6] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). [7] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [8] S. Zhang, Phys. Rev. Lett. 85, 393 (2000 ). [9] S. O. Valenzuela and M. Tinkham, Nature (London) 442, 176 (2006). [10] T. Kimura, Y . Otani , T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). [11] K. Ando , S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601 (2008). [12] T. Seki , Y . Hasegawa, S. Mitani, S. Takahashi, H. Im amura, S. Maekawa, J. Nitta, and K. Takahashi , Nat. Mater. 7, 125 (2008). [13] O. Mosendz, J. E. Pearson, F. Y . Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffman, Phys. Rev. Lett. 104, 046601 (2010). [14] L. Liu , T. Moriyama, D. C. Ralph, and R. A. Buhrm an, Phys. Rev. Lett. 106, 106602 (2011). [15] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, arXiv:1110.6846 . [16] V. E. Demidov , S. Urazhdin, E. R. J. Edwards, M. D. Stiles, R. D. McMichael, and S. O. Demokritov , Phys. Rev. Lett. 107, 107204 (2011). [17] Z. Wang, Y . Sun, M. Wu, V . Tiberkevich, and A. Slavin, Phys. Rev. Lett. 107, 146602 (2011). [18] E. Padr ón-Hern ández, A. Azevedo, and S. M. Rezende, Appl. Phys. Lett. 99, 19251 1 (2011). [19] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004) . [20] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004) . [21] A. Thiaville , Y . Nakatani, J. Miltat, and Y . Suzuki , Europhys. Lett. 69, 990 (2005). [22] S.-W. Jung , W. Kim, T. -D. Lee, K. -J. Lee, and H. -W. Lee , Appl. Phys. Lett. 92, 2025 08 (2008). [23] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). [24] K. Sekiguchi, K. Yamda, S. -M. Seo, K. -J. Lee, D. Chiba, K. Kobayashi, and T. Ono, Phys. Rev. Lett. 108, 017203 (2012) . [25] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). [26] A. Mougin , M. Cormier, J. P. Adam, P. J. Metaxas, and J. Ferré , Europhys. Lett. 78, 57007 (2007). [27] Y. Tserkovnyak and A, Brataas, and G. E. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [28] T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auffret, B. Rodmacq, A. Schul, S. Pizzini, J. V ogel, and M. Bonfim, Appl. Phys. Lett. 93, 262504 (2008); ibid 95, 179902 (2009). [29] I. M. Miron , T. Moore, H. Szambolics, L. D. Buda -Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. V ogel, M. Bonfim, A. Schul, and G. Gaudin, Nat. Mater. 10, 189 (2011). [31] I. M. Miron, private communication. [32] K.-W. Kim, S. -M. Seo, J. Ryu, K. -J. Lee, and H. -W. Lee, arXiv:1111.3422v2. [33] A. Manchon and K. -J. Lee, Appl. Phys. Lett. 99, 022504 (2011); ibid 99, 229905 (2011). FIGURE CAPTION FIG. 1. (Color online) Schematic s of FM/NM bi-layer nanowire . (top) Structure. (lower left) Spatial profile of DW. The color ed contour shows x component of the magnetization for 2 -D micromagnetics . (lower rig ht) Width -averaged magnetization components . FIG. 2. (Color online) Domain wall velocity for > . (a) DW velocity ( DWv ) as a function of the total current density of bi -layer ( J0) for three values of θSH (= +0.1, 0 .0, 0.1) and β = 0.01 ( = 0.02) . Symbols are modeling results, whereas solid lines correspond to Eq. (6) . (b) DW tilt angle ( DW ) as a function of J0. Filled symbols above the chiral switching threshold (| J0| = 1.1×1012 A/m2) are shifted from their original values by –180° (filled green triangles) and +180° (filled red circles) . (c) Normalized y component of the magnetization at the DW center (my) as a function of J0. FIG. 3. (Color online) Domain wall velocity for < . (a) DW velocity ( DWv ) as a function of the total current density of bi -layer ( J0) for three values of θSH (= +0.1, 0.0, 0.1) and β = 0.0 3 ( = 0.02) . Symbols are modeling results, whereas solid lines correspond to Eq. (6). (b) DW tilt angle ( DW ) as a function of J0. Filled symbols represent the cases that the chirality of DW switches from its initial tilt angle 00 . (c) N ormalized y component of the magnetization at the DW center (my) as a function of J0. FIG. 1. Seo et al. FIG. 2. Seo et al. FIG. 3. Seo et al.

2012-02-15

We theoretically study the current-induced dynamics of a transverse magnetic domain wall in bi-layer nanowires consisting of a ferromagnet on top of a nonmagnet having strong spin-orbit coupling. Domain wall dynamics is characterized by two threshold current densities, $J_{th}^{WB}$ and $J_{th}^{REV}$, where $J_{th}^{WB}$ is a threshold for the chirality switching of the domain wall and $J_{th}^{REV}$ is another threshold for the reversed domain wall motion caused by spin Hall effect. Domain walls with a certain chirality may move opposite to the electron-flow direction with high speed in the current range $J_{th}^{REV} < J < J_{th}^{WB}$ for the system designed to satisfy the conditions $J_{th}^{WB} > J_{th}^{REV}$ and \alpha > \beta, where \alpha is the Gilbert damping constant and \beta is the nonadiabaticity of spin torque. Micromagnetic simulations confirm the validity of analytical results.

Current-induced motion of a transverse magnetic domain wall in the presence of spin Hall effect

1202.3450v1

arXiv:1203.0495v2 [astro-ph.EP] 11 Oct 2013Damping-Antidamping Effect on Comets Motion G.V. L´ opez∗and E. M. Ju´ arez Departamento de F´ ısica de la Universidad de Guadalajara, Blvd. Marcelino Garc´ ıa Barrag´ an 1421, esq. Calzada Ol´ ım pica, 44430 Guadalajara, Jalisco, M´ exico PACS: 45.20.D−,45.20.3j, 45.50.Pk, 95.10.Ce, 95.10.Eg, 96.30.Cw, 03.67.Lx, 03.67.Hr, 03.67.-a, 03.65w September, 2013 Abstract We make an observation about Galilean transformation on a 1- D mass variable systems which leads us to the right way to deal with m ass vari- able systems. Then using this observation, we study two-bod ies gravitational problem where the mass of one of the bodies varies and suffers a d amping- antidamping effect due to star wind during its motion. For this system, a constant of motion, a Lagrangian and a Hamiltonian are given for the radial motion, and the period of the body is studied using the consta nt of motion of the system. Our theoretical results are applied to Halley’s comet. ∗gulopez@udgserv.cencar.udg.mx 11 Introduction There is not doubt that mass variable systems have been relevant s ince the founda- tion of the classical mechanics and modern physics (L´ opez et al 20 04). These type of systems have been known as Gylden-Meshcherskii problems (Gylde n 1884; Meshch- erskii 1893, 1902; Lovett 1902; Jeans 1924; Berkovich 1981; Be kov 1989; Prieto and Docobo 1997), and among these type of systems one could mention : the motion of rockets (Sommerfeld 1964), the kinetic theory of dusty plasma (Zagorodny et al 2000), propagation of electromagnetic waves in a dispersive-nonlin ear media (Seri- maa et al 1986), neutrinos mass oscillations (Bethe 1986; Commins a nd Bucksbaum 1983), black holes formation (Helhl et al 1998), and comets intera cting with solar wind (Daly1989). This last system belong to the so called ”gravitation al two-bodies problem” which is one of the most studied and well known system in clas sical me- chanics (Goldstein 1950). In this type of system, one assumes nor mally that the masses of the bodies are fixed and unchanged during the dynamical motion. How- ever,when one is dealing with comets, beside to consider its mass varia tion due to the interaction with the solar wind, one would like to have an estimation of the the effect of the solar wind pressure on the comet motion. This pressur e may produces a dissipative-antidissipative effect on its motion. The dissipation effec t must be felt by the comet when this one is approaching to the sun (or star), and the antidissi- pation effect must be felt by the comet when this one is moving away fr om the sun. To deal with these type of mass variation problem, it has been propo sed that the Newton equation must be modified (Sommerfeld 1964; Plastino and Mu zzio 1992) since the system becomes non invariant under change of inertial sy stems (Galileo transformation). In this paper, we will make first an observation about this statemen t which in- dicates the such a proposed modification of Newton’s equation has s ome problems and rather the use of the original Newton equation is the right appr oach to deal with mass variation systems, which it was used in previous paper (L´ o pez 2007) to study two-bodies gravitational problem with mass variation in one of them, where we were interested in the difference of the trajectories in the spac es (x,v) and (x,p). As a consequence, there is an indication that mass variation problem s must be dealt as non invariant under Galilean transformation. Second, we study t he two-bodies gravitational problem taking into consideration the mass variation o f one of them and its damping-antidamping effect due to the solar wind. The mass of the other body is assumed big and fixed , and the reference system of motion is chosen just in this body. In addition, we will assume that the mass lost is expelled fr om the 2bodyradially to its motion. Doing this, the three-dimensional two-bo dies problem is reduced to a one-dimensional problem. Then, a constant of motion , the Lagrangian, and the Hamiltonian are deduced for this one-dimensional problem, w here a radial dissipative-antidissipative force proportional to the velocity squa re is chosen. A model for the mass variation is given, and the damping-antidamping e ffect is stud- ied on the period of the trajectories, the trajectories themselve s, and the aphelion distance of a comet. We use the parameters associated to comet H alley to illustrate the application of our results. 2 Mass variation problem and Galileo transfor- mation. To simplify our discussion and without losing generality, we will restrict myself to one degree of freedom,. Newton equation of motion is given by d dt/parenleftbig m(t)v/parenrightbig =F(x,v,t), (1) wherem(t)vis the quantity of movement, Fis the total external force acting on the object,m(t) andv=dx/dtare its time depending mass and velocity of the body (motion of the mass lost is not considered). Galileo transformations to another inertial frame ( S′) which is moving with a constant velocity urespect our original frameSare defined as x′=x−ut (2a) t′=t (2b) whichimplies thefollowing relationbetween thevelocity seenintherefe rencesystem S,v, and the velocity seen in the reference system S′,v′, v′=v−u. (3) Multiplying the last term by m(t′) and making the differentiation with respect to t′, one gets d dt′/parenleftbig m(t′)v′/parenrightbig =F′(x′,v′,t′), (4) 3whereF′is given by F′(x′,v′,t′) =F(x′+ut′,v′+u,t′)−udm(t′) dt′. (5) Therefore, Eq. 1 and Eq. 4 have the same form but the force is diffe rent since in addition to the transformed force term F(x′+ut′,v′+u,t′), one has the term udm(t′)/dt′. This non invariant form of the force under Galilean transformation has leadto propose(Sommerfeld 1964; Plastino andMuzzio 1992)that N ewtonequation (1) to modify Newton’s equation of motion for mass variation object s, to keep the principle of invariance of equation under Galilean transformations, o f the form m(t)dv dt=F(x,v,t)+wdm(t) dt, (6) wherewis the relative velocity of the escaping mass with respect the center of mass of the object. When one does a Galilean transformation on this equa tion, one gets m(t′)dv′ dt′=F′(x′,v′,t′), (7) whereF′is given by F′(x′,v′,t′) =F(x′+ut′,v′+u,t′)+wdm(t′) dt′, (8) which has the same form as Eq. 6. However, assume for the moment thatw= constant andF= 0. So, from Eq. 6, it follows that v(t) =v0+ln/parenleftbiggm(t) m0/parenrightbiggw , (9) wherem0=m(0). In this way, if we have a mass variation of the for m(t) =m0e−αt (for example), one would have a velocity behavior like v(t) =v0−wαt, (10) which is not acceptable since one can have v >0,v= 0 and v <0 depending on the value wαt. Even more, since for F= 0, the equation resulting in the reference systemS′is the same, i.e. in S′one gets the same type of solution, v′(t′) =v0+ln/parenleftbiggm(t′) m0/parenrightbigg (11) 4which is independent on the relative motion of the reference frames , and this must not be possible due to relation (3). In addition, it worths to mention that special theory of relativity ca n be seen as the motion of mass variation problem, where the mass depends on th e velocity of the particle of the form m(v) =m0(1−v2/c2)−1/2, withcbeing the speed of light. This system is obviously not invariant under Galilean transformation, and given the force, Newton’s equation motion is always kept in the same form t o solve a relativistic problem, d/parenleftbig m(v)v/parenrightbig /dt=F(x,v,t), (C. Møller 1952, L´ opez et al 2004). 3 Mass variation and equations of motion. Having explained and clarify the problem of mass variation (Spivak 201 0), Newton’s equations of motion for two bodies interacting gravitationally, seen from arbitrary inertial reference system, and with radial dissipative-antidissipat ive force acting in one of them are given by d dt/parenleftbigg m1dr1 dt/parenrightbigg =−Gm1m2 |r1−r2|3(r1−r2) (12a) and d dt/parenleftbigg m2dr2 dt/parenrightbigg =−Gm1m2 |r2−r1|3(r2−r1)−γ |r1−r2|/bracketleftbiggd|r1−r2| dt/bracketrightbigg2 (r2−r1),(12b) wherem1andm2are the masses of the two bodies, r1= (x1,y1,z1) and r2= (x2,y2,z2) are their vectors positions from the reference system, Gis the gravitational constant ( G= 6.67×10−11m3/Kg s2),γis the nonnegative constant parameter of the dissipative-antidissipative force, and |r1−r2|=|r2−r1|=/radicalbig (x2−x1)2+(y2−y1)2+(z2−z1)2 is the Euclidean distance between the two bodies. Note that if γ >0 and d|r1−r2|/dt>0one has dissipation since the force acts against the motion of the body, and for d|r1−r2|/dt<0one has anti-dissipation since the force pushes the body. Ifγ <0 this scheme is reversed and corresponds to our actual situation with the comet mass lost. It will be assumed the mass m1of the first body is constant and that the mass 5m2of the second body varies. Now, It is clear that the usual relative, r, and center of mass,R, coordinates defined as r=r2−r1andR= (m1r1+m2r2)/(m1+m2) are not so good to describe the dynamics of this system. However, one can consider the case form1≫m2(which is the case star-comet), and consider to put our referenc e system just on the first body ( r1=˜0). In this case, Eq. (12a) and Eq. (12b) are reduced to the equation m2d2r dt2=−Gm1m2 r3r−˙m2˙r−γ/bracketleftbiggdr dt/bracketrightbigg2 ˆr, (13) where one has made the definition r=r2= (x,y,z),ris its magnitude, r=/radicalbig x2+y2+z2, andˆr=r/ris the unitary radial vector. Using spherical coordinates (r,θ,ϕ), x=rsinθcosϕ , y=rsinθsinϕ , z=rcosθ , (14) one obtains the following coupled equations m2(¨r−r˙θ2−r˙ϕ2sin2θ) =−Gm1m2 r2−˙m2˙r−γ˙r2, (15a) m2(2˙r˙θ+r¨θ−r˙ϕ2sinθcosθ) =−˙m2r˙θ , (15b) and m2(2˙r˙ϕsinθ+r¨ϕsinθ+2r˙ϕ˙θcosθ) =−˙m2r˙ϕsinθ . (15c) Taking ˙ϕ= 0 as solution of this last equation, the resulting equations are m2(¨r−r˙θ2) =−Gm1m2 r2−˙m2˙r−γ˙r2, (16a) and m2(2˙r˙θ+r¨θ)+ ˙m2r˙θ= 0. (16b) From this last expression, one gets the following constant of motion (usual angular momentum of the system) lθ=m2r2˙θ , (17) and with this constant of motion substituted in Eq. (16a), one obta ins the following one-dimensional equation of motion for the radial part d2r dt2=−Gm1 r2−˙m2 m2/parenleftbiggdr dt/parenrightbigg −γ m2˙r2+l2 θ m2 2r3. (18) 6Now, let us assume that m2is a function of the distance between the first and the second body, m2=m2(r). Therefore, it follows that ˙m2=m′ 2˙r , (19) wherem′ 2is defined as m′ 2=dm2/dr. Thus, Eq. (18) is written as d2r dt2=−Gm1 r2+l2 θ m2 2r3−m′ 2+γ m2/parenleftbiggdr dt/parenrightbigg2 , (20) which, in turns, can be written as the following autonomous dynamica l system dr dt=v;dv dt=−Gm1 r2+l2 θ m2 2r3−m′ 2+γ m2v2. (21) Note from this equation that m′ 2is always a non-positive function of rsince it represents the mass lost rate. On the other hand, γis a negative parameter in our case. 4 Constant of Motion, Lagrangian and Hamilto- nian A constant of motion for the dynamical system (21) is a function K=K(r,v) which satisfies the partial differential equation (L´ opez 1999) v∂K ∂r+/bracketleftbigg−Gm1 r2+l2 θ m2 2r3−m′ 2+γ m2v2/bracketrightbigg∂K ∂v= 0. (22) The general solution of this equation is given by (John 1974) K(x,v) =F(c(r,v)), (23) whereFis an arbitrary function of the characteristic curve c(r,v) which has the following expression c(r,v) =m2 2(r)e2γλ(r)v2+/integraldisplay/parenleftbigg2Gm1 r2−2l2 θ m2 2r3/parenrightbigg m2 2(r)e2γλ(r)dr, (24) and the function λ(r) has been defined as λ(r) =/integraldisplaydr m2(r). (25) 7During a cycle of oscillation, the function m2(r) can be different for the comet approaching the sun and for the comet moving away from the sun. L et us denote m2+(r) for the first case and m2−(r) for the second case. Therefore, one has two cases to consider in Eq. (23) which will denoted by ( ±). Now, if mo 2±denotes the mass at aphelium (+) or perielium (-) of the comet, F(c) =c±/2mo 2±represents the functionality in Eq. (23) such that for m2constant and γequal zero, this constant of motion is the usual gravitational energy. Thus, the constant o f motion can be chosen as K±=c(r,v)/2m0 2±, that is, K±=m2 2±(r) 2mo 2±e2γλ±(r)v2+V± eff(r), (26a) where the effective potential Veffhas been defined as V± eff(r) =Gm1 mo 2±/integraldisplaym2 2±(r)e2γλ±(r)dr r2−l2 θ mo 2±/integraldisplaye2γλ±(r)dr r3(26b) This effective potential has an extreme at the point r∗defined by the relation r∗m2 2(r∗) =l2 θ Gm1(27) which is independent on the parameter γand depends on the behavior of m2(r). This extreme point is a minimum of the effective potential since one has /parenleftBigg d2V± eff dr2/parenrightBigg r=r∗>0. (28) Using the known expression (Kobussen 1979; Leubner 1981; L´ op ez 1996) for the Lagrangian in terms of the constant of motion, L(r,v) =v/integraldisplayK(r,v)dv v2, (29) the Lagrangian, generalized linear momentum and the Hamiltonian are given by L±=m2 2±(r) 2mo 2±e2γλ±(r)v2−V± eff(r), (30) p=m2 2±(r)v mo 2±e2γλ±(r), (31) 8and H±=mo 2±p2 2m2 2±(r)e−2γλ±(r)+V± eff(r). (32) The trajectories in the space ( x,v) are determined by the constant of motion (26a). Given the initial condition ( ro,vo), the constant of motion has the specific value K± o=m2 2±(ro) 2mo 2±e2γλ±(ro)v2 o+V± eff(ro), (33) and the trajectory in the space ( r,v) is given by v=±/radicalBigg 2mo 2± m2 2±(r)e−γλ±(r)/bracketleftbigg K± o−V± eff(r)/bracketrightbigg1/2 . (34) Note that one needs to specify ˙θoalso to determine Eq. (17). In addition, one normallywants toknowthetrajectoryintherealspace, thatis, t heacknowledgment ofr=r(θ). Since one has that v=dr/dt= (dr/dθ)˙θand Eqs. (17) and (34), it follows that θ(r) =θo+l2 θ/radicalbig2mo 2±/integraldisplayr rom2±(r)eγλ±(r)dr r2/radicalBig K±o−V± eff(r). (35) The half-time period (going from aphelion to perihelion (+), or backwa rd (-)) can be deduced from Eq. (34) as T± 1/2=1/radicalbig2mo 2±/integraldisplayr2 r1m2±(r)eγλ±(r)dr/radicalBig K±o−V± eff(r), (36) wherer1andr2are the two return points resulting from the solution of the following equation V± eff(ri) =K± oi= 1,2. (37) Ontheother hand, thetrajectory inthespace ( r,p) isdetermine by theHamiltonian (32), and given the same initial conditions, the initial poandH± oare obtained from Eqs. (32) and (31). Thus, this trajectory is given by p=±/radicalBigg 2m2 2±(r) mo 2±eγλ±(r)/bracketleftbigg H± o−V± eff(r)/bracketrightbigg1/2 . (38) 9It is clear just by looking the expressions (34) and (38) that the tr ajectories in the spaces (r,v) and (r,p) must be different due to complicated relation (31) between v andp(L´ opez 2007). 5 Mass-Variable Model and Results As a possible application, consider that a comet looses material as a r esult of the interaction with star wind in the following way (for one cycle of oscillatio n) m2±(r) = m2−(r2(i−1))/parenleftbigg 1−e−αr/parenrightbigg incoming (+)v <0 m2+(r2i−1)−b/parenleftbigg 1−e−α(r−r2i−1)/parenrightbigg outgoing (−)v >0(39) where the parameters b >0 andα >0 can be chosen to math the mass loss rate in the incoming and outgoing cases. The index ”i” represent the ith-se mi-cycle, being r2(i−1)andr2i−1the aphelion( ra) and perihelion( rp) points ( rois given by the initial conditions, and one has that m2−(ro) =mo). For this case, the functions λ+(r) and λ−(r) are given by λ+(r) =1 αmaln/parenleftbigg eαr−1/parenrightbigg , (40a) and λ−(r) =−1 α(b−mp)/bracketleftBigg αr+ln/parenleftbig mp−b(1−e−α(r−rp))/parenrightbig/bracketrightBigg . (40b) where we have defined ma=m2(ra) andmp=m2(rp). Using the Taylor expansion, one gets e2γλ+(r)=e2γr/ma/bracketleftbigg 1−2γ αmae−αr+1 22γ αma/parenleftbigg2γ αma−1/parenrightbigg e−2αr+.../bracketrightbigg ,(41a) and e2γλ−(r)=e−2γr (b−mp) (mp−b)2γ α(mp−b)/bracketleftbigg 1+2γ α(mp−b)e−α(r−rp) mp−b +1 22γ α(mp−b)/parenleftbigg2γ α(mp−b)−1/parenrightbigge−2α(r−rp) (mp−b)2+.../bracketrightbigg .(41b) 10The effective potential for the incoming comet can be written as V+ eff(r) =/bracketleftbigg −Gm1ma r+l2 θ 2ma1 r2/bracketrightbigg e2γr/ma+W1(γ,α,r), (42a) and for the outgoing comet as V− eff(r) =/bracketleftbigg −Gm1ma r+l2 θ 2ma1 r2/bracketrightbigge2γr (mp−b) (mp−b)2γ α(mp−b)+W2(γ,α,r),(42b) whereW1andW2are given in the appendix A. We will use the data corresponding to the sun mass (1 .9891×1030Kg) and the Halley comet ( Cevolani et al 1987, Brandy 1982, Jewitt 2002) mc≈2.3×1014Kg, r p≈0.6au, r a≈35au, l θ≈10.83×1029Kg·m2/s,(34) with a mass lost of about δm≈2.8×1011Kgper cycle of oscillation. Although, the behavior of Halley comet seem to be chaotic (Chirikov and Veches lavov 1989), but we will neglect this fine detail here. Now, the parameters αand ”b” appearing on the mass lost model, Eq. (39), are determined by the chosen mas s lost of the comet during the approaching to the sun and during the moving away from the sun (we have assumed the same mass lost in each half of the cycle of oscilla tion of the comet around the sun). Using Eq. (42a) and Eq. (42b) in the expre ssion (34), the trajectories can be calculated in the spaces ( r,v) . Fig. 1 shows these trajectories usingδm= 2×1010Kg(orδm/m= 0.0087%) for γ= 0 and (continuos line), and forγ=−3Kg/m(dashed line), starting both cases from the same aphelion dis- tance. As one can see on the minimum, dissipation causes to reduce a little bit the velocity of the comet , and the antidissipation increases the comet v elocity, reaching a further away aphelion point. Also, when only mass lost is considered (γ= 0) the comet returns to aphelion point a little further away from the initial o ne during the cycle of oscillation. Something related with this effect is the chang e of period as a function of mass lost ( γ= 0). This can be see on Fig. 2, where the period is calculated starting always from the same aphelion point ( ra). Note that with a mass lost of the order 2 .8×1011Kg(Halley comet), which correspond to δm/m=.12%, the comet is well within 75 years period. The variation of the ratio of t he change of aphelion distance as a function of mass lost ( γ= 0) is shown on Fig.3. On Fig. 4, the mass lost rate is kept fixed to δm/m= 0.0087%, and the variation of the period of the comet is calculated as a function of the dissipative-ant idissipative pa- rameterγ <0 (using |γ|for convenience). As one can see, antidissipation always 11wins to dissipation, bringing about the increasing of the period as a fu nction of this parameter. The reason seems to be that the antidissipation ac ts on the comet when this ones is lighter than when dissipation was acting (dissipation a cts when the comet approaches to the sun, meanwhile antidissipation acts wh en the comet goes away from the sun). Since the period of Halley comets has not c hanged much during many turns, we can assume that the parameter γmust vary in the interval (−0.01,0]Kg/m. Finally, Fig. 5 shows the variation, during a cycle of oscillation, of the ratio of the new aphelion ( r′ a) to old aphelion ( ra) as a function of the parameter γ. 6 Conclusions and comments We have shown that the proposed modified Newton equation for mas s variation systems has some problems. Therefore, we have considered that it is better to keep Newton’s equations of motion for mass variable systems to have a co nsistent ap- proach to these problems. Having this in mind, the Lagrangian, Hamilt onian and a constant of motion of the gravitational attraction of two bodies were given when one of the bodies has variable mass and the dissipative-antidissipativ e effect of the solar wind is considered. By choosing the reference system in the ma ssive body, the system of equations is reduce to 1-D problem. Then, the const ant of motion, Lagrangian and Hamiltonian were obtained consistently. A model for comet-mass- variation was given, and with this model, a study was made of the varia tion of the period of one cycle of oscillation of the comet when there are mas s variation and dissipation-antidissipation. When mass variation is only considere d, the comet trajectory is moving away from the sun, the mass lost is reduced as the comet is fartheraway (according toour model), andtheperiodofoscillation s becomes bigger. When dissipation-antidissipation is added, this former effect become s higher as the parameter γbecomes higher. 127 Appendix A Expression for W1andW2: W1=Gm2 2− mo 2+/braceleftBigg −p(p−1)e(−4+p)αr 2r+αpEi(αpr)−2αp(p−1)Ei/parenleftbig (−4+p)αr/parenrightbig +αp2(p−1) 2Ei/parenleftbig (−4+p)αr/parenrightbig +p(p−1) r/bracketleftbig e(p−3)αr+3α(1−p)rEi/parenleftbig (p−3)αr/parenrightbig/bracketrightbig +p(p+3) 2/bracketleftbigg −e(p−2)αr r+α(p−2)Ei/parenleftbig (p−2)αr/parenrightbig/bracketrightbigg +p+2 r/bracketleftbig e(p−2)αr+α(p−1)rEi/parenleftbig (p−1)αr/parenrightbig/bracketrightbig/bracerightBigg +l2 θ 2m2 2+r2/braceleftBigg p(p−1) 2e(p−2)αr−pe(p−1)αr−αp(p−1)e(p−2)αr+αp(p−1) 2epαr +α2p(p−1)r 2e(p−2)αr+pαre(p−1)αr−p2αre(p−1)αr−p2α2r2Ei/parenleftbig pαr/parenrightbig −α2(p−2)2p(p−1)r2 2Ei/parenleftbig (p−2)αr/parenrightbig +pα2r2Ei/parenleftbig (p−1)αr/parenrightbig −2α2p2r2Ei/parenleftbig (p−1)αr/parenrightbig +p3α2r2Ei/parenleftbig (p−1)αr/parenrightbig/bracerightBigg (A1) wheremais the mass of the body at the aphelion, and we have made the definitio ns p=2γ αma(A2) and the function Eiis the exponential integral, Ei(z) =/integraldisplay∞ −ze−t tdt (A3) 13W2=Gm2 2− mo 2+/braceleftBigg e(q−2)αr r/bracketleftbigg 1+q(q−1) 2(mp+αq)e2qαr+2q mp+αqeqαr/bracketrightbigg +qαEi/parenleftbig qαr/parenrightbig −q(q−1)e2qαr (mp+αq)2r/bracketleftbig e(q−3)αr−α(q−3)rEi/parenleftbig (q−3)αr/parenrightbig/bracketrightbig +qeqαr (mp+αq)r/bracketleftbig e(q−3)αr−α(q−3)rEi/parenleftbig (q−3)αr/parenrightbig/bracketrightbig −2αEi/parenleftbig (q−2)αr/parenrightbig +αqEi/parenleftbig (q−2)αr/parenrightbig −q(q−1)αe2qαr (mp+αq)2Ei/parenleftbig (q−2)αr/parenrightbig +q2(q−1)αe2qαr 2(mp+αq)2Ei/parenleftbig (q−2)αr/parenrightbig −4αeqαr mp+αqEi/parenleftbig (q−2)αr/parenrightbig +2q2αeqαr (mp+αq)rEi/parenleftbig (q−2)αr/parenrightbig +2 r/bracketleftbig e(q−1)αr−(q−1)αrEi/parenleftbig (q−1)αr/parenrightbig/bracketrightbig +qeqαr (mp+αq)r/bracketleftbig e(q−1)αr−(q−1)αrEi/parenleftbig (q−1)αr/parenrightbig/bracketrightbig/bracerightBigg +l2 θ 2m2 2+(mp+αq)q/braceleftBigg −qαeqαr r+q2α2Ei/parenleftbig qαr/parenrightbig +q(q−1)e(3q−2)αr 2(mp+αq)2r2/bracketleftbig −1+2αr−qαr+(2−q)2α2r2e(2−q)αrEi/parenleftbig (q−2)αr/parenrightbig/bracketrightbig −qe(2q−1)αr (mp+αq)r2/bracketleftbig −1+αr+qαr+(q−1)2α2r2e(1−q)αrEi/parenleftbig (q−1)αr/parenrightbig/bracketrightbig/bracerightBigg (A4) wherempis the mass of the body at the perihelion, and we have made the definit ion q=2γ α(mp−b)(A5) 14References Bekov A.A., 1989, Astron. Zh., 66, 135 Berkovich L.M., 1981, Celestial Mechanics, 24 ,407 Bethe H.A., 1986, Phys. Rev. Lett., 56, 1305 Brandy J.L., 1982, J. Brit. Astron. Assoc., 92, no. 5, 209 Cevolani G., Bortolotti G. and Hajduk A., 1987, IL Nuo. Cim. C, 10, n o.5, 587 Chirikov B.V. and Vecheslavov V.V., 1989, Astron. Astrophys., 221, 146 Commins E.D. and Bucksbaum P.H., 1983, Weak Interactions of Lepto ns and Quarks, Cambridge University Press Daly P.W., 1989, Astron. Astrophys., 226, 318 Goldstein H., 1950, Classical Mechanics, Addison-Wesley, M.A. Gylden, H., 1884, Astron. Nachr., 109, no. 2593,1 Helhl F.W., Kiefer C. and Metzler R.J.K., 1998, Black Holes: Theory and Observation, Springer-Verlag Jeans J.H., 1924, MNRAS, 85, no. 1, 2 Jewitt D.C., 2002, Astron. Jour., 123, 1039 John F., 1974, Partial Differential Equations, Springer-Verlag, Ne w York Kobussen J.A., 1979, Acta Phys. Austr. 51,193 Leubner C., 1981, Phys. Lett. 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A, 33, 2 913 Spivak M.,2010, Physics for Mathematicians, Mechanics I Publish or Pe rish Inc., chapter 3 Sommerfeld A., 1964, Lectures on Theoretical Physics, Vol. I, Aca demic Press 15Zagorodny A.G., Schram P.P.J.M., and Trigger S.A., 2000, Phys. Rev. Le tt., 84, 3594 161/Multiply10122/Multiply10123/Multiply10124/Multiply10125/Multiply10126/Multiply1012r/LParen1m/RParen1 /Minus20000/Minus100001000020000v/LParen3m s/RParen3 Figure 1: Trajectories in the ( r,v) space with δm/m= 0.009.0 0.1 0.2 0.3 0.4 0.5 0.6 δm/m (%)100150200250300350400450500T (years) Figure 2: Period of the comet as a function of the mass lost ratio. 170 0.1 0.2 0.3 0.4 0.5 0.6 δm/m (%)012345δ ra/ra Figure 3: Ratio of aphelion distance change as a function of the mass lost rate.0 1 2 3 4 5 6 7 | γ |708090100110120130140T (years)δm=2x1010 Kg(δm/m=0.0087%) Figure 4: Period of the comet as a function of the parameter γ. 180 1 2 3 4 5 6 7 | γ |11.21.41.61.82ra'/raδm/m=0.0087% Figure 5: Ratio of the aphelion increasing as a function of the parame terγ. 19

2012-03-02

We make an observation about Galilean transformation on a 1-D mass variable systems which leads us to the right way to deal with mass variable systems. Then using this observation, we study two-bodies gravitational problem where the mass of one of the bodies varies and suffers a damping-antidamping effect due to star wind during its motion. For this system, a constant of motion, a Lagrangian and a Hamiltonian are given for the radial motion, and the period of the body is studied using the constant of motion of the system. Our theoretical results are applied to Halley's comet.

Damping-Antidamping Effect on Comets Motion

1203.0495v2